2008全国赛A题全国二等奖(赛才网)(1)
2008年数学建模A题全国一等奖论文
2008年全国大学生数学建模竞赛A题全国一等奖论文数码相机定位摘要本文通过对数码相机的靶标和像平面相互之间关系的分析，利用选取相关对应点和坐标转换的方法，确定靶标圆心在像平面的投影位置，进而完成了系统标定模型，解决了相机的单目定位问题。
对于问题1，为确定靶标上圆的圆心在一个相机像平面的像坐标，需要得到相机像平面中点与靶标上点的对应关系。
通过将相机外部参数和内部参数联立可以建立模型1。
对于问题2，内部参数通过焦距可以得到，而外部参数的获得则需要事先确定一组特殊点。
由于靶标上两条线的交点在像平面上的投影点即为这两条线在像平面上的投影图线的交点，因此我们首先对图像进行边缘提取和椭圆拟合，然后利用程序选择靶标上A 、C 两个圆的外公共切线的切点作为特殊点。
将对应特殊点带入（1）式，就可以求得外部参数。
最后利用几何关系得出靶标上圆心的坐标，带入得到它们在该相机像平面的坐标。
结果为：vA O （-4.4324，-6.7785，0）、vB O （-2.3，-6.4456，0）、vC O （3.39，-5.9757，0）、vD O （-4.5471，3.7096，0）、vE O （2.1965，3.2275，0）。
见图3。
对于问题3，为了检验模型，本文通过计算机模拟数据，可以得到一个内外参数都已知的图像。
进而可以确定这四个顶点在像平面的准确坐标。
根据（1）式可以得到这四个顶点的计算坐标，把计算坐标与准确坐标的距离为对角线的矩形面积称为误差面积，误差率=误差面积/相纸面积。
计算误差率分别为：0.017591%、0.01777%、0.01532%、0.01557%。
从而可知用此模型精确度高，稳定性强。
对于问题4，类似于问题3，进行计算机模拟，得到空间两不同角度拍摄图像，进而得到在此数码相机坐标系下的特殊点坐标。
由于在求像坐标时考虑到了数码相机的透视效应，也就是内部参数，而两个数码相机的空间位置关系仅仅是外部参数的关系，因此可以求得仅考虑外部参数时两个像平面上的坐标，进而做差求出两个数码相机的相对位置坐标。
08年全国高中数学联赛试题及答案
2008年全国高中数学联赛受中国数学会委托，2008年全国高中数学联赛由重庆市数学会承办。
中国数学会普及工作委员会和重庆市数学会负责命题工作。
2008年全国高中数学联赛一试命题范围不超出教育部2000年《全日制普通高级中学数学教学大纲》中所规定的教学要求和内容，但在方法的要求上有所提高。
主要考查学生对基础知识和基本技能的掌握情况，以及综合和灵活运用的能力。
全卷包括6道选择题、6道填空题和3道大题，满分150分。
答卷时间为100分钟。
全国高中数学联赛加试命题范围与国际数学奥林匹克接轨，在知识方面有所扩展，适当增加一些竞赛教学大纲的内容。
全卷包括3道大题，其中一道平面几何题，试卷满分150分。
答卷时问为120分钟。
一 试一、选择题（每小题6分，共36分）1．函数254()2x x f x x-+=-在(,2)-∞上的最小值是 （ ）。
（A ）0 （B ）1 （C ）2 （D ）32．设[2,4)A =-，2{40}B x x ax =--≤，若B A ⊆，则实数a 的取值范围为（ ）。
（A ）[1,2)- （B ）[1,2]- （C ）[0,3] （D ）[0,3)3．甲、乙两人进行乒乓球比赛，约定每局胜者得1分，负者得0分，比赛进行到有一人比对方多2分或打满6局时停止．设甲在每局中获胜的概率为23，乙在每局中获胜的概率为13，且各局胜负相互独立，则比赛停止时已打局数ξ的期望E ξ为 （ ）。
（A ）24181 （B ）26681 （C ）27481（D ） 6702434．若三个棱长均为整数（单位：cm ）的正方体的表面积之和为564 cm 2，则这三个正方体的体积之和为 （ ）。
（A ）764 cm 3或586 cm 3 （B ） 764 cm 3（C ）586 cm 3或564 cm 3 （D ） 586 cm 3 5．方程组0,0,0x y z xyz z xy yz xz y ++=⎧⎪+=⎨⎪+++=⎩的有理数解(,,)x y z 的个数为 （ ）。
2008年全国大学生英语竞赛样题（A级）及答案
2008年全国大学生英语竞赛样题（A级）及答案2008年全国大学生英语竞赛样题(A级)Part I Listening Comprehension (25 minutes, 30 marks)Section A (10 marks)Directions: In this section, you will hear 10 short conversations. At the end of each conversation, a question will be asked about what was said. Both the conversation and the question will be read only once. After each question, there will be a pause. During the pause, you must read the three choices marked A, B and C, and decide which is the best answer. Then mark the corresponding letter on the Answer Sheet with a single line through the centre.1. A. At a bank. B. At a department store. C. At a lawyer's office.2. A. Jealous. B. Indifferent. C. Negative.3. A. A party invitation.B. A tour of the house.C. A dinner party.4. A. Policeman and driver.B. Teacher and pupil.C. Driver and actress.5. A. To attend a party.B. To the beauty salon.C. To the school.6. A. 1,090. B. 530. C. 1,060.7. A. Dec. 13th, 1906. B. Dec. 30th, 1906. C. Nov. 13th, 1916.8. A. Nothing. B. Dinner. C. A snack.9. A. In less than three days.B. In more than three days.C. In three days.10. A. To a friend's house.B. To the kindergarten.C. Do some shopping.Section B (5 marks)Directions: In this section, you will hear one long conversation. The conversation will be read only once. At the end of the conversation, there will be a one-minute pause. During the pause, you must read the five questions, each with three choices marked A, B and C, and decide which is the best answer. Then mark the corresponding letter on the Answer Sheet with a single line through the centre.11. What does Susan have to do on Saturday morning?A. Walk the dog.B. Clean the house.C. Go to the doctor's.12. What will Susan do at noon?A. Go shopping.B. Watch TV.C. Go to the dentist's.13. What time is Susan meeting with Julie?A. 12:30 am.B. 2:00 pm.C. 4:30 pm.14. How does Susan feel about her schedule on Saturday?A. Exciting.B. Disappointing.C. Exhausting.15. Why can't they watch a video at Susan's house?A. The machine doesn't work.B. They can't decide on a video.C. Susan's mother is going to use it.Section C (5 marks)Directions: In this section, you will hear 5 short news items. After each item, there will be a pause. During the pause, you must read the question and then the three choices marked A, B and C, and decide which is the best answer. Then mark the corresponding letter on the Answer Sheet with a single line through the centre.16. Who got to move to the next step?A. Andre Agassi.B. Rogen Fidow.C. Roger Federer.17. What did the gunman force the charity to do?A. Delay the operation in this area.B. Operate on their own staff.C. Attack the Sans Frontieres.18. Why were the Iraqi detainees made to pick up garbage?A. Because the serviceman abused the detainees.B. Because the commander gave the order to the serviceman.C. Because the detainees were accused.19. Where did the clash happen?A. In the West Bank.B. Near Israeli.C. Near Ariel.20. Why do the Israeli forces threaten to carry out a massive offensive against the Palestinians?A. Because Israeli troops were besieged by the Palestinians.B. Because the military operation began since last November.C. Because Palestinians killed Israelis in a suicide attack.Section D (10 marks)Directions: In this section, you will hear a short passage. There are ten missing words or phrases in it. Fill in the blanks with the exact words or phrases you hear on the tape. Remember to write the answers on the Answer Sheet.There are two types of people in the world. Although they have (21)________ degrees of health and wealth and other comforts of life, one becomes happy, and the other becomes (22) ________. This arises from the different ways in which they consider things, persons, and events, and the resulting effects upon their minds.People who are to be happy fix their attention on the (23) ________ of things, the pleasant part of convention, the fine weather. They enjoy all the (24) ________ things. Those who are to be unhappy think and speak only of the (25) ________ things. Therefore, they are continually (26) ________. By their remarks, they sour the pleasure of society, offend many people, and make themselves (27) ________ everywhere. If this turn of mind were found in nature, such unhappy persons would be the more to be pitied. The tendency to criticize and to be disgusted is perhaps taken up originally (28) ________. It grows into a habit, unknown to its possessors. The habit may be strong, but itmay be cured when those who have it are convinced of its bad effects (29) ________. I hope this little warning may be of service to them, and help them change this habit. Although in fact it is chiefly (30) ________, it has serious consequences in life since it brings on deep sorrow and bad luck.Part II Vocabulary and Structure (10 minutes, 15 marks) Directions: There are 15 incomplete sentences in this section. For each blank there are four choices marked A, B, C and D. Choosethe one that bestcompletes the sentence. Then mark the corresponding letter on the Answer Sheet with a single line through the centre.31. Essentially, a theory is an abstract, symbolic representation of________ reality.A. what it is conceivedB. that is conceivedC. what is conceived to beD. that is being conceived of32. New York City collects garbage ________ $209 per family per year.A. at the cost ofB. at sacrifice ofC. in demand ofD. in support of33. I would just as soon ________ rudely to her.A. you not speakB. you won't speakC. you not speakingD. you didn't speak34. ________ so many people been out of work as today.A. More than ever beforeB. Never before haveC. In the past, there never haveD. Formerly, there never were35. We got home late last night, ________, early this morning.A. namelyB. in other wordsC. or ratherD. in any case36. The miners escaped the danger by ________.A. a hair breadthB. hair breadthC. hair's breadthD. a hair's breadth37. Praise is like warm sunshine. It not only brings joy to ________ who are praised but also adds pleasure to ________ who praise.A. these; theseB. we; usC. them; themD. those; those38. When our imports ________ our exports, we say we have a favorable trade balance; however, when our imports and exports are the other way round we say we have an unfavorable trade balance.A. fall shortB. fall short ofC. run out ofD. are lack of39. ________ that distinguish human beings from other primates are related to the ability of people to stand upright and walk on two legs.A. Many of the physical characteristicsB. Of the many physical characteristicsC. The physical characteristicsD. There are many physical characteristics40. Classical logic is characterized by a concern for the structure and elements of argument ________ that thought,language, and reality are interrelated.A. based on the beliefB. on the belief basedC. belief based on theD. the based belief on41. ________ fashioned from a wick floating in a bowl of oil functioned according to the principle of capillary action.A. All lamps earlyB. Lamps all earlyC. All early lampsD. Early all lamps42. In 1964 ________ of Henry Osaka Tanner's paintings was shown at the Smithsonian Institution.A. was a major collectionB. that a major collectionC. a collection was majorD. a major collection43. Traditionally, ________ on Thanksgiving Day in New England.A. when served is sweet ciderB. when sweet cider is servedC. is served sweet ciderD. sweet cider is served44. Richard: This class is so boring. When will it end?Drew: I don't know. I don't think I can take it much longer.Teacher: ________Richard: Oh, nothing. We were discussing the text.A. Hey you two! What are you talking about?B. Do one of you know what the answer to this problem is?C. Am I boring you two?D. The two of you go to the principal's office right now!45. Delivery man: I have a package here for Beverly Beaver.Beverly: That is me. ________Delivery man: Yes, please sign here on the line.Beverly: Thank you. These must be the new drapes I ordered.A. Where is the receipt?B. Do you have the sender's address?C. Do you take a credit card?D. Do I have to sign for it?Part III Reading Comprehension (20 minutes, 40 marks)Section A (5 marks)Directions: There is one passage in this section with 5 questions. For each question, there are four choices marked A, B, C and D. You should decide on the best choice. Then mark the corresponding letter on theAnswer Sheet with a single line through the centre.Questions 46-50 are based on the following passage.No matter what our work is, we all have to face that choice - in my field no less than in others. There may not be a more important field for the dissemination of values in our country than the entire communicationsindustry - most strikingly, television. Networks are very sensitive to that fact, and they employ dozens of censors to prevent all of us from using language on television that an eight year old might have to explain to his parents. But the point that censors miss, I think, is that it is not so much what we say that teaches as what we don't say. Even programs that attempt to make a moral point don't always make the point that they intended to. Because when we sense we are being sole something, we automatically defend ourselves against it. I thinkit may be the unspoken assumptions that mold an audience.Look at the way, for instance, that violence is treated on television. It is not only the quantity that offends. There probably is no more violence on television than there is in a Shakespearean tragedy. But on television you find unfelt violence and in Shakespeare you tend to find felt violence. In Shakespeare the characters react with a human response: They fear, they hurt, they mourn. Most of the time on television, violence is dealt with by sweeping it under the rug as fast as possible and by having people go on about their business as if nothing had happened. (If I can't have less violence, I want at least a better grade of violence.) One of the unspoken assumptions is that violence can be tolerated as long as you ignore it and have no reaction to it. But that seems to me to be dangerously close to psychopathic behavior.I wonder if there is any connection between the long acceptance by our people of the Vietnam War and the thousands and thousands of deaths that we have seen on television over the years that were never mourned, never even paused for except to sell shampoo for sixty seconds.Maybe our greatest problem is that we have two separate sets of value systems that we use - the one we talk about and the one we live by. We seem to place a very high value on fairness and on human concerns. And yet we stillhave widespread discrimination based on race, sex and religion. You still don't find Jews, blacks or other minorities in any significant numbers in decision-making positions in the banking industry, for instance. You think that's an accident? I think somebody puts a value on that. And you don't find women in any significant numbers in decision-making capacities in any industry.Why? Because we place a higher value on appeasing the fragile male ego than we do on fairness and decency. Maybe what we need is a declaration of interdependence.Questions:46. It can be concluded from the first paragraph that ________.A. we won't buy things that are recommendedB. we become defensive whenever we are taught to do somethingC. television is the most important part of the communication industryD. TV programs can never successfully convey a moral point47. What is the author's attitude toward violence on television?A. He thinks that violence on television should be totally banned.B. Violence on television is more tolerable than the one in Shakespeare's plays.C. Violence can be tolerated if you pretend that it doesn't exist.D. Violence, if unavoidable, should be presented on television in a more appropriate way.48. The word “appeasing” (line 6 in paragraph 3) most probably means ________.A. attackingB. accusingC. soothingD. satisfying49. By citing the examples of the Jews, blacks, and other minorities in the last paragraph, the author aims to show that ________.A. it's an accident that they are out-numbered in the decision-making positions in the banking industryB. they stand for one set of value systemC. what we talk about and what we actually do might be totally differentD. fairness is well respected in our life50. The author's tone in writing the text is ________.A. neutralB. matter-of-factC. criticalD. questioningSection B (15 marks)Directions: There is one passage in this section with 10 questions. Go over the passage quickly and answer the questions on the Answer Sheet.For questions 51 - 55, markY (for YES) if the statement agrees with the information given in the passage;N (for NO) if the statement contradicts the information given in the passage;NG (for NOT GIVEN) if the information is not given in the passage.For questions 56 - 60, complete the sentences with the information given in the passage.You hear the same complaint all the time as people get older: “My memory is terrible.” Is it all in the mind, or do real changes take place in the brain with age to justify such grumbling (抱怨)? The depressing answer is that the brain's cells, the neurons, die and decline in efficiency with age.Professor Arthur Shimamura, of the University of Californiaat Berkeley, says there are three main ways in which mental function changes. The first is mental speed, for example how quickly you can react to fast-moving incidents on the road. Drivers in their late teens react quickly but tend to drive too fast, while the over sixties are more cautious but react more slowly. The near-inevitable slowing with age also partly explains why soccer players are seen as old in their thirties, while golf professionals are still in their prime at that age. This type of mental slowing results from a reduction in the efficiency with which the brain's neurons work.The fact that adults find it harder to learn musical instruments than children points to a second type of mental loss with age - a reduction in learning capacity. The parts of the brain known as the temporal lobes control new learning, and are particularly vulnerable to the effects of aging. This means that, as we get older, we take longer to learn a new language, and slower to master new routines and technologies at work, and we have to rely more on diaries and other mental aids.“Working memory” is the third brain system which is vulnerable to the effects of aging. Working memory is the brain's “blackboard”,where we juggle from moment to moment the things we have to keep in mind when solving problems, planning tasks and generally organizing our day-to-day life. Absent-mindedness occurs at all ages because of imperfections in the working memory system - so, for instance, you may continually lose your glasses, or find yourself walking into a room of your house only to find that you cannot remember what you came for.Such absent-mindedness tends to creep up on us as we age and occurs because our plans and intentions, which are chalkedup on the mental blackboard, are easily wiped out by stray thoughts and other distractions. Stress and preoccupation can also cause such absent-mindedness, in addition to age-related changes in the brain. The frontal lobes of the brain - located behind the forehead and above the eyes - are where the working memory system is located. Like the temporal lobes, which handle new learning, the frontal lobes are more vulnerable to the aging process than other parts of the brain.The news, however, is not all bleak. Although neurons reduce in number with age, the remaining neurons send out new and longer connection fibers (dendrites) to maintain connections and allow us to function reasonably well with only relatively small drops in ability.This and other evidence suggests that the principle “use it or lost it” might apply to the aging brain. Professor Shimamura studied a group of university professors who were still intellectually active, and compared their performance on neuropsychological tests with that of others of their age group, as well as with younger people. He found that on several tests of memory, the mentally active professors in their sixties and early seventies were superior to their contemporaries, and as good as the younger people.Research on animals provide even stronger evidence of the effects of stimulation on the brain structure. Professor Bryan Kolb, of the University of Lethbridge in Canada, has shown that animals kept in stimulating environments show sprouting (生长) and lengthening of the connecting nerve fibers in comparison with animals kept in unstimulating environments.The beneficial effects of continued mental activity are shown by the fact that older contestants in quiz shows are just as fastand accurate in responding to general knowledge questions as younger competitors, suggesting that at least part of their intellectual apparatus is spared the effects of aging because of practice and skill.Such findings lead to the intriguing possibility of “mental fitnesstra ining” to accompany jogging and workouts for the health conscious. Research in Stockholm by Professor Lars Backman and his colleague has shown that older people can be trained to use their memory better, with the effects of this training lasting several years.Just as people go bald or grey at different rates, so the same is true for their mental faculties. Why this should be the case for memory and other mental functions is not yet clear. If Professor Shimamura is right, then the degree to which people use and stretch their mental faculties may also have a role to play.Questions:51. The passage gives a description of several methods of testing mental ability.52. Professor Shimamura identified a number of areas in which mental function may change.53. As the temporal lobes of the brain are affected by aging, it becomes harder to pick up new skills.54. Research indicates that physical training can help to improve memory.55. Taking part in quizzes is the best way to stimulate the brain.56. Absent-mindedness is not necessarily ________.57. Scientists do not understand why people's ________ decline at different rates.58. If a person is under stress, ________ may become more frequent.59. When the frontal lobes of the brain are affected by aging, there is a gradual deterioration in ________.60. If an animal's brain is kept active and stimulated, ______________. Section C (10 marks)Directions: In this section, there is one passage followed by 5 questions. Read the passage carefully, then answer the questions in a maximum of 10 words. Remember to write the answers on the Answer Sheet.Questions 61 - 65 are based on the following passage.Alarmed by a 20-year decline in student achievement, American schools are considering major upheavals in the career structure of teachers, school boards around the country are planning to abandon traditional salary schedules and single out outstanding teachers for massive pay rise.The idea is regarded with deep suspicion by the United States' biggest teachers' unions, the National Education Association and the American Federation of Teachers. They say the creation of a cadre of elite teachers will sour professional relationships and encourage teachers to compete instead of cooperate; they also question whether a fair way can be devised to tell which teachers really do perform better than their colleagues.But heightened public anxiety about secondary education appears to have given the master teacher concept unstoppable political momentum. Florida and Tennessee are racing to introduce ambitious statewide master teacher schemes before the end of the year. Less grandiose proposals to pay teachers on the basis of merit instead of seniority have already beenimplemented in countless school districts. And the Secretary of Education, Mr. Terrel Bell, recently promised substantial incentive grants to states which intend to follow their example.Low pay is believed to be the single most important reason for the flight from teaching. The average salary of a teacher in the United States is just under $19,000, much less than that of an engineer ($34,700) and not much more than that of a secretary($16,500). To make ends meet it is common for teachers to take second jobs in the evening and in their summer holidays, and women, who used to make up the bulk of teacher candidates are turning to better paid professions.The unions insist that the answer to this problem is to increase the basic pay of all teachers, but most states would find that too expensive, they would be better able to afford schemes that confine pay increases to a small number of exceptional teachers. Champions of the idea say it would at least hold out the promise of high pay and status to bright graduates who are confident of their ability to do well in the classroom, but are deterred by the present meager opportunities for promotion.One of the first large-scale tests of this approach will come in Tennessee, where a year of painstaking negotiations has just overcome bitterunion opposition to a wide-ranging master teacher scheme. Tennessee promised that they will allocate $300 million as education budget. In return for a chance to earn bigger salaries and faster promotion, teachers will subject themselves to closer scrutiny.The Tennessee plan will make it harder for poor teachers to join the profession. Beginners will have to serve a probationary year before qualifying, and another three apprentice years beforereceiving tenure. Apprentice teachers who fail to reach a required standard will not be allowed to stay on. Survivors will be designated “career teachers” and given a chance to climb through three career rugs and earn bonuses of up to $7,000. Advancement will not be automatic. The performance of each teacher will be closely assessed by committees of teachers drawn from other districts.Questions:61. What support is the federal government offering to states that set upa master teacher scheme?62. What's the purpose of the master teacher scheme?63. In the state of Tennessee, how will teachers be assessed?64. What is the main idea of the text?65. What can be inferred from the text about the master teacher scheme?Section D (10 marks)Directions: In this section, there is one passage followed by a summary. Read the passage carefully and complete the summary below by choosing no more than three words from the passage. Remember to write the answers on the Answer Sheet.Questions 66 - 70 are based on the following passage.The United KingdomBritain (or Great Britain) is an island that lies off the north-west coast of Europe. The nearest country is France, which is 20 miles away and from which Britain is separated by the English Channel. The island is surrounded by the Atlantic Ocean to the west, and the North Sea to the east. It comprises the mainlands of England, Wales and Scotland, that is, three countries. Scotland is in the north, while Wales is in the west. Ireland, which is alsoan island, lies off the west coast of Britain. It consists of Northern Ireland and the Irish Republic. Britain together with Northern Ireland constitute the United Kingdom (UK). Thus, the United Kingdom is composed of four countries, the largest of which is England. The capital city is London, which is situated in south-east England.The UK has a total area of about 244,100 square kilometres (94,248 square miles). About 70% of the land area is devoted to agriculture, about 7% is wasteland, moorland and mountains, about 13% is devoted to urban development, and 10% is forest and woodland. The northern and western regions of Britain, that is Scotland and Vales, are mainly mountainous and hilly. Parts of the north-west and centre of England also consist of mountains and hills.Britain has a generally mild and temperate climate. It is, however, subject to frequent changes. It has an average annual rainfall of about 120 centimetres (47 inches).In 1998 the population of the United Kingdom was nearly 59 million. The density of population was approximately 240 people per square kilometre. However, in England, where 83% of the population live, the density was much higher, about 363 per square kilometre.In the UK, English is the first language of the vast majority of people. However, in western Wales, Welsh is the first language for many of the people. In Scotland only a small number of people speak Gaelic.In Britain about 66% of the population say that they are Christian, while fewer than 5% say that they belong to other religions.Summary:Britain is an island that (66) ________ the Atlantic Ocean and the North Sea. It comprises the mainlands of England, Wales and Scotland. Ireland(67) ________ the west coast of Britain. It (68) ________ Northern Ireland and the Irish Republic. The United Kingdom consists of Britain together with Northern Ireland. The capital city is London which (69) ________ south-east England.In 1998 the population of the UK was nearly 59 million. The density of population was 240 people per square kilometre. In the UK English is the first language of most people. In western Wales, Welsh (70) ________ many of the people, but few people in Scotland speak Gaelic.Part IV Cloze (15 minutes, 15 marks)Section A (5 marks)Directions: There are 5 blanks in the passage. For each blank, some letters of the word have been given (not exceeding 3 letters). Read the passage below and think of the word which best fits each blank. Use only one word in each blank. Remember to write the answers on the Answer Sheet.Language is learned primarily through communication with other people. Research shows that the more communication children (71) exp________ at home, the better developed their language skills will be. Children need to talk well in order to learn well. This is especially important for the development of native language skills since the language is often reinforced by the child's environment outside school.However, the (72) qu________ of communication is just as or more important than quantity alone. The language adults use helps children become aware of the many different aspects of objects and events around them. For example, during a shoppingtrip to the market or store, adults can develop children's concepts by bringing their attention to the shapes, colors, sounds, textures, and sizes of objects and events around them. Parents can do this naturally through conversation without (73) di________ teaching. In other words, conversation with children in everyday situations expands their minds and develops their thinking skills.In addition to conversing with children, adults can help prepare their children to succeed in school by (74) en________ them to take an active interest in books and in the print that surrounds them in the environment. The child's first major task in school is learning to read. Children who come to school with knowledge that the print aroundthem carries important meaning and with an interest in books an d stories will usually (75) su________ in learning to read rapidly. Section B (10 marks)Directions: There are 10 blanks in the passage. Use the word given in each blanket to form a word that fits in each blank. Remember to write the answers on the Answer Sheet.Forensic science (法庭科学) means (76) ________ (use) science to solve crime. But what exactly does a forensic scientist do?At the scene of crime, the crime scene (77) ________ (investigator) will thoroughly examine the area. They will look for fingerprints and palm prints; marks of tools and weapons; marks from shoes; fibers from (78) ______ (cloth); body fluids.Increasingly, CSIs (= crime scene investigators) are using specialized techniques, on site, to determine which areas to concentrate their investigation on.A forensic pathologist (法医)specializes in examining dead bodies to determine how and when death occurred. As well as (79) ________ (examine) the body,。
2008年全国中学生语文能力竞赛决赛八年级Word版
2008年全国中学生语文能力竞赛决赛八年级（初二）组试题（试题总分：１5０分〓答题时间：１5０分钟）温馨提示：亲爱的同学们，经过一轮的比拼，你终于攀登上了新的高峰。
这张试卷是你在新的高峰上全面展示自己学识与才华的平台，相信你独特的思考、个性化的体验、富有创意的表达，一定会取得最优秀的成绩。
千万别忘了，把答案写在答题纸上啊！一、听力（20分）答题提示：第1—3小题的录音内容播放两遍，第4小题的录音内容播放三遍，考生可以边听边答题。
1.听词语，读音不正确的一项是（〓）（2分）A．锐不可当〓瓦砾 B. 风雪载途〓惧惮C．相辅相成〓镂空 D. 粗制滥造〓狩猎2.把你听到的词语归类。
（4分）A. 表示动作行为的词语：〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓B. 形容自然景物的词语：〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓3.听提示材料，写出相关的诗句。
（6分）A．〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓B．〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓C．〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓4.听短文，完成下面的问题。
（8分）（1）根据文章内容填空。
（2分）“成”和“功”有着密切的关系，“成”是“功”的〓〓〓〓，“功”是“成”的〓〓〓〓。
成功对于懒汉来说是〓〓〓〓的，对于奋斗者来说又是〓〓〓〓的。
（2）文中举了哪几个例子论述“功到自然成”的？（3分）〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓（3）听了这篇短文，你懂得了什么道理？打算怎么做？（3分）〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓〓二、益智（10分）5.成语连环套。
（4分）示例：热火代天（改朝换代）【本来应该是“热火朝天”，“朝”却被“代”代替了，所以是“改朝换代”。
2008年全国高中数学联赛试题及解答
2008全国高中数学联合竞赛一试试题参考答案及评分标准（A 卷）说明：1．评阅试卷时，请依据本评分标准．选择题只设6分和0分两档，填空题只设9分和0分两档；其他各题的评阅，请严格按照本评分标准的评分档次给分，不要增加其他中间档次．2．如果考生的解答方法和本解答不同，只要思路合理、步骤正确，在评卷时可参考本评分标准适当划分档次评分，解答题中5分为一个档次，不要增加其他中间档次． 一、选择题（本题满分36分，每小题6分） 1．函数254()2x x f x x-+=-在(,2)-∞上的最小值是(C)A ．0B ．1C ．2D ．3 [解] 当2x <时，20x ->，因此21(44)1()(2)22x x f x x x x+-+==+---2≥2=，当且仅当122x x=--时上式取等号．而此方程有解1(,2)x =∈-∞，因此()f x 在(,2)-∞上的最小值为2．2．设[2,4)A =-，2{40}B x x ax =--≤，若B A ⊆，则实数a 的取值范围为 （ D ）A ．[1,2)-B ．[1,2]-C ．[0,3]D ．[0,3) [解] 因240x ax --=有两个实根12a x =-22a x =故B A ⊆等价于12x ≥-且24x <，即22a ≥-且42a ， 解之得03a ≤<．3．甲乙两人进行乒乓球比赛，约定每局胜者得1分，负者得0分，比赛进行到有一人比对方多2分或打满6局时停止．设甲在每局中获胜的概率为23，乙在每局中获胜的概率为13，且各局胜负相互独立，则比赛停止时已打局数ξ的期望E ξ为 （ B ） A.24181B.26681C.27481D.670243[解法一] 依题意知，ξ的所有可能值为2，4，6.设每两局比赛为一轮，则该轮结束时比赛停止的概率为22215()()339+=．若该轮结束时比赛还将继续，则甲、乙在该轮中必是各得一分，此时，该轮比赛结果对下轮比赛是否停止没有影响．从而有 5(2)9P ξ==， 4520(4)()()9981P ξ===，2416(6)()981P ξ===，故520162662469818181E ξ=⨯+⨯+⨯=．[解法二] 依题意知，ξ的所有可能值为2，4，6.令k A 表示甲在第k 局比赛中获胜，则k A 表示乙在第k 局比赛中获胜．由独立性与互不相容性得12125(2)()()9P P A A P A A ξ==+=， 1234123412341234(4)()()()()P P A A A A P A A A A P A A A A P A A A A ξ==+++332112202[()()()()]333381=+=， 1234123412341234(6)()()()()P P A A A A P A A A A P A A A A P A A A A ξ==+++2221164()()3381==， 故520162662469818181E ξ=⨯+⨯+⨯=．4．若三个棱长均为整数（单位：cm ）的正方体的表面积之和为564 cm 2，则这三个正方体的体积之和为 （ A ）A. 764 cm 3或586 cm 3B. 764 cm 3C. 586 cm 3或564 cm 3D. 586 cm 3 [解] 设这三个正方体的棱长分别为,,a b c ，则有()2226564a b c ++=，22294a b c ++=，不妨设110a b c ≤≤≤<，从而2222394c a b c ≥++=，231c >．故610c ≤<．c 只能取9，8，7，6．若9c =，则22294913a b +=-=，易知2a =，3b =，得一组解(,,)(2,3,9)a b c =．若8c =，则22946430a b +=-=，5b ≤．但2230b ≥，4b ≥，从而4b =或5．若5b =，则25a =无解，若4b =，则214a =无解．此时无解．若7c =，则22944945a b +=-=，有唯一解3a =，6b =．若6c =，则22943658a b +=-=，此时222258b a b ≥+=，229b ≥．故6b ≥，但6b c ≤=，故6b =，此时2583622a =-=无解．综上，共有两组解2,3,9a b c =⎧⎪=⎨⎪=⎩或3,6,7.a b c =⎧⎪=⎨⎪=⎩体积为3331239764V =++=cm 3或3332367586V =++=cm 3．5．方程组0,0,0x y z xyz z xy yz xz y ++=⎧⎪+=⎨⎪+++=⎩的有理数解(,,)x y z 的个数为（ B ）A. 1B. 2C. 3D. 4[解] 若0z =，则00.x y xy y +=⎧⎨+=⎩，解得00x y =⎧⎨=⎩，或11.x y =-⎧⎨=⎩，若0z ≠，则由0xyz z +=得1xy =-． ① 由0x y z ++=得z x y =--． ②将②代入0xy yz xz y +++=得220x y xy y ++-=． ③ 由①得1x y=-，代入③化简得3(1)(1)0y y y ---=.易知310y y --=无有理数根，故1y =，由①得1x =-，由②得0z =，与0z ≠矛盾，故该方程组共有两组有理数解0,0,0x y z =⎧⎪=⎨⎪=⎩或1,1,0.x y z =-⎧⎪=⎨⎪=⎩6．设ABC ∆的内角A B C ,,所对的边,,a b c 成等比数列， 则sin cot cos sin cot cos A C A B C B ++的取值范围是A. (0,)+∞B.C.D.)+∞ [解] 设,,a b c 的公比为q ，则2,b aq c aq ==，而sin cot cos sin cos cos sin sin cot cos sin cos cos sin A C A A C A CB C B B C B C++=++sin()sin()sin sin()sin()sin A C B B bq B C A A aππ+-=====+-．因此，只需求q 的取值范围．因,,a b c 成等比数列，最大边只能是a 或c ，因此,,a b c 要构成三角形的三边，必需且只需a b c +>且b c a +>．即有不等式组22,a aq aq aq aq a ⎧+>⎪⎨+>⎪⎩即2210,10.q q q q ⎧--<⎪⎨+->⎪⎩解得q q q <<⎨⎪><⎪⎩从而1122q <<，因此所求的取值范围是． 二、填空题（本题满分54分，每小题9分）7．设()f x ax b =+，其中,a b 为实数，1()()f x f x =，1()(())n n f x f f x +=，1,2,3,n = ，若7()128381f x x =+，则a b += 5 . [解] 由题意知12()(1)n n n n f x a x a a a b --=+++++11n na a xb a -=+⋅-，由7()128381f x x =+得7128a =，713811a b a -⋅=-，因此2a =，3b =，5a b +=．8．设()cos 22(1cos )f x x a x =-+的最小值为12-，则a=[解] 2()2cos 122cos f x x a a x =---2212(cos )2122a x a a =----，(1) 2a >时，()f x 当cos 1x =时取最小值14a -；(2) 2a <-时，()f x 当cos 1x =-时取最小值1；(3)22a -≤≤时，()f x 当cos 2ax =时取最小值21212a a ---． 又2a >或2a <-时，()f x 的最小值不能为12-，故2112122a a ---=-，解得2a =-2a =-舍去)．9．将24个志愿者名额分配给3个学校，则每校至少有一个名额且各校名额互不相同的分配方法共有 222 种． [解法一] 用4条棍子间的空隙代表3个学校，而用*表示名额．如||||********表示第一、二、三个学校分别有4，18，2个名额．若把每个“*”与每个“|”都视为一个位置，由于左右两端必须是“｜”，故不同的分配方法相当于24226+=个位置（两端不在内）被2个“｜”占领的一种“占位法”．“每校至少有一个名额的分法”相当于在24个“*”之间的23个空隙中选出2个空隙插入“｜”，故有223C 253=种． 又在“每校至少有一个名额的分法”中“至少有两个学校的名额数相同”的分配方法有31种．综上知，满足条件的分配方法共有253－31＝222种． [解法二] 设分配给3个学校的名额数分别为123,,x x x ，则每校至少有一个名额的分法数为不定方程12324x x x ++=．的正整数解的个数，即方程12321x x x ++=的非负整数解的个数，它等于3个不同元素中取21个元素的可重组合：2121232323H C C 253===． 又在“每校至少有一个名额的分法”中“至少有两个学校的名额数相同”的分配方法有31种．综上知，满足条件的分配方法共有253－31＝222种． 10．设数列{}n a 的前n 项和n S 满足：1(1)n n n S a n n -+=+，1,2,n = ，则通项n a =112(1)n n n -+． [解]1111(1)(2)(1)n n n n n n n a S S a a n n n n +++-=-=--++++，即 2n n a n n n n n n a ++++-++-+=+)1(111)2)(1(221=)1(1)2)(1(2+++++-n n a n n n ， 由此得 2)1(1))2)(1(1(1++=++++n n a n n a n n ．令1(1)n n b a n n =++，111122b a =+= (10a =)， 有112n n b b +=，故12n nb =，所以)1(121+-=n n a nn ． 11．设()f x 是定义在R 上的函数，若(0)2008f = ，且对任意x ∈R ，满足(2)()32x f x f x +-≤⋅，(6)()632x f x f x +-≥⋅，则)2008(f =200822007+．[解法一] 由题设条件知(2)()((4)(2))((6)(4))((6)())f x f x f x f x f x f x f x f x +-=-+-+-+-+++-24323263232x x x x ++≥-⋅-⋅+⋅=⋅，因此有(2)()32x f x f x +-=⋅，故(2008)(2008)(2006)(2006)(2004)(2)(0)(0)f f f f f f f f =-+-++-+2006200423(2221)(0)f =⋅+++++10031413(0)41f +-=⋅+-200822007=+．[解法二] 令()()2x g x f x =-，则2(2)()(2)()2232320x x x x g x g x f x f x ++-=+--+≤⋅-⋅=， 6(6)()(6)()226326320x x x x g x g x f x f x ++-=+--+≥⋅-⋅=，即(2)(),(6)()g x g x g x g x +≤+≥，答12图1故()(6)(4)(2)()g x g x g x g x g x ≤+≤+≤+≤， 得()g x 是周期为2的周期函数，所以200820082008(2008)(2008)2(0)222007f g g =+=+=+． 12．一个半径为1的小球在一个内壁棱长为内可向各个方向自由运动，则该小球永远不可能接触到的容器内壁的面积是．[解] 如答12图1，考虑小球挤在一个角时的情况，记小球半径为r ，作平面111A B C //平面ABC ，与小球相切于点D ，则小球球心O 为正四面体111P A B C -的中心，111PO A B C ⊥面，垂足D 为111A B C 的中心．因11111113P A B CA B C V S PD -∆=⋅1114O A B C V -=⋅111143A B C S OD ∆=⋅⋅⋅， 故44PD OD r==，从而43PO PD OD r r r =-=-=．记此时小球与面PAB 的切点为1P ，连接1OP ，则2211PP PO OP =-==． 考虑小球与正四面体的一个面(不妨取为PAB )相切时的情况，易知小球在面PAB 上最靠近边的切点的轨迹仍为正三角形，记为答13图答12图21PEF ，如答12图2．记正四面体的棱长为a ，过1P 作1PM PA ⊥于M ． 因16MPP π∠=，有11cos PM PP MPP =⋅==，故小三角形的边长12PE PA PM a =-=-．小球与面PAB 不能接触到的部分的面积为（如答12图2中阴影部分）1PAB P EFS S ∆∆-22())a a =--2=-．又1r =，a =1PAB P EF S S ∆∆-==由对称性，且正四面体共4个面，所以小球不能接触到的容器内壁的面积共为三、解答题（本题满分60分，每小题20分） 13．已知函数|sin |)(x x f =的图像与直线y kx =)0(>k 有且仅有三个交点，交点的横坐标的最大值为α，求证： 2cos 1sin sin 34ααααα+=+．[证]()f x 的图象与直线y kx = )0(>k 的三个交点如答13图所示，且在3(,)2ππ内相切，其切点为(,sin )A αα-，3(,)2παπ∈． …5分由于()cos f x x '=-，3(,)2x ππ∈，所以sin cos ααα-=-，即tan αα=． …10分 因此cos cos sin sin 32sin 2cos αααααα=+14sin cos αα=…15分22cos sin 4sin cos αααα+=21tan 4tan αα+=214αα+=． …20分14．解不等式121086422log (3531)1log (1)x x x x x ++++<++．[解法一] 由44221log (1)log (22)x x ++=+，且2log y 在(0,)+∞上为增函数，故原不等式等价于1210864353122x x x x x ++++>+．即 1210864353210x x x x x +++-->．…5分分组分解12108x x x +-1086222x x x ++-864444x x x ++-642x x x ++-4210x x ++->，864242(241)(1)0x x x x x x +++++->， …10分所以 4210x x +->，2211(022x x ---+-->． …15分所以2x ，即x <x >故原不等式解集为(,)-∞+∞ ． …20分[解法二] 由44221log (1)log (22)x x ++=+，且2log y 在(0,)+∞上为增函数，故原不等式等价于1210864353122x x x x x ++++>+．…5分即6422232262133122(1)2(1)x x x x x x x x+<+++++=+++， )1(2)1()1(2)1(232232+++<+x x xx ， …10分令3()2g t t t =+，则不等式为221()(1)g g x x<+，显然3()2g t t t =+在R 上为增函数，由此上面不等式等价于答15图2211x x <+， …15分即222()10x x +->，解得2x >(2x <舍去)，故原不等式解集为(,)-∞+∞ ． …20分15．如题15图，P 是抛物线22y x =上的动点，点B C ,在y 轴上，圆22(1)1x y -+=内切于PBC ∆，求PBC ∆面积的最小值．[解] 设00(,),(0,),(0,)P x y B b C c ，不妨设b c >．直线PB 的方程:00y b y b x x --=，化简得000()0y b x x y x b --+=．又圆心(1,0)到PB 的距离为1，1= ， …5分故22222000000()()2()y b x y b x b y b x b -+=-+-+， 易知02x >，上式化简得2000(2)20x b y b x -+-=，同理有2000(2)20x c y c x -+-=． …10分 所以0022y b c x -+=-，002x bc x -=-，则 22200020448()(2)x y x b c x +--=-．因00(,)P x y 是抛物线上的点，有2002y x =，则22204()(2)x b c x -=-，0022x b c x -=-． …15分所以00000014()(2)4222PBC x S b c x x x x x ∆=-⋅=⋅=-++--48≥=．当20(2)4x -=时，上式取等号，此时004,x y ==±．因此PBC S ∆的最小值为8． …20分2008年全国高中数学联合竞赛加试（A 卷）试题参考答案及评分标准说明：1．评阅试卷时，请严格按照本评分标准的评分档次给分； 2．如果考生的解答方法和本解答不同，只要思路合理、步骤正确，在评卷时可参考本评分标准适当划分档次评分，10分为一个档次，不要增加其他中间档次． 一、（本题满分50分）如题一图，给定凸四边形ABCD ，180B D ∠+∠< ，P 是平面上的动点，令()f P PA BC PD CA PC AB =⋅+⋅+⋅．（Ⅰ）求证：当()f P 达到最小值时，P A B C ,,,四点共圆； （Ⅱ）设E 是ABC ∆外接圆O 的 AB 上一点，满足：AE AB=，1BC EC=，12ECB ECA ∠=∠，又,DA DC 是O的切线，AC 求()f P 的最小值．[解法一] （Ⅰ）如答一图1，由托勒密不等式，对平面上的任意点P ，有PA BC PC AB PB AC ⋅+⋅≥⋅．因此 ()f P PA BC PC AB PD CA =⋅+⋅+⋅PB CA PD CA ≥⋅+⋅()PB PD CA =+⋅．因为上面不等式当且仅当,,,P A B C 顺次共圆时取等号，因此当且仅当P 在ABC ∆的外接圆且在AC 上时， ()()f P PB PD CA =+⋅．…10分又因PB PD BD +≥，此不等式当且仅当,,B P D 共线且P 在BD 上时取等号．因此当且仅当P 为ABC ∆的外接圆与BD 的交点时，()f P 取最小值min ()f P AC BD =⋅．故当()f P 达最小值时，,,,P A B C 四点共圆．…20分（Ⅱ）记ECB α∠=，则2ECA α∠=，由正弦定理有sin 2sin 3AE ABαα==，从而32sin 2αα=34sin )4sin cos αααα-=，所以2cos )4cos 0αα--=，整理得24cos 0αα-，…30分解得cosα=cos α=， 故30α= ，60ACE ∠= ．由已知1BC EC=()0sin 30sin EAC EAC∠-∠,有sin(30)1)sin EAC EAC ∠-=∠ ，1cos 1)sin 2EAC EAC EAC ∠-∠=∠，整理得1cos2EAC EAC ∠=∠，故tan 2EAC ∠== 可得75EAC ∠= ， …40分 从而45E ∠= ，45DAC DCA E ∠=∠=∠= ，ADC ∆为等腰直角三角形．因AC =1CD =．又ABC∆也是等腰直角三角形，故BC =，212215BD =+-⋅=，BD =故min ()f P BD AC =⋅=．…50分[解法二] （Ⅰ）如答一图2，连接BD 交ABC ∆的外接圆O 于0P 点（因为D 在O 外，故0P 在BD 上）．过,,A C D 分别作000,,P A P C P D 的垂线，两两相交得111A B C ∆，易知0P 在ACD ∆内，从而在111A B C ∆内，记ABC∆之三内角分别为x y z ，，，则0180AP C y z x ∠=︒-=+，又因110B C P A ⊥，110B A P C ⊥，得1B y ∠=，同理有1A x ∠=，1C z ∠=，所以111A B C ∆∽ABC ∆． …10分设11B C BC λ=，11C A CA λ=，11A B AB λ=， 则对平面上任意点M ，有0000()()f P P A BC P D CA P C AB λλ=⋅+⋅+⋅ 011011011P A B C P D C A PC A B =⋅+⋅+⋅ 1112A B C S ∆=111111MA B C MD C A MC A B ≤⋅+⋅+⋅ ()MA BC MD CA MC AB λ=⋅+⋅+⋅()f M λ=，从而0()()f P f M ≤．由M 点的任意性，知0P 点是使()f P 达最小值的点． 由点0P 在O 上，故0,,,P A B C 四点共圆． …20分 （Ⅱ）由（Ⅰ），()f P 的最小值11102()A B C f P S λ∆=2ABC S λ∆=，记ECB α∠=，则2ECA α∠=，由正弦定理有sin 2sin 3AE AB αα==，从而答一图232sin 2αα=34sin )4sin cos αααα-=，所以2cos )4cos 0αα--=，整理得24cos 0αα-=， …30分解得cosα=cos α=，故30α= ，60ACE ∠= ． 由已知1BC EC=()0sin 30sin EAC EAC∠-∠,有sin(30)1)sin EAC EAC ∠-=∠ ，即1cos 1)sin 2EAC EAC EAC ∠-∠=∠，整理得1cos2EAC EAC ∠=∠，故tan 2EAC ∠== 可得75EAC ∠= ， …40分 所以45E ∠=︒，ABC ∆为等腰直角三角形，AC 1ABC S ∆=，因为145AB C ∠=︒，1B 点在O 上，190AB B ∠=︒，所以11B BDC 为矩形，11B C BD ===故λ=min ()21f P == …50分[解法三] （Ⅰ）引进复平面，仍用,,A B C 等代表,,A B C 所对应的复数．由三角形不等式，对于复数12,z z ，有1212z z z z +≥+，当且仅当1z 与2z （复向量）同向时取等号．有 PA BC PC AB PA BC PC AB⋅+⋅≥⋅+⋅ ，所以 ()()()()A P C B C P B A --+--()()()()A P CBC P B A ≥--+-- （1）P C A B C B P A =-⋅-⋅+⋅+⋅()()B P C A PB AC=--=⋅，从而PA BC PC AB PD CA⋅+⋅+⋅PB AC PD AC ≥⋅+⋅()PB PD AC =+⋅BD AC≥⋅ ． （2） …10分（1）式取等号的条件是 复数 ()()A P C B --与()()C P B A --同向，故存在实数0λ>，使得()()()()A P C B C P B A λ--=--，A PB AC P C Bλ--=--， 所以 arg()arg()A P B A C P C B--=--，向量PC 旋转到PA 所成的角等于BC旋转到AB 所成的角，从而,,,P A B C 四点共圆．（2）式取等号的条件显然为,,B P D 共线且P 在BD 上． 故当()f P 达最小值时P 点在ABC ∆之外接圆上，,,,P A B C 四点共圆． …20分（Ⅱ）由（Ⅰ）知min ()f P BD AC =⋅． 以下同解法一．二、（本题满分50分）设()f x 是周期函数，T 和1是()f x 的周期且01T <<．证明： （Ⅰ）若T 为有理数，则存在素数p ，使1p是()f x 的周期； （Ⅱ）若T 为无理数，则存在各项均为无理数的数列{}n a 满足110n n a a +>>> (1,2,)n =⋅⋅⋅，且每个(1,2,)na n =⋅⋅⋅都是()f x 的周期．[证] （Ⅰ）若T 是有理数，则存在正整数,m n 使得n T m=且(,)1m n =，从而存在整数,a b ，使得 1ma nb +=．于是11ma nb a bT a b T m m+==+=⋅+⋅是()f x 的周期． …10分 又因01T <<，从而2m ≥．设p 是m 的素因子，则m pm '=，m *'∈N ，从而11m p m'=⋅ 是()f x 的周期． …20分（Ⅱ）若T 是无理数，令111a T T ⎡⎤=-⎢⎥⎣⎦，则101a <<，且1a 是无理数，令21111a a a ⎡⎤=-⎢⎥⎣⎦，……111n n n a a a +⎡⎤=-⎢⎥⎣⎦，……． …30分由数学归纳法易知n a 均为无理数且01n a <<．又111n n a a ⎡⎤-<⎢⎥⎣⎦，故11n n n a a a ⎡⎤<+⎢⎥⎣⎦，即111n n n n a a a a +⎡⎤=-<⎢⎥⎣⎦．因此{}n a 是递减数列． …40分最后证：每个n a 是()f x 的周期．事实上，因1和T 是()f x 的周期，故111a TT ⎡⎤=-⎢⎥⎣⎦亦是()f x 的周期．假设ka 是()f x 的周期，则111k k k a a a +⎡⎤=-⎢⎥⎣⎦也是()f x 的周期．由数学归纳法，已证得n a 均是()f x 的周期． …50分三、（本题满分50分）设0k a >，1,2,,2008k = ．证明：当且仅当200811k k a =>∑时，存在数列{}n x满足以下条件：（1）010n n x x x +=<<，1,2,3,n = ； （2）lim n n x →∞存在； （3）20082007111n n k n k k n k k k x x a x a x -+++==-=-∑∑，1,2,3,n = ．[证] 必要性：假设存在{}n x 满足（1），（2），（3）．注意到（3）中式子可化为2008111()n n k n k n k k x x a x x -++-=-=-∑，n ∈*N，其中00x =．将上式从第1项加到第n 项，并注意到00x =得111222200820082008()()()n n n n x a x x a x x a x x +++=-+-++- ．…10分由（1）可设lim n n b x →∞=，将上式取极限得 112220082008()()()b a b x a b x a b x =-+-++-20081122200820081()k k b a a x a x a x ==⋅-+++∑20081kk b a =<⋅∑，因此200811k k a =>∑． …20分充分性：假设200811k k a =>∑．定义多项式函数如下：20081()1kk k f s a s ==-+∑，[0,1]s ∈，则()f s 在[0,1]上是递增函数，且(0)10f =-<，20081(1)10k k f a ==-+>∑．因此方程()0f s =在[0,1]内有唯一的根0s s =，且001s <<，即0()0f s =．…30分下取数列{}n x 为01n k n k x s ==∑，1,2,n = ，则明显地{}n x 满足题设条件（1），且100101n nkn k s s x s s +=-==-∑．因001s <<，故1lim 0n n s+→∞=，因此10000lim lim 11n n n n s s sx s s +→∞→∞-==--，即{}n x 的极限存在，满足（ⅱ）．…40分最后验证{}n x 满足（3），因0()0f s =，即2008011k k k a s ==∑，从而200820082008101111()()nk n n k n n k k k n k n k k k k x x s a s s a sa x x +-++-===-====-∑∑∑．综上，存在数列{}n x 满足（1），（2），（3）． …50分。
2008年全国初中数学联合竞赛试题参考答案及评分标准
2008年全国初中数学联合竞赛试题参考答案及评分标准说明：评阅试卷时，请依据本评分标准.第一试，选择题和填空题只设7分和0分两档；第二试各题，请按照本评分标准规定的评分档次给分.如果考生的解答方法和本解答不同，只要思路合理，步骤正确，在评卷时请参照本评分标准划分的档次，给予相应的分数.第一试一、选择题（本题满分42分，每小题7分）本题共有6小题，每题均给出了代号为的四个答案，其中有且仅有一个是正确的.将你所选择的答案的代号填在题后的括号内.每小题选对得7分；不选、选错或选出的代号字母超过一个（不论是否写在括号内），一律得0分.D C B A ,,,1．设，213a a +=213b b +=，且，则代数式a b ≠211a b 2+的值为 （ ） )(A 5. 7. 9. 11.)(B )(C )(D 【答】B .解 由题设条件可知2310a a −+=，，且2310b b −+=a b ≠，所以是一元二次方程的两根，故，，因此,a b 2310x x −+=3a b +=1ab =222222222211()23217()1a b a b ab a b a b ab ++−−×+====. 故选B . 2．如图，设AD ，BE ，CF 为三角形的三条高，若ABC 6AB =，5BC =，，则线段3EF =BE 的长为 （ ）)(A 185. 4. )(B )(C 215. )(D 245. 【答】. D 解 因为AD ，BE ，CF 为三角形的三条高，易知ABC ,,,B C E F 四点共圆，于是△AEF ∽△，故ABC 35AF EF AC BC ==，即3cos 5BAC ∠=，所以4sin 5BAC ∠=. 在Rt △ABE 中，424sin 655BE AB BAC =∠=×=. 故选. D 3．从分别写有数字1，2，3，4，5的5张卡片中任意取出两张，把第一张卡片上的数字作为十位数字，第二张卡片上的数字作为个位数字，组成一个两位数，则所组成的数是3的倍数的概率是 （ ）)(A 15. )(B 310. )(C 25. )(D 12. 【答】. C 解 能够组成的两位数有12，13，14，15，21，23，24，25，31，32，34，35，41，42，43，45，51，52，53，54，共20个，其中是3的倍数的数为12，15，21，24，42，45，51，54，共8个.所以所组成的数是3的倍数的概率是82205=. 故选C. 4．在△中，，∠=，ABC 12ABC ∠=°132ACB °BM 和CN 分别是这两个角的外角平分线，且点,M N 分别在直线和直线AC AB 上，则 （ ）)(A BM CN >. )(B BM CN =.)(C . BM CN <)(D BM 和CN 的大小关系不确定.【答】B .解 ∵，12ABC ∠=°M 为的外角平分线，∴ABC ∠1(18012)842MBC ∠=°−°=B °°. 又，∴180********BCM ACB ∠=°−∠=°−°=180844848BMC ∠=°−°−°=°，∴.BM BC =又11(180)(180132)2422ACN ACB ∠=°−∠=°−°=°， ∴18018012()BNC ABC BCN ACB ACN ∠=°−∠−∠=°−°−∠+∠168(13224)=°−°+°12ABC =°=∠，∴. 因此，CN CB =BM BC CN ==.故选B .5．现有价格相同的5种不同商品，从今天开始每天分别降价10％或20％，若干天后，这5种商品的价格互不相同，设最高价格和最低价格的比值为，则的最小值为 （ ）r r )(A 39()8. )(B 49(8. )(C 59(8. )(D 98. 【答】 B .解 容易知道，4天之后就可以出现5种商品的价格互不相同的情况.设5种商品降价前的价格为a ，过了天. 天后每种商品的价格一定可以表示为n n 98(110%)(120%)()()1010k n k k a a −n k ⋅−⋅−=⋅⋅k −，其中为自然数，且0. k n ≤≤要使r 的值最小，五种商品的价格应该分别为：98()()1010i n i a −⋅⋅，1198()(1010i n i a +−−⋅⋅， 2298()(1010i n a +−i ⋅⋅−，3398()()1010i n i a +−⋅⋅−，4498()()1010i n i a ，其中i 为不超过的自然数. n +−−⋅⋅所以r 的最小值为44498()()91010(988()()1010i n i i n ia a +−−−⋅⋅=⋅⋅. 故选B . 6． 已知实数,x y满足(，则2008x y −=223233x y x y −+−2007−的值为 （ ）)(A . 2008. 2008−)(B )(C 1−. 1.)(D 【答】. D解 ∵(2008x y −=，∴x y ==+，y x −==+由以上两式可得x y =. 所以2(2008x =，解得，所以22008x =22222323320073233200720071x y x y x x x x x −+−−=−+−−=−=.故选.D二、填空题（本题满分28分，每小题7分）1．设12a −=，则5432322a a a a a a a +−−−+=−2−.解 ∵2213(122a a −===−，∴21a a +=， ∴543232323222()2(a a a a a a a a a a a a a a a a+−−−++−−++=−⋅−)2 33332221211(1)(11)2(1)1a a a a a a a a a a a−−+−−===−=−++=−+=−⋅−−−−.2．如图，正方形的边长为1，ABCD ,M N 为BD 所在直线上的两点，且AM =∠，则四边形的面积为135MAN =°AMCN 52解 设正方形的中心为O ，连，则ABCD AO AO BD ⊥，2AO OB ==,2MO ===, ∴MB MO OB =−=. 又， 135ABM NDA ∠=∠=°13590NAD MAN DAB MAB MAB ∠=∠−∠−∠=°−°−∠45=°−MAB AMB ∠=∠，所以△∽△ADN MBA ，故AD DNMB BA =，从而12AD DN BA MB =⋅==. 根据对称性可知，四边形的面积 AMCN11222222MAN S S MN AO ==×××=×××=△522. 3．已知二次函数2y x ax b =++的图象与x 轴的两个交点的横坐标分别为，，且m n 1m n +≤.设满足上述要求的b 的最大值和最小值分别为p ，q ，则p q +=12解 根据题意，是一元二次方程的两根，所以,m n 20x ax b ++=m n a +=−，. mn b =∵1m n +≤，∴1m n m n +≤+≤，1m n m n −≤+≤.∵方程的判别式，∴20x ax b ++=24a b Δ=−≥022()44a m n b +14≤=≤. 22244()()()1b mn m n m n m n ==+−−≥+−≥−1，故14b ≥−，等号当且仅当12m n 时取得； =−=122244()()1()b mn m n m n m n ==+−−≤−−≤，故14b ≤，等号当且仅当12m n 时取得. ==所以14p =，14q =−，于是12p q +=. 4．依次将正整数1，2，3，…的平方数排成一串：149162536496481100121144…，排在第1个位置的数字是1，排在第5个位置的数字是6，排在第10个位置的数字是4，排在第2008个位置的数字是 1 .解 1到3，结果都只各占1个数位，共占13223×=个数位； 24到，结果都只各占2个数位，共占个数位；292612×=210到，结果都只各占3个数位，共占23132266×=个数位；232到，结果都只各占4个数位，共占299468272×=个数位；2100到，结果都只各占5个数位，共占523162171085×=个数位；此时还差2008个数位.(312662721085)570−++++=2317到，结果都只各占6个数位，共占2411695570×=个数位.所以，排在第2008个位置的数字恰好应该是的个位数字，即为1. 2411第二试 （A ）一．（本题满分20分） 已知，对于满足条件221a b +=01x ≤≤的一切实数x ，不等式(1)(1)()0a x x ax bx b x bx −−−−−−≥ （1）恒成立.当乘积ab 取最小值时，求的值.,a b 解 整理不等式（1）并将代入，得221a b +=2(1)(21)0a b x a x a ++−++≥ （2）在不等式（2）中，令0x =，得；令0a ≥1x =，得.0b ≥易知10，a b ++>21012(1)a ab +<<++，故二次函数2(1)(21)y a b x a x a =++−++的图象（抛物线）的开口向上，且顶点的横坐标在0和1之间.由题设知，不等式（2）对于满足条件01x ≤≤的一切实数x 恒成立，所以它的判别式，即2(21)4(1)0a a b Δ=+−++⋅≤a 14ab ≥. 由方程组 221,14a b ab ⎧+=⎪⎨=⎪⎩ （3） 消去b ，得16，所以421610a a −+=224a =或224a +=.又因为，所以0a≥4a =或4a +=，于是方程组（3）的解为,4,4a b ⎧=⎪⎪⎨⎪=⎪⎩或,4.4a b ⎧=⎪⎪⎨−⎪=⎪⎩ 所以的最小值为ab 14，此时的值有两组，分别为 ,ab ,44a b +==和44a b +==. 二．（本题满分25分） 如图，圆与圆相交于O D ,A B 两点，为圆的切线，点在圆上，且.BC D C O AB BC =（1）证明：点O 在圆的圆周上.D （2）设△的面积为，求圆的的半径的最小值.ABC S D r 解 （1）连，因为为圆心，,,,OA OB OC AC O AB BC =，所以△OBA ∽△，从而OBC OBA OBC ∠=∠.因为OD ，所以,AB DB BC ⊥⊥9090DOB OBA OBC DBO ∠=°−∠=°−∠=∠，所以DB DO =，因此点O 在圆的圆周上.D （2）设圆的半径为a ，的延长线交于点O BO ACE ，易知BE AC ⊥.设2AC y =(0)y a <≤，，，则，，OE x =AB l =22a x 2y =+()S y a x =+22222222()2222()aS l y a x y a ax x a ax a a x y=++=+++=+=+=. 因为,22ABC OBA OAB BDO ∠=∠=∠=∠AB BC =,DB DO =，所以△∽△，所以BDO ABC BD BO AB AC=，即2r a l y =，故2al r y =. 所以22223222()4422a l a aS S a S y y y y ==⋅=⋅≥r ，即2≥r ，其中等号当a y =时成立，这时是圆O 的AC直径.所以圆的的半径的最小值为Dr 2. 三．（本题满分25分）设为质数，b 为正整数，且a 29(2)509(4511)ab a b +=+ （1）求，b 的值.a 解 （1）式即2634511()509509ab a b ++=，设634511,509509a b a b m n ++==，则 50965094351m a n a b 1−−== （2） 故，又，所以35116n m a −+=02n m =2351160m m a −+= （3）由（1）式可知，(2能被509整除，而509是质数，于是2)a b +2a b +能被509整除，故为整数，即关于的一元二次方程（3）有整数根，所以它的判别式为完全平方数.m m 251172a Δ=−不妨设（t 为自然数），则722251172a t Δ=−=511(511)(511)a t t t 22=−=+−.由于511和511的奇偶性相同，且511t +t −511t +≥，所以只可能有以下几种情况：①两式相加，得36，没有整数解.51136,5112,t a t +=⎧⎨−=⎩21022a +=②两式相加，得18，没有整数解. 51118,5114,t a t +=⎧⎨−=⎩41022a +=③两式相加，得12，没有整数解. 51112,5116,t a t +=⎧⎨−=⎩61022a +=④两式相加，得6，没有整数解.5116,51112,t a t +=⎧⎨−=⎩121022a +=⑤两式相加，得4，解得5114,51118,t a t +=⎧⎨−=⎩181022a +=251a =. ⑥两式相加，得2，解得5112,51136,t a t +=⎧⎨−=⎩361022a +=493a =，而4931729=×不是质数，故舍去. 综合可知.251a =此时方程（3）的解为3m =或5023m =（舍去）. 把，代入（2）式，得251a =3m =5093625173b ×−×==. 第二试 （B ）一．（本题满分20分）已知，对于满足条件221a b +=1,0x y xy +=≥的一切实数对(,)x y ，不等式220ay xy bx −+≥ （1）恒成立.当乘积ab 取最小值时，求的值.,a b 解 由可知011,0x y xy +=≥,01x y ≤≤≤≤.在（1）式中，令0,1x y ==，得；令0a ≥1,0x y ==，得b .0≥x 将代入（1）式，得，即1y =−22(1)(1)0a x x x bx −−−+≥2(1)(21)0a b x a x a ++−++≥ （2）易知10，a b ++>21012(1)a ab +<<++，故二次函数2(1)(21)y a b x a x a =++−++的图象（抛物线）的开口向上，且顶点的横坐标在0和1之间.由题设知，不等式（2）对于满足条件01x ≤≤的一切实数x 恒成立，所以它的判别式，即2(21)4(1)0a a b Δ=+−++⋅≤a 14ab ≥. 由方程组 221,14a b ab ⎧+=⎪⎨=⎪⎩ （3） 消去b ，得16，所以421610a a −+=224a −=或224a +=，又因为，所以0a≥4a =或4a =.于是方程组（3）的解为,4,4a b ⎧=⎪⎪⎨⎪=⎪⎩或,4.4a b ⎧=⎪⎪⎨−⎪=⎪⎩所以满足条件的的值有两组，分别为,ab ,44a b +==和44a b +==c ). 二．（本题满分25分）题目和解答与（A ）卷第二题相同.三．（本题满分25分）题目和解答与（A ）卷第三题相同.第二试 （C ）一．（本题满分20分）题目和解答与（B ）卷第一题相同.二．（本题满分25分）题目和解答与（A ）卷第二题相同.三．（本题满分25分）设为质数，b 为正整数，且满足a ,29(22)509(41022511)2a b c a b c b c ⎧+−=+−⎨−=⎩ (1)(2)求的值.(a b c +解 （1）式即2()509509=， 设66341022511,509509a b c a b c m n +−+−==，则 5096509423511m a n a b c −−−== （3） 故，又，所以35116n m a −+=02n m =2351160m m a −+= （4）由（1）式可知，(2能被509整除，而509是质数，于是22)a b c +−c 22a b +−能被509整除，故为整数，即关于的一元二次方程（4）有整数根，所以它的判别式为完全平方数.m m 251172a Δ=−不妨设（t 为自然数），则722251172a t Δ=−=511(511)(511)a t t t 22=−=+−.由于511和511的奇偶性相同，且511t +t −511t +≥，所以只可能有以下几种情况：①两式相加，得36，没有整数解. 51136,5112,t a t +=⎧⎨−=⎩21022a +=②两式相加，得18，没有整数解.51118,5114,t a t +=⎧⎨−=⎩41022a +=③两式相加，得12，没有整数解. 51112,5116,t a t +=⎧⎨−=⎩61022a +=④两式相加，得6，没有整数解.5116,51112,t a t +=⎧⎨−=⎩121022a +=⑤两式相加，得4，解得5114,51118,t a t +=⎧⎨−=⎩181022a +=251a =. ⑥两式相加，得，解得5112,51136,t a t +=⎧⎨−=⎩236102a +=2493a =，而4931729=×不是质数，故舍去.综合可知，此时方程（4）的解为或251a =3m =5023m （舍去）. =把，代入（3）式，得251a =3m =50936251273b c ×−×−==，即27c b =−. 代入（2）式得，所以b ，(27)2b b −−=5=3c =，因此()251(53)2008a b c +=×+=.。
12页--2008年全国大学生英语竞赛样题(A级)及答案--来自赛才网
2008年全国大学生英语竞赛样题(A级)Part I Listening Comprehension (25 minutes, 30 marks)Part II Vocabulary and Structure (10 minutes, 15 marks31. Essentially, a theory is an abstract, symbolic representation of ________ reality.A. what it is conceivedB. that is conceivedC. what is conceived to beD. that is being conceived of32. New York City collects garbage ________ $209 per family per year.A. at the cost ofB. at sacrifice ofC. in demand ofD. in support of33. I would just as soon ________ rudely to her.A. you not speakB. you won't speakC. you not speakingD. you didn't speak34. ________ so many people been out of work as today.A. More than ever beforeB. Never before haveC. In the past, there never haveD. Formerly, there never were35. We got home late last night, ________, early this morning.A. namelyB. in other wordsC. or ratherD. in any case36. The miners escaped the danger by ________.A. a hair breadthB. hair breadthC. hair's breadthD. a hair's breadth37. Praise is like warm sunshine. It not only brings joy to ________ who are praised but also adds pleasure to ________ who praise.A. these; theseB. we; usC. them; themD. those; those38. When our imports ________ our exports, we say we have a favorable trade balance; however, when our imports and exports are the other way round we say we have an unfavorable trade balance.A. fall shortB. fall short ofC. run out ofD. are lack of39. ________ that distinguish human beings from other primates are related to the ability of people to stand upright and walk on two legs.A. Many of the physical characteristicsB. Of the many physical characteristicsC. The physical characteristicsD. There are many physical characteristics40. Classical logic is characterized by a concern for the structure and elements of argument ________ that thought, language, and reality are interrelated.A. based on the beliefB. on the belief basedC. belief based on theD. the based belief on41. ________ fashioned from a wick floating in a bowl of oil functioned according to the principle of capillary action.A. All lamps earlyB. Lamps all earlyC. All early lampsD. Early all lamps42. In 1964 ________ of Henry Osaka Tanner's paintings was shown at the Smithsonian Institution.A. was a major collectionB. that a major collectionC. a collection was majorD. a major collection43. Traditionally, ________ on Thanksgiving Day in New England.A. when served is sweet ciderB. when sweet cider is servedC. is served sweet ciderD. sweet cider is served44. Richard: This class is so boring. When will it end?Drew: I don't know. I don't think I can take it much longer.Teacher: ________Richard: Oh, nothing. We were discussing the text.A. Hey you two! What are you talking about?B. Do one of you know what the answer to this problem is?C. Am I boring you two?D. The two of you go to the principal's office right now!45. Delivery man: I have a package here for Beverly Beaver.Beverly: That is me. ________Delivery man: Yes, please sign here on the line.Beverly: Thank you. These must be the new drapes I ordered.A. Where is the receipt?B. Do you have the sender's address?C. Do you take a credit card?D. Do I have to sign for it?Part III Reading Comprehension (20 minutes, 40 marks)Section A (5 marks)Directions: There is one passage in this section with 5 questions. For each question, there are four choices marked A, B, C and D. You should decide on the best choice. Then mark the corresponding letter on theAnswer Sheet with a single line through the centre.Questions 46-50 are based on the following passage.No matter what our work is, we all have to face that choice - in my field no less than in others. There may not be a more important field for the dissemination of values in our country than the entire communicationsindustry - most strikingly, television. Networks are very sensitive to that fact, and they employ dozens of censors to prevent all of us from using language on television that an eight year old might have to explain to his parents. But the point that censors miss, I think, is that it is not so much what we say that teaches as what we don't say. Even programs that attempt to make a moral point don't always make the point that they intended to. Because when we sense we are being sole something, we automatically defend ourselves against it. I think it may be the unspoken assumptions that mold an audience.Look at the way, for instance, that violence is treated on television. It is not only the quantity that offends. There probably is no more violence on television than there is in a Shakespearean tragedy. But on television you find unfelt violence and in Shakespeare you tend to find felt violence. In Shakespeare the characters react with a human response: They fear, they hurt, they mourn. Most of the time on television, violence is dealt with by sweeping it under the rug as fast as possible and by having people go on about their business as if nothing had happened. (If I can't have less violence, I want at least a better grade of violence.) One of the unspoken assumptions is that violence can be tolerated as long as you ignore it and have no reaction to it. But that seems to me to be dangerously close to psychopathic behavior. I wonder if there is any connection between the long acceptance by our people of the Vietnam War and the thousands and thousands of deaths that we have seen on television over the years that were never mourned, never even paused for except to sell shampoo for sixty seconds.Maybe our greatest problem is that we have two separate sets of value systems that we use - the one we talk about and the one we live by. We seem to place a very high value on fairness and on human concerns. And yet we still have widespread discrimination based on race, sex and religion. You still don't find Jews, blacks or other minorities in any significant numbers in decision-making positions in the banking industry, for instance. You think that's an accident? I think somebody puts a value on that. And you don't find women in any significant numbers in decision-making capacities in any industry. Why? Because we place a higher value on appeasing the fragile male ego than we do on fairness and decency. Maybe what we need is a declaration of interdependence.Questions:46. It can be concluded from the first paragraph that ________.A. we won't buy things that are recommendedB. we become defensive whenever we are taught to do somethingC. television is the most important part of the communication industryD. TV programs can never successfully convey a moral point47. What is the author's attitude toward violence on television?A. He thinks that violence on television should be totally banned.B. Violence on television is more tolerable than the one in Shakespeare's plays.C. Violence can be tolerated if you pretend that it doesn't exist.D. Violence, if unavoidable, should be presented on television in a more appropriate way.48. The word “appeasing” (line 6 in paragraph 3) most probably means ________.A. attackingB. accusingC. soothingD. satisfying49. By citing the examples of the Jews, blacks, and other minorities in the last paragraph, the author aims to show that ________.A. it's an accident that they are out-numbered in the decision-making positions in the banking industryB. they stand for one set of value systemC. what we talk about and what we actually do might be totally differentD. fairness is well respected in our life50. The author's tone in writing the text is ________.A. neutralB. matter-of-factC. criticalD. questioningSection B (15 marks)Directions: There is one passage in this section with 10 questions. Go over the passage quickly and answer the questions on the Answer Sheet.For questions 51 - 55, markY (for YES) if the statement agrees with the information given in the passage;N (for NO) if the statement contradicts the information given in the passage;NG (for NOT GIVEN) if the information is not given in the passage.For questions 56 - 60, complete the sentences with the information given in the passage.You hear the same complaint all the time as people get older: “My memory is terrible.” Is it all in the mind, or do real changes take place in the brain with age to justify such grumbling (抱怨)? The depressing answer is that the brain's cells, the neurons, die and decline in efficiency with age.Professor Arthur Shimamura, of the University of California at Berkeley, says there are three main ways in which mental function changes. The first is mental speed, for examplehow quickly you can react to fast-moving incidents on the road. Drivers in their late teens react quickly but tend to drive too fast, while the over sixties are more cautious but react more slowly. The near-inevitable slowing with age also partly explains why soccer players are seen as old in their thirties,while golf professionals are still in their prime at that age. This type of mental slowing results from a reduction in the efficiency with which the brain's neurons work.The fact that adults find it harder to learn musical instruments than children points to a second type of mental loss with age - a reduction in learning capacity. The parts of the brain known as the temporal lobes control new learning, and are particularly vulnerable to the effects of aging. This means that, as we get older, we take longer to learn a new language, and slower to master new routines and technologies at work, and we have to rely more on diaries and other mental aids.“Working memory” is the third brain system which is vulnerable to the effects of aging. Working memory is the brain's “blackboard”, where we juggle from moment to moment the things we have to keep in mind when solving problems, planning tasks and generally organizing our day-to-day life. Absent-mindedness occurs at all ages because of imperfections in the working memory system - so, for instance, you may continually lose your glasses, or find yourself walking into a room of your house only to find that you cannot remember what you came for.Such absent-mindedness tends to creep up on us as we age and occurs because our plans and intentions, which are chalked up on the mental blackboard, are easily wiped out by stray thoughts and other distractions. Stress and preoccupation can also cause such absent-mindedness, in addition to age-related changes in the brain. The frontal lobes of the brain - located behind the forehead and above the eyes - are where the working memory system is located. Like the temporal lobes, which handle new learning, the frontal lobes are more vulnerable to the aging process than other parts of the brain.The news, however, is not all bleak. Although neurons reduce in number with age, the remaining neurons send out new and longer connection fibers (dendrites) to maintain connections and allow us to function reasonably well with only relatively small drops in ability.This and other evidence suggests that the principle “use it or lost it” might apply to the aging brain. Professor Shimamura studied a group of university professors who were still intellectually active, and compared their performance on neuropsychological tests with that of others of their age group, as well as with younger people. He found that on several tests of memory, the mentally active professors in their sixties and early seventies were superior to their contemporaries, and as good as the younger people.Research on animals provide even stronger evidence of the effects of stimulation on the brain structure. Professor Bryan Kolb, of the University of Lethbridge in Canada, has shown that animals kept in stimulating environments show sprouting (生长) and lengthening of the connecting nerve fibers in comparison with animals kept in unstimulating environments.The beneficial effects of continued mental activity are shown by the fact that older contestants in quiz shows are just as fast and accurate in responding to general knowledge questions as younger competitors, suggesting that at least part of their intellectual apparatus is spared the effects of aging because of practice and skill.Such findings lead to the intriguing possibility of “mental fitness tra ining” to accompany jogging and workouts for the health conscious. Research in Stockholm by Professor Lars Backman and his colleague has shown that older people can be trained to use their memory better, with the effects of this training lasting several years.Just as people go bald or grey at different rates, so the same is true for their mental faculties. Why this should be the case for memory and other mental functions is not yet clear. If Professor Shimamura is right, then the degree to which people use and stretch their mental faculties may also have a role to play.Questions:51. The passage gives a description of several methods of testing mental ability.52. Professor Shimamura identified a number of areas in which mental function may change.53. As the temporal lobes of the brain are affected by aging, it becomes harder to pick up new skills.54. Research indicates that physical training can help to improve memory.55. Taking part in quizzes is the best way to stimulate the brain.56. Absent-mindedness is not necessarily ________.57. Scientists do not understand why people's ________ decline at different rates.58. If a person is under stress, ________ may become more frequent.59. When the frontal lobes of the brain are affected by aging, there is a gradual deterioration in ________.60. If an animal's brain is kept active and stimulated, ______________.Section C (10 marks)Directions: In this section, there is one passage followed by 5 questions. Read the passage carefully, then answer the questions in a maximum of 10 words. Remember to write the answers on the Answer Sheet.Questions 61 - 65 are based on the following passage.Alarmed by a 20-year decline in student achievement, American schools are considering major upheavals in the career structure of teachers, school boards around the country are planning to abandon traditional salary schedules and single out outstanding teachers for massive pay rise.The idea is regarded with deep suspicion by the United States' biggest teachers' unions, the National Education Association and the American Federation of Teachers. They say the creation of a cadre of elite teachers will sour professional relationships and encourage teachers to compete instead of cooperate; they also question whether a fair way can be devised to tell which teachers really do perform better than their colleagues.But heightened public anxiety about secondary education appears to have given the master teacher concept unstoppable political momentum. Florida and Tennessee are racing to introduce ambitious statewide master teacher schemes before the end of the year. Less grandiose proposals to pay teachers on the basis of merit instead of seniority have already been implemented in countless school districts. And the Secretary of Education, Mr. Terrel Bell, recently promised substantial incentive grants to states which intend to follow their example.Low pay is believed to be the single most important reason for the flight from teaching. The average salary of a teacher in the United States is just under $19,000, much less than that of an engineer ($34,700) and not much more than that of a secretary($16,500). To make ends meet it is common for teachers to take second jobs in the evening and in their summer holidays, and women, who used to make up the bulk of teacher candidates are turning to better paid professions.The unions insist that the answer to this problem is to increase the basic pay of all teachers, but most states would find that too expensive, they would be better able to afford schemes that confine pay increases to a small number of exceptional teachers. Champions of the idea say it would at least hold out the promise of high pay and status to bright graduates who are confident of their ability to do well in the classroom, but are deterred by the present meager opportunities for promotion.One of the first large-scale tests of this approach will come in Tennessee, where a year of painstaking negotiations has just overcome bitterunion opposition to a wide-ranging master teacher scheme. Tennessee promised that they will allocate $300 million as education budget. In return for a chance to earn bigger salaries and faster promotion, teachers will subject themselves to closer scrutiny.The Tennessee plan will make it harder for poor teachers to join the profession. Beginners will have to serve a probationary year before qualifying, and another three apprentice years before receiving tenure. Apprentice teachers who fail to reach a required standard will not be allowed to stay on. Survivors will be designated “career teachers” and given a chance to climb through three career rugs and earn bonuses of up to $7,000. Advancement will not be automatic. The performance of each teacher will be closely assessed by committees of teachers drawn from other districts.Questions:61. What support is the federal government offering to states that set up a master teacher scheme?62. What's the purpose of the master teacher scheme?63. In the state of Tennessee, how will teachers be assessed?64. What is the main idea of the text?65. What can be inferred from the text about the master teacher scheme?Section D (10 marks)Directions: In this section, there is one passage followed by a summary. Read the passage carefully and complete the summary below by choosing no more than three words from the passage. Remember to write the answers on the Answer Sheet.Questions 66 - 70 are based on the following passage.The United KingdomBritain (or Great Britain) is an island that lies off the north-west coast of Europe. The nearest country is France, which is 20 miles away and from which Britain is separated by the English Channel. The island is surrounded by the Atlantic Ocean to the west, and the North Sea to the east. It comprises the mainlands of England, Wales and Scotland, that is, three countries. Scotland is in the north, while Wales is in the west. Ireland, which is also an island, lies off the west coast of Britain. It consists of Northern Ireland and the Irish Republic. Britain together with Northern Ireland constitute the UnitedKingdom (UK). Thus, the United Kingdom is composed of four countries, the largest of which is England. The capital city is London, which is situated in south-east England.The UK has a total area of about 244,100 square kilometres (94,248 square miles). About 70% of the land area is devoted to agriculture, about 7% is wasteland, moorland and mountains, about 13% is devoted to urban development, and 10% is forest and woodland. The northern and western regions of Britain, that is Scotland and Vales, are mainly mountainous and hilly. Parts of the north-west and centre of England also consist of mountains and hills.Britain has a generally mild and temperate climate. It is, however, subject to frequent changes. It has an average annual rainfall of about 120 centimetres (47 inches).In 1998 the population of the United Kingdom was nearly 59 million. The density of population was approximately 240 people per square kilometre. However, in England, where 83% of the population live, the density was much higher, about 363 per square kilometre.In the UK, English is the first language of the vast majority of people. However, in western Wales, Welsh is the first language for many of the people. In Scotland only a small number of people speak Gaelic.In Britain about 66% of the population say that they are Christian, while fewer than 5% say that they belong to other religions.Summary:Britain is an island that (66) ________ the Atlantic Ocean and the North Sea. It comprises the mainlands of England, Wales and Scotland. Ireland (67) ________ the west coast of Britain. It (68)________ Northern Ireland and the Irish Republic. The United Kingdom consists of Britain together with Northern Ireland. The capital city is London which (69) ________ south-east England.In 1998 the population of the UK was nearly 59 million. The density of population was 240 people per square kilometre. In the UK English is the first language of most people. In western Wales, Welsh (70) ________ many of the people, but few people in Scotland speak Gaelic.Part IV Cloze (15 minutes, 15 marks)Section A (5 marks)Directions: There are 5 blanks in the passage. For each blank, some letters of the word have been given (not exceeding 3 letters). Read the passage below and think of the word which best fits each blank. Use only one word in each blank. Remember to write the answers on the Answer Sheet.Language is learned primarily through communication with other people. Research shows that the more communication children (71) exp________ at home, the better developed their language skills will be. Children need to talk well in order to learn well. This is especially important for the development of native language skills since the language is often reinforced by the child's environment outside school.However, the (72) qu________ of communication is just as or more important than quantity alone. The language adults use helps children become aware of the many different aspects of objects and events around them. For example, during a shopping trip to the market or store, adults can develop children's concepts by bringing their attention to the shapes, colors, sounds, textures, and sizes of objectsand events around them. Parents can do this naturally through conversation without (73) di________ teaching. In other words, conversation with children in everyday situations expands their minds and develops their thinking skills.In addition to conversing with children, adults can help prepare their children to succeed in school by (74) en________ them to take an active interest in books and in the print that surrounds them in the environment. The child's first major task in school is learning to read. Children who come to school with knowledge that the print aroundthem carries important meaning and with an interest in books and stories will usually (75) su________ in learning to read rapidly.Section B (10 marks)Directions: There are 10 blanks in the passage. Use the word given in each blanket to form a word that fits in each blank. Remember to write the answers on the Answer Sheet.Forensic science (法庭科学) means (76) ________ (use) science to solve crime. But what exactly does a forensic scientist do?At the scene of crime, the crime scene (77) ________ (investigator) will thoroughly examine the area. They will look for fingerprints and palm prints; marks of tools and weapons; marks from shoes; fibers from (78) ______ (cloth); body fluids.Increasingly, CSIs (= crime scene investigators) are using specialized techniques, on site, to determine which areas to concentrate their investigation on.A forensic pathologist (法医)specializes in examining dead bodies to determine how and when death occurred. As well as (79) ________ (examine) the body, either on site or through photos (80) ________ (take) at the scene, the forensic pathologist may also use X-ray imaging(X光成像技术) and will usually conduct an autopsy (尸体解剖). The pathologist will also take samples of body tissues, like blood, liver or hair, for further (81) ________ (analyse).A new form of identification (82) ________ (rely) on DNA, which carries the genetic (基因的) information of each person. Everyone's DNA is different (except for identical twins). DNA fingerprinting allows police to identify an individual in the same way as fingerprints do.DNA profiles (DNA片断) are a very powerful (83) ________ (mean)of determining whether two or more samples may or may not have come from the same person. If DNA profiles do not match, they (84) ________ (come) from different people.However, if they do match, there is still a very slight chance that they may have come from different people.DNA analysis does not enable scientists to build up a picture of a person from DNA. The only (85) ______ (character) that the DNA tells us is the sex of the person.Part V Translation (15 minutes, 15 marks)Section A (8 marks)Directions: Translate the underlined sentences of the following passage into Chinese. Remember to write the answers on the Answer Sheet.If a nation is essentially disunited, it is left to the government to hold it together. (86) This increases the expense of government, and reduces correspondingly the amount of economic resources that could be used for developing the country. And it should not be forgotten how small those resources are in a poor and backward country. Where the cost of government is high, resources for development are correspondingly low.This may be illustrated by comparing the position of a nation with that of a private business enterprise. An enterprise has to incur certain costs and expenses in order to stay in business. For our purposes, we are concerned only with one kind of cost - the cost of managing and administering the business. Such administrative overhead of a business is analogous to the cost of government in a nation.(87) The administrative overhead of a business is low to the extent that everyone working in the business can be trusted to behave in a way that best promotes the interests of the firm. If they can each be trusted to take such responsibilities, and to exercise such initiative as falls within their sphere, then administrative overhead will be low. It will be low because it will be necessary to have only one man looking after each job, without having another man to check upon what he is doing, keep him in line, and report on him to someone else. But if no one can be trusted to act in a loyal and responsible manner towards his job, then the business will require armies of administrators, checkers, and foremen, and administrative overhead will rise correspondingly. (88) As administrative overhead rises, so the earnings of the business, after meeting the expenses of administration, will fall; and the business will have less money to distribute as dividends or invest directly in its future progress and development.It is precisely the same with a nation. To the extent that the people can be relied upon to behave in a loyal and responsible manner, the government does not require armies of police and civil servants to keep them in order. (89) But if a nation is disunited, the government cannot be sure that the actions of the people will be in the interests of the nation; and it will have to watch, check, and control the people accordingly. A disunited nation therefore has to incur unduly high costs of government.Section B (7 marks)Directions: Translate the following sentences into English, using the and expressions given in the brackets. Remember to write your answers on the Answer Sheet.90. 这些要求有点过头，一系列的争端也由此而起。
2008年大学生数学建模竞赛A题优秀论文(1)
我们参赛选择的题号是（从 A/B/C/D 中选择一项填写）：
我们的参赛报名号为（如果赛区设置报名号的话）：
所属学校（请填写完整的全名）：
杭州电子科技大学
参赛队员 (打印并签名) ：1. 宋飞杰
2. 张佳喜
3. 司继春
指导教师或指导教师组负责人 (打印并签名)： 数模组
日期： 2008 年 9 月 21 日
在如何检验模型的问题上，需要分两方面进行检验，一是精度，而是稳定性。 按照以上的方法求圆心在像平面上的坐标，并没有充分利用像平面上所有轮廓点的 信息，因此可以利用这些点来检验模型的精度。 对于稳定性问题，可以采用计算机模拟的方法，随机修改图形的轮廓，并用以上的 方法再次进行求解，通过比较修改前后的结果来分析模型的稳定性。 最后，考虑另外一台相机的定位相对位置问题。根据前面模型，我们应能够对任意 一台相机确定靶标相对它的位置，因此可以以这个靶标作为参照物，建立一个世界坐标 系，将这两台相机的位置在这个坐标系里面表示出来，以此确定两台相机的相对位置。
摘要
本文研究数码相机定位中有关系统标定的相关问题。 首先，本文建立了三个坐标系：像素平面坐标系、像物理平面坐标系和相机坐标系 。 其中像素平面坐标系和像物理平面坐标系是同一个平面针对不同需要而建立的；相机坐 标系是一个世界坐标系，它以相机为参照物。 然后针对第一问确定圆心在像平面上的坐标的问题，本文建立了两个子模型：针孔 相机模型和确定靶标相对相机位置的模型，然后提出了运用以上两个子模型求解坐标的 方法。 在第一个子模型针孔相机模型中，本文对数码相机进行了适当的简化，即把数码相 机看成是一个针孔相机的结构，利用射影几何的有关知识建立了从相机坐标到像物理坐 标的转换关系模型。 在第二个子模型确定靶标相对相机位置的模型中，本文利用像平面上四个图形公切 线的交点建立了与靶标平面的联系，并结合靶标的尺寸、形状，建立起了确定靶标位置 的模型。 在建立了以上两个子模型后，通过第二个子模型可以求出靶标上圆心在相机坐标系 中的坐标，再利用第一个子模型的转换关系，就可以得到圆心在像平面上的坐标。 针对模型的求解，本文使用模拟退火算法计算出了像平面上四条公切线交点的坐 标，并使用基于最小二乘法的 Matlab 优化工具箱的工具求解出靶标的位置，进一步求 出 了 圆 心 在 像 平 面 上 的 坐 标 ， 五 个 坐 标 分 别 为 ： A0(-190.26,-196.77),B0 (-88.88,189.15),C0 (129.74,-172.72),D0 (72.85,119.30),E0 (-229.13,119.21) 在模型的检验模型中，本文分别讨论了以上模型的精度和稳定性。在精度检验中， 我们将像平面上未被利用的图形的轮廓上的点映射回靶标平面上，并在靶标平面上检验 这个轮廓是否与相应的圆形重合。经过检验，轮廓上的点与相应的圆形之间的平均偏差 在 1 像素以内，说明以上模型的精度很高。 在随后的稳定性检验中，我们通过计算机模拟的方式，随机改变了像平面上图形的 轮廓，并对这些轮廓求解圆心，结果即使在轮廓损失了近 30%的信息量时，圆心的平均 偏移距离也只有 0.2682，不到一个像素，说明模型具有很好的稳定性。 最后，本文通过改变世界坐标系，以靶标作为参照物，给出了计算两台相机光学中 心、像平面中心坐标的方法，得出了两台相机相对位置的模型。
2008年全国大学生英语竞赛样题及答案(ABCD各级)
2008年全国大学生英语竞赛样题及答案（A级）Part I Listening Comprehension (25 minutes, 30 marks)Section A (10 marks)Directions: In this section, you will hear 10 short conversations. At the end of each conversation, a question will be asked about what was said. Both the conversation and the question will be read only once. After each question, there will be a pause. During the pause, you must read the three choices marked A, B and C, and decide which is the best answer. Then mark the corresponding letter on the Answer Sheet with a single line through the centre.1. A. At a bank. B. At a department store. C. At a lawyer's office.2. A. Jealous. B. Indifferent. C. Negative.3. A. A party invitation.B. A tour of the house.C. A dinner party.4. A. Policeman and driver.B. Teacher and pupil.C. Driver and actress.5. A. To attend a party.B. To the beauty salon.C. To the school.6. A. 1,090. B. 530. C. 1,060.7. A. Dec. 13th, 1906. B. Dec. 30th, 1906. C. Nov. 13th, 1916.8. A. Nothing. B. Dinner. C. A snack.9. A. In less than three days.B. In more than three days.C. In three days.10. A. To a friend's house.B. To the kindergarten.C. Do some shopping.Section B (5 marks)Directions: In this section, you will hear one long conversation. The conversation will be read only once. At the end of the conversation, there will be a one-minute pause. During the pause, you must read the five questions, each with three choices marked A, B and C, and decide which is the best answer. Then mark the corresponding letter on the Answer Sheet with a single line through the centre.11. What does Susan have to do on Saturday morning?A. Walk the dog.B. Clean the house.C. Go to the doctor's.12. What will Susan do at noon?A. Go shopping.B. Watch TV.C. Go to the dentist's.13. What time is Susan meeting with Julie?A. 12:30 am.B. 2:00 pm.C. 4:30 pm.14. How does Susan feel about her schedule on Saturday?A. Exciting.B. Disappointing.C. Exhausting.15. Why can't they watch a video at Susan's house?A. The machine doesn't work.B. They can't decide on a video.C. Susan's mother is going to use it.Section C (5 marks)Directions: In this section, you will hear 5 short news items. After each item, there will be a pause. During the pause, you must read the question and then the three choices marked A, B and C, and decide which is the best answer. Then mark the corresponding letter on the Answer Sheet with a single line through the centre.16. Who got to move to the next step?A. Andre Agassi.B. Rogen Fidow.C. Roger Federer.17. What did the gunman force the charity to do?A. Delay the operation in this area.B. Operate on their own staff.C. Attack the Sans Frontieres.18. Why were the Iraqi detainees made to pick up garbage?A. Because the serviceman abused the detainees.B. Because the commander gave the order to the serviceman.C. Because the detainees were accused.19. Where did the clash happen?A. In the West Bank.B. Near Israeli.C. Near Ariel.20. Why do the Israeli forces threaten to carry out a massive offensive against the Palestinians?A. Because Israeli troops were besieged by the Palestinians.B. Because the military operation began since last November.C. Because Palestinians killed Israelis in a suicide attack.Section D (10 marks)Directions: In this section, you will hear a short passage. There are ten missing words or phrases in it. Fill in the blanks with the exact words or phrases you hear on the tape. Remember to write the answers on the Answer Sheet.There are two types of people in the world. Although they have (21) ________ degrees of health and wealth and other comforts of life, one becomes happy, and the other becomes (22) ________. This arises from the different ways in which they consider things, persons, and events, and the resulting effects upon their minds. People who are to be happy fix their attention on the (23) ________ of things, the pleasant part of convention, the fine weather. They enjoy all the (24) ________ things. Those who are to be unhappy think and speak only of the (25) ________ things. Therefore, they are continually (26) ________. By their remarks, they sour the pleasure of society, offend many people, and make themselves (27) ________ everywhere. If this turn of mind were found in nature, such unhappy persons would be the more to be pitied. The tendency to criticize and to be disgusted is perhaps taken up originally (28) ________. It grows into a habit, unknown to its possessors. The habit may be strong, but itmay be cured when those who have it are convinced of its bad effects (29) ________. I hope this little warning may be of service to them, and help them change this habit. Although in fact it is chiefly (30) ________, it has serious consequences in life since it brings on deep sorrow and bad luck.Part II Vocabulary and Structure (10 minutes, 15 marks)Directions: There are 15 incomplete sentences in this section. For each blank there are four choices marked A, B, C and D. Choose the one that best completes the sentence. Then mark the corresponding letter on the Answer Sheet with a single line through the centre.31. Essentially, a theory is an abstract, symbolic representation of ________ reality.A. what it is conceivedB. that is conceivedC. what is conceived to beD. that is being conceived of32. New York City collects garbage ________ $209 per family per year.A. at the cost ofB. at sacrifice ofC. in demand ofD. in support of33. I would just as soon ________ rudely to her.A. you not speakB. you won't speakC. you not speakingD. you didn't speak34. ________ so many people been out of work as today.A. More than ever beforeB. Never before haveC. In the past, there never haveD. Formerly, there never were35. We got home late last night, ________, early this morning.A. namelyB. in other wordsC. or ratherD. in any case36. The miners escaped the danger by ________.A. a hair breadthB. hair breadthC. hair's breadthD. a hair's breadth37. Praise is like warm sunshine. It not only brings joy to ________ who are praised but also adds pleasure to ________ who praise.A. these; theseB. we; usC. them; themD. those; those38. When our imports ________ our exports, we say we have a favorable trade balance; however, when our imports and exports are the other way round we say we have an unfavorable trade balance.A. fall shortB. fall short ofC. run out ofD. are lack of39. ________ that distinguish human beings from other primates are related to the ability of people to stand upright and walk on two legs.A. Many of the physical characteristicsB. Of the many physical characteristicsC. The physical characteristicsD. There are many physical characteristics40. Classical logic is characterized by a concern for the structure and elements of argument ________ that thought, language, and reality are interrelated.A. based on the beliefB. on the belief basedC. belief based on theD. the based belief on41. ________ fashioned from a wick floating in a bowl of oil functioned according to the principle of capillary action.A. All lamps earlyB. Lamps all earlyC. All early lampsD. Early all lamps42. In 1964 ________ of Henry Osaka Tanner's paintings was shown at the Smithsonian Institution.A. was a major collectionB. that a major collectionC. a collection was majorD. a major collection43. Traditionally, ________ on Thanksgiving Day in New England.A. when served is sweet ciderB. when sweet cider is servedC. is served sweet ciderD. sweet cider is served44. Richard: This class is so boring. When will it end?Drew: I don't know. I don't think I can take it much longer.Teacher: ________Richard: Oh, nothing. We were discussing the text.A. Hey you two! What are you talking about?B. Do one of you know what the answer to this problem is?C. Am I boring you two?D. The two of you go to the principal's office right now!45. Delivery man: I have a package here for Beverly Beaver.Beverly: That is me. ________Delivery man: Yes, please sign here on the line.Beverly: Thank you. These must be the new drapes I ordered.A. Where is the receipt?B. Do you have the sender's address?C. Do you take a credit card?D. Do I have to sign for it?Part III Reading Comprehension (20 minutes, 40 marks)Section A (5 marks)Directions: There is one passage in this section with 5 questions. For each question, there are four choices marked A, B, C and D. You should decide on the best choice. Then mark the corresponding letter on theAnswer Sheet with a single line through the centre.Questions 46-50 are based on the following passage.No matter what our work is, we all have to face that choice - in my field no less than in others. There may not be a more important field for the dissemination of values in our country than the entire communications industry - most strikingly, television. Networks are very sensitive to that fact, and they employ dozens of censors to prevent all of us from using language on television that an eight year old might have to explain to his parents. But the point that censors miss, I think, is that it is not so much what we say that teaches as what we don't say. Even programs that attempt to make a moral point don't always make the point that they intended to. Because when we sense we are being sole something, we automatically defend ourselves against it. I think it may be the unspoken assumptions that mold an audience.Look at the way, for instance, that violence is treated on television. It is not only the quantity that offends. There probably is no more violence on television than there is in a Shakespearean tragedy. But on television you find unfelt violence and in Shakespeare you tend to find felt violence. In Shakespeare the characters react with a human response: They fear, they hurt, they mourn. Most of the time on television, violence is dealt with by sweeping it under the rug as fast as possible and by having people go on about their business as if nothing had happened. (If I can't have less violence, I want at least a better grade of violence.) One of the unspoken assumptions is that violence can be tolerated as long as you ignore it and have no reaction to it. But that seems to me to be dangerously close to psychopathic behavior. I wonder if there is any connection between the longacceptance by our people of the Vietnam War and the thousands and thousands of deaths that we have seen on television over the years that were never mourned, never even paused for except to sell shampoo for sixty seconds.Maybe our greatest problem is that we have two separate sets of value systems that we use - the one we talk about and the one we live by. We seem to place a very high value on fairness and on human concerns. And yet we still have widespread discrimination based on race, sex and religion. You still don't find Jews, blacks or other minorities in any significant numbers in decision-making positions in the banking industry, for instance. You think that's an accident? I think somebody puts a value on that. And you don't find women in any significant numbers in decision-making capacities in any industry. Why? Because we place a higher value on appeasing the fragile male ego than we do on fairness and decency. Maybe what we need is a declaration of interdependence.Questions:46. It can be concluded from the first paragraph that ________.A. we won't buy things that are recommendedB. we become defensive whenever we are taught to do somethingC. television is the most important part of the communication industryD. TV programs can never successfully convey a moral point47. What is the author's attitude toward violence on television?A. He thinks that violence on television should be totally banned.B. Violence on television is more tolerable than the one in Shakespeare's plays.C. Violence can be tolerated if you pretend that it doesn't exist.D. Violence, if unavoidable, should be presented on television in a more appropriate way.48. The word “appeasing” (line 6 in paragraph 3) most probably means ________.A. attackingB. accusingC. soothingD. satisfying49. By citing the examples of the Jews, blacks, and other minorities in the last paragraph, the author aims to show that ________.A. it's an accident that they are out-numbered in the decision-making positions in the banking industryB. they stand for one set of value systemC. what we talk about and what we actually do might be totally differentD. fairness is well respected in our life50. The author's tone in writing the text is ________.A. neutralB. matter-of-factC. criticalD. questioningSection B (15 marks)Directions: There is one passage in this section with 10 questions. Go over the passage quickly and answer the questions on the Answer Sheet.For questions 51 - 55, markY (for YES) if the statement agrees with the information given in the passage;N (for NO) if the statement contradicts the information given in the passage;NG (for NOT GIVEN) if the information is not given in the passage.For questions 56 - 60, complete the sentences with the information given in the passage.You hear the same complaint all the time as people get older: “My memory is terrible.” Is it all in the m ind, or do real changes take place in the brain with age to justify such grumbling (抱怨)? The depressing answer is that the brain's cells, the neurons, die and decline in efficiency with age.Professor Arthur Shimamura, of the University of California at Berkeley, says there are three main ways in which mental function changes. The first is mental speed, for example how quickly you can react tofast-moving incidents on the road. Drivers in their late teens react quickly but tend to drive too fast, while the over sixties are more cautious but react more slowly. The near-inevitable slowing with age also partly explains why soccer players are seen as old in their thirties, while golf professionals are still in their prime at that age. This type of mental slowing results from a reduction in the efficiency with which the brain's neurons work.The fact that adults find it harder to learn musical instruments than children points to a second type of mental loss with age - a reduction in learning capacity. The parts of the brain known as the temporal lobes control new learning, and are particularly vulnerable to the effects of aging. This means that, as we get older, we take longer to learn a new language, and slower to master new routines and technologies at work, and we have to rely more on diaries and other mental aids.“Working memory” is the third brain system which is vulnerable to the effects of aging. Working memory is the brain's “blackboard”, where we juggle from moment to moment the things we have to keep in mind when solving problems, planning tasks and generally organizing our day-to-day life. Absent-mindedness occurs at all ages because of imperfections in the working memory system - so, for instance, you may continually lose your glasses, or find yourself walking into a room of your house only to find that you cannot remember what you came for.Such absent-mindedness tends to creep up on us as we age and occurs because our plans and intentions, which are chalked up on the mental blackboard, are easily wiped out by stray thoughts and other distractions. Stress and preoccupation can also cause such absent-mindedness, in addition to age-related changes in the brain. The frontal lobes of the brain - located behind the forehead and above the eyes - are where the working memory system is located. Like the temporal lobes, which handle new learning, the frontal lobes are more vulnerable to the aging process than other parts of the brain.The news, however, is not all bleak. Although neurons reduce in number with age, the remaining neurons send out new and longer connection fibers (dendrites) to maintain connections and allow us to function reasonably well with only relatively small drops in ability.This and other evidence sugge sts that the principle “use it or lost it” might apply to the aging brain. Professor Shimamura studied a group of university professors who were still intellectually active, and compared their performance on neuropsychological tests with that of others of their age group, as well as with younger people. He found that on several tests of memory, the mentally active professors in their sixties and early seventies were superior to their contemporaries, and as good as the younger people.Research on animals provide even stronger evidence of the effects of stimulation on the brain structure. Professor Bryan Kolb, of the University of Lethbridge in Canada, has shown that animals kept in stimulating environments show sprouting (生长) and lengthening of the connecting nerve fibers in comparison with animals kept in unstimulating environments.The beneficial effects of continued mental activity are shown by the fact that older contestants in quiz shows are just as fast and accurate in responding to general knowledge questions as younger competitors, suggesting that at least part of their intellectual apparatus is spared the effects of aging because of practice and skill.Such findings lead to the intriguing possibility of “mental fitness training” to acco mpany jogging and workouts for the health conscious. Research in Stockholm by Professor Lars Backman and his colleague has shown that older people can be trained to use their memory better, with the effects of this training lasting several years.Just as people go bald or grey at different rates, so the same is true for their mental faculties. Why this should be the case for memory and other mental functions is not yet clear. If Professor Shimamura is right, then the degree to which people use and stretch their mental faculties may also have a role to play.Questions:51. The passage gives a description of several methods of testing mental ability.52. Professor Shimamura identified a number of areas in which mental function may change.53. As the temporal lobes of the brain are affected by aging, it becomes harder to pick up new skills.54. Research indicates that physical training can help to improve memory.55. Taking part in quizzes is the best way to stimulate the brain.56. Absent-mindedness is not necessarily ________.57. Scientists do not understand why people's ________ decline at different rates.58. If a person is under stress, ________ may become more frequent.59. When the frontal lobes of the brain are affected by aging, there is a gradual deterioration in ________.60. If an animal's brain is kept active and stimulated, ______________.Section C (10 marks)Directions: In this section, there is one passage followed by 5 questions. Read the passage carefully, then answer the questions in a maximum of 10 words. Remember to write the answers on the Answer Sheet. Questions 61 - 65 are based on the following passage.Alarmed by a 20-year decline in student achievement, American schools are considering major upheavals in the career structure of teachers, school boards around the country are planning to abandon traditional salary schedules and single out outstanding teachers for massive pay rise.The idea is regarded with deep suspicion by the United States' biggest teachers' unions, the National Education Association and the American Federation of Teachers. They say the creation of a cadre of eliteteachers will sour professional relationships and encourage teachers to compete instead of cooperate; they also question whether a fair way can be devised to tell which teachers really do perform better than their colleagues.But heightened public anxiety about secondary education appears to have given the master teacher concept unstoppable political momentum. Florida and Tennessee are racing to introduce ambitious statewide master teacher schemes before the end of the year. Less grandiose proposals to pay teachers on the basis of merit instead of seniority have already been implemented in countless school districts. And the Secretary of Education, Mr. Terrel Bell, recently promised substantial incentive grants to states which intend to follow their example.Low pay is believed to be the single most important reason for the flight from teaching. The average salary of a teacher in the United States is just under $19,000, much less than that of an engineer ($34,700) and not much more than that of a secretary($16,500). To make ends meet it is common for teachers to take second jobs in the evening and in their summer holidays, and women, who used to make up the bulk of teacher candidates are turning to better paid professions.The unions insist that the answer to this problem is to increase the basic pay of all teachers, but most states would find that too expensive, they would be better able to afford schemes that confine pay increases to a small number of exceptional teachers. Champions of the idea say it would at least hold out the promise of high pay and status to bright graduates who are confident of their ability to do well in the classroom, but are deterred by the present meager opportunities for promotion.One of the first large-scale tests of this approach will come in Tennessee, where a year of painstaking negotiations has just overcome bitter union opposition to a wide-ranging master teacher scheme. Tennessee promised that they will allocate $300 million as education budget. In return for a chance to earn bigger salaries and faster promotion, teachers will subject themselves to closer scrutiny.The Tennessee plan will make it harder for poor teachers to join the profession. Beginners will have to serve a probationary year before qualifying, and another three apprentice years before receiving tenure. Apprentice teachers who fail to reach a required standard will not be allowed to stay on. Survivors will be designated “career teachers” and given a chance to climb through three career rugs and earn bonuses of up to $7,000. Advancement will not be automatic. The performance of each teacher will be closely assessed by committees of teachers drawn from other districts.Questions:61. What support is the federal government offering to states that set up a master teacher scheme?62. What's the purpose of the master teacher scheme?63. In the state of Tennessee, how will teachers be assessed?64. What is the main idea of the text?65. What can be inferred from the text about the master teacher scheme?Section D (10 marks)Directions: In this section, there is one passage followed by a summary. Read the passage carefully and complete the summary below by choosing no more than three words from the passage. Remember to write the answers on the Answer Sheet.Questions 66 - 70 are based on the following passage.The United KingdomBritain (or Great Britain) is an island that lies off the north-west coast of Europe. The nearest country is France, which is 20 miles away and from which Britain is separated by the English Channel. The island is surrounded by the Atlantic Ocean to the west, and the North Sea to the east. It comprises the mainlands of England, Wales and Scotland, that is, three countries. Scotland is in the north, while Wales is in the west. Ireland, which is also an island, lies off the west coast of Britain. It consists of Northern Ireland and the Irish Republic. Britain together with Northern Ireland constitute the United Kingdom (UK). Thus, the United Kingdom is composed of four countries, the largest of which is England. The capital city is London, which is situated in south-east England.The UK has a total area of about 244,100 square kilometres (94,248 square miles). About 70% of the land area is devoted to agriculture, about 7% is wasteland, moorland and mountains, about 13% is devoted to urbandevelopment, and 10% is forest and woodland. The northern and western regions of Britain, that is Scotland and Vales, are mainly mountainous and hilly. Parts of the north-west and centre of England also consist of mountains and hills.Britain has a generally mild and temperate climate. It is, however, subject to frequent changes. It has an average annual rainfall of about 120 centimetres (47 inches).In 1998 the population of the United Kingdom was nearly 59 million. The density of population was approximately 240 people per square kilometre. However, in England, where 83% of the population live, the density was much higher, about 363 per square kilometre.In the UK, English is the first language of the vast majority of people. However, in western Wales, Welsh is the first language for many of the people. In Scotland only a small number of people speak Gaelic.In Britain about 66% of the population say that they are Christian, while fewer than 5% say that they belong to other religions.Summary:Britain is an island that (66) ________ the Atlantic Ocean and the North Sea. It comprises the mainlands of England, Wales and Scotland. Ireland (67) ________ the west coast of Britain. It (68) ________ Northern Ireland and the Irish Republic. The United Kingdom consists of Britain together with Northern Ireland. The capital city is London which (69) ________ south-east England.In 1998 the population of the UK was nearly 59 million. The density of population was 240 people per square kilometre. In the UK English is the first language of most people. In western Wales, Welsh (70) ________ many of the people, but few people in Scotland speak Gaelic.Part IV Cloze (15 minutes, 15 marks)Section A (5 marks)Directions: There are 5 blanks in the passage. For each blank, some letters of the word have been given (not exceeding 3 letters). Read the passage below and think of the word which best fits each blank. Use only one word in each blank. Remember to write the answers on the Answer Sheet.Language is learned primarily through communication with other people. Research shows that the more communication children (71) exp________ at home, the better developed their language skills will be. Children need to talk well in order to learn well. This is especially important for the development of native language skills since the language is often reinforced by the child's environment outside school.However, the (72) qu________ of communication is just as or more important than quantity alone. The language adults use helps children become aware of the many different aspects of objects and events around them. For example, during a shopping trip to the market or store, adults can develop children's concepts by bringing their attention to the shapes, colors, sounds, textures, and sizes of objects and events around them.Parents can do this naturally through conversation without (73) di________ teaching. In other words, conversation with children in everyday situations expands their minds and develops their thinking skills.In addition to conversing with children, adults can help prepare their children to succeed in school by (74) en________ them to take an active interest in books and in the print that surrounds them in the environment. The child's first major task in school is learning to read. Children who come to school with knowledge that the print aroundthem carries important meaning and with an interest in books and stories will usually (75)su________ in learning to read rapidly.Section B (10 marks)Directions: There are 10 blanks in the passage. Use the word given in each blanket to form a word that fits in each blank. Remember to write the answers on the Answer Sheet.Forensic science (法庭科学) means (76) ________ (use) science to solve crime. But what exactly does a forensic scientist do?At the scene of crime, the crime scene (77) ________ (investigator) will thoroughly examine the area. They will look for fingerprints and palm prints; marks of tools and weapons; marks from shoes; fibers from (78)______ (cloth); body fluids.。
2008全国大学生数学建模竞赛 题目汇总
2008高教社杯全国大学生数学建模竞赛题目 （请先阅读“全国大学生数学建模竞赛论文格式规范”）A题数码相机定位数码相机定位在交通监管（电子警察）等方面有广泛的应用。
所谓数码相机定位是指用数码相机摄制物体的相片确定物体表面某些特征点的位置。
最常用的定位方法是双目定位，即用两部相机来定位。
对物体上一个特征点，用两部固定于不同位置的相机摄得物体的像，分别获得该点在两部相机像平面上的坐标。
只要知道两部相机精确的相对位置，就可用几何的方法得到该特征点在固定一部相机的坐标系中的坐标，即确定了特征点的位置。
于是对双目定位，精确地确定两部相机的相对位置就是关键，这一过程称为系统标定。
标定的一种做法是：在一块平板上画若干个点，同时用这两部相机照相，分别得到这些点在它们像平面上的像点，利用这两组像点的几何关系就可以得到这两部相机的相对位置。
然而，无论在物平面或像平面上我们都无法直接得到没有几何尺寸的“点”。
实际的做法是在物平面上画若干个圆（称为靶标），它们的圆心就是几何的点了。
而它们的像一般会变形，如图1所示，所以必须从靶标上的这些圆的像中把圆心的像精确地找到，标定就可实现。
图 1 靶标上圆的像有人设计靶标如下，取1个边长为100mm的正方形，分别以四个顶点（对应为A、C、D、E）为圆心，12mm为半径作圆。
以AC边上距离A点30mm处的B为圆心，12mm为半径作圆，如图2所示。
图 2 靶标示意图用一位置固定的数码相机摄得其像，如图3所示。
图3 靶标的像请你们：（1） 建立数学模型和算法以确定靶标上圆的圆心在该相机像平面的像坐标, 这里坐标系原点取在该相机的光学中心，x-y平面平行于像平面；（2） 对由图2、图3分别给出的靶标及其像，计算靶标上圆的圆心在像平面上的像坐标, 该相机的像距（即光学中心到像平面的距离）是1577个像素单位(1毫米约为3.78个像素单位)，相机分辨率为1024×768；（3） 设计一种方法检验你们的模型，并对方法的精度和稳定性进行讨论；（4） 建立用此靶标给出两部固定相机相对位置的数学模型和方法。
2008年全国高中数学联合竞赛加试(A卷)
2008年全国高中数学联合竞赛加试（A 卷）说明：1．评阅试卷时，请严格按照本评分标准的评分档次给分；2．如果考生的解答方法和本解答不同，只要思路合理、步骤正确，在评卷时可参考本评分标准适当划分档次评分，10分为一个档次，不要增加其他中间档次．一、（本题满分50分）如题一图，给定凸四边形ABCD ，180B D ∠+∠< ，P 是平面上的动点，令()f P PA BC PD CA PC AB =⋅+⋅+⋅．（Ⅰ）求证：当()f P 达到最小值时，P A B C ,,,四点共圆；（Ⅱ）设E 是ABC ∆外接圆O 的 AB 上一点，满足：AE AB =，1BC EC =， 12ECB ECA ∠=∠，又,DA DC 是O 的切线，AC ()f P 的最小值．二、（本题满分50分）设()f x 是周期函数，T 和1是()f x 的周期且01T <<．证明：（Ⅰ）若T 为有理数，则存在素数p ，使1p是()f x 的周期； （Ⅱ）若T 为无理数，则存在各项均为无理数的数列{}n a 满足110n n a a +>>>(1,2,)n =⋅⋅⋅，且每个(1,2,)n a n =⋅⋅⋅都是()f x 的周期．三、（本题满分50分）设0k a >，1,2,,2008k = ．证明：当且仅当200811k k a =>∑时，存在数列{}n x 满足以下条件：（ⅰ）010n n x x x +=<<，1,2,3,n = ； （ⅱ）lim n n x →∞存在；（ⅲ）20082007111n n k n k k n k k k x x a x a x -+++==-=-∑∑，1,2,3,n = ．参考答案一、[解法一] （Ⅰ）如答一图1，由托勒密不等式，对平面上的任意点P ，有PA BC PC AB PB AC ⋅+⋅≥⋅． 因此 ()f P PA BC PC AB PD CA =⋅+⋅+⋅ PB CA PD CA ≥⋅+⋅()PB PD CA =+⋅．因为上面不等式当且仅当,,,P A B C 顺次共圆时取等号，因此当且仅当P 在ABC ∆的外接圆且在 AC 上时，()()f P PB PD CA =+⋅． …10分又因PB PD BD +≥，此不等式当且仅当,,B P D 共线且P 在BD 上时取等号．因此当且仅当P 为ABC ∆的外接圆与BD 的 交点时，()f P 取最小值min ()f P AC BD =⋅．故当()f P 达最小值时，,,,P A B C 四点共圆． …20分（Ⅱ）记E C B α∠=，则2E C Aα∠=，由正弦定理有sin 2sin 3AE AB αα==，从而s i n 32s i n 2αα=34sin )4sin cos αααα-=，所以2cos )4cos 0αα--=，整理得24cos 0αα-=， …30分解得cos αcos α=， 故30α= ，60ACE ∠= ．由已知1BCEC ==()0sin 30sin EAC EAC ∠-∠,有sin(30)1)sin EAC EAC ∠-=∠，即1cos 1)sin 2EAC EAC EAC∠-∠=∠，整理得1cos 2EAC EAC ∠=∠，故t a n 23EAC ∠==，可得75EAC ∠= ， …40分从而45E ∠= ，45DAC DCA E ∠=∠=∠= ，ADC ∆为等腰直角三角形．因AC =，则1CD =．又ABC ∆也是等腰直角三角形，故BC =212215BD =+-⋅=，BD ．故min ()f P BD AC =⋅ …50分 [解法二] （Ⅰ）如答一图2，连接BD 交ABC ∆的外接圆O 于0P 点（因为D 在O 外，故0P 在BD 上）． 过,,A C D 分别作000,,P A PC P D 的垂线，两两相交得111A B C ∆，易知0P 在ACD ∆内，从而在111A B C ∆内，记ABC ∆之三内角分别为x y z ，，，则0180APC y z x ∠=︒-=+，又因110B C P A ⊥，110B A PC ⊥，得1B y ∠=，同理有1A x ∠=，1C z ∠=， 所以111A B C ∆∽ABC ∆． …10分设11B C BC λ=，11C A CA λ=，11A B AB λ=，则对平面上任意点M ，有答一图10000()()f P P A BC P D CA PC AB λλ=⋅+⋅+⋅ 011011011P A B C P D C A PC A B =⋅+⋅+⋅ 1112A B C S ∆=111111MA BC MD C A MC A B ≤⋅+⋅+⋅ ()MA BC MD CA MC AB λ=⋅+⋅+⋅ ()f M λ=， 从而 0()()f P f M ≤． 由M 点的任意性，知0P 点是使()f P 达最小值的点． 由点0P 在O 上，故0,,,P A B C 四点共圆． …20分 （Ⅱ）由（Ⅰ），()f P 的最小值11102()A B C f P S λ∆=2ABC S λ∆=，记ECB α∠=，则2ECA α∠=，由正弦定理有sin 2sin 3AE AB αα==2sin 2αα=34sin )4sin cos αααα-=，所以2cos )4cos 0αα--=，整理得24cos 0αα-=， …30分解得cosαcos α=，故30α= ，60ACE ∠= ．由已知1BCEC ==()0sin 30sin EAC EAC ∠-∠,有sin(30)1)sin EAC EAC ∠-=∠ ，即1cos 1)sin 2EAC EAC EAC ∠-∠=∠，整理得1cos2EAC EAC ∠=∠，故tan 2EAC ∠== 75EAC ∠=， …40分所以45E ∠=︒，ABC ∆为等腰直角三角形，AC 1ABC S ∆=，因为145AB C ∠=︒，1B 点在O 上，190AB B ∠=︒，所以11B BDC 为矩形，11B C BD ==λ=，所以min ()21f P = …50分[解法三] （Ⅰ）引进复平面，仍用,,A B C 等代表,,A B C 所对应的复数． 由三角形不等式，对于复数12,z z ，有 1212z z z z +≥+，当且仅当1z 与2z （复向量）同向时取等号．有 PA BC PC AB PA BC PC AB ⋅+⋅≥⋅+⋅ ，所以 ()()()()A P C B C P B A --+--()()()()A P C B C P B A ≥--+-- （1）P C A B C B P A =-⋅-⋅+⋅+⋅()()B P C A PB AC =--=⋅，从而 PA BC PC AB PD CA ⋅+⋅+⋅PB AC PD AC ≥⋅+⋅()PB PD AC =+⋅ BD AC ≥⋅． （2） …10分 （1）式取等号的条件是复数 ()()A P C B --与()()C P B A --同向，故存在实数0λ>，使得()()()()A P C B C P B A λ--=--，A PB AC P C B λ--=--， 所以 arg()arg()A P B AC P C B--=--，向量PC 旋转到PA 所成的角等于BC旋转到AB 所成的角， 从而,,,P A B C 四点共圆．（2）式取等号的条件显然为,,B P D 共线且P 在BD 上．故当()f P 达最小值时P 点在ABC ∆之外接圆上，,,,P A B C 四点共圆． …20分 （Ⅱ）由（Ⅰ）知min ()f P BD AC =⋅．以下同解法一．二、[证] （Ⅰ）若T 是有理数，则存在正整数,m n 使得nT m=且(,)1m n =，从而存 在整数,a b ，使得1ma nb +=．于是11ma nb a bT a b T m m+==+=⋅+⋅ 是()f x 的周期． …10分 又因01T <<，从而2m ≥．设p 是m 的素因子，则m pm '=，m *'∈N ，从而11m p m'=⋅是()f x 的周期． …20分 （Ⅱ）若T 是无理数，令111a T T ⎡⎤=-⎢⎥⎣⎦，则101a <<，且1a 是无理数，令21111a a a ⎡⎤=-⎢⎥⎣⎦，…… 111n n n a a a +⎡⎤=-⎢⎥⎣⎦，……． …30分由数学归纳法易知n a 均为无理数且01n a <<．又111n n a a ⎡⎤-<⎢⎥⎣⎦，故11n n n a a a ⎡⎤<+⎢⎥⎣⎦，即111n n n n a a a a +⎡⎤=-<⎢⎥⎣⎦．因此{}n a 是递减数列． …40分最后证：每个n a 是()f x 的周期．事实上，因1和T 是()f x 的周期，故111a T T ⎡⎤=-⎢⎥⎣⎦亦 是()f x 的周期．假设k a 是()f x 的周期，则111k k k a a a +⎡⎤=-⎢⎥⎣⎦也是()f x 的周期．由数学归纳法，已证得n a 均是()f x 的周期． …50分三、[证] 必要性：假设存在{}n x 满足（ⅰ），（ⅱ），（iii ）．注意到（ⅲ）中式子可化为2008111()n n k n k n k k x x a x x -++-=-=-∑，n ∈*N ，其中00x =．将上式从第1项加到第n 项，并注意到00x =得111222200820082008()()()n n n n x a x x a x x a x x +++=-+-++- ． …10分由（ⅱ）可设lim n n b x →∞=，将上式取极限得112220082008()()()b a b x a b x a b x =-+-++- 20081122200820081()k k b a a x a x a x ==⋅-+++∑20081k k b a =<⋅∑，因此200811k k a =>∑． …20分充分性：假设200811k k a =>∑．定义多项式函数如下：20081()1k k k f s a s ==-+∑，[0,1]s ∈，则()f s 在[0,1]上是递增函数，且(0)10f =-<，20081(1)10k k f a ==-+>∑．因此方程()0f s =在[0,1]内有唯一的根0s s =，且001s <<，即0()0f s =． …30分下取数列{}n x 为01nkn k x s ==∑，1,2,n = ，则明显地{}n x 满足题设条件（ⅰ），且1000101n nk n k s s x s s +=-==-∑．因001s <<，故10lim 0n n s +→∞=，因此100000lim lim 11n n n n s s s x s s +→∞→∞-==--，即{}n x 的极限存在，满 足（ⅱ）． …40分 最后验证{}n x 满足（ⅲ），因0()0f s =，即2008011k k k a s ==∑，从而200820082008100001111()()n k n n k n n k k k n k n k k k k x x s a s s a s a x x +-++-===-====-∑∑∑．综上，存在数列{}n x 满足（ⅰ），（ⅱ），（ⅲ）． …50分。
2008年全国大学生英语竞赛真题及答案
2008年全国大学生英语竞赛真题及答案Part I Listening Comprehension (25 minutes, 30 marks)Section A (7 marks)1-7 BBAABACSection B (8 marks)8-15 BBCCAABCSection C (5 marks)16-20 CCBBASection D (10 marks)21. share 22. like 23. resource 24. possible 25. basic concepts 26. analyse 27. require your attention 28. urgent needs 29. Lastly 30. availablePart II Vocabulary and Structures (10 minutes, 15 marks)31-35 BADDC 36-40 DCCCA 41-45 ADDBBPart III Reading Comprehension (20 minutes, 40 marks)Section A (5 marks)46-50 DACDDSection B (5 marks)51-55 FTFFTSection C (16 marks)56. Saving interesting old buildings. / Conservation of old towns / cities / San Antonio.57. Buildings.58. In the early 1700s.59. Many missions or churches built by the Spanish.2008 National English Contest for College Students(Level D - Preliminary)参考答案及作文评分标准- 13 -60. Preserved.61. By giving tea parties and selling cakes.62. They wanted to preserve the beauty of San Antonio.63. They want people to know about San Antonio蒺s history.Section D (14 marks)64. for treating 65. music therapy / music 66. illnesses 67. complex sounds 68. stimulate 69. fun 70. classical musicPart IV Cloze (15 minutes, 15 marks)Section A (5 marks)71. belief 72. Apparently 73. unwilling 74. making 75. thirdSection B (10 marks)76. meant 77. building 78. collections 79. whole 80. developed 81. existence 82. animals 83. glimpse 84. for 85. singlePart V Translation (15 minutes, 15 marks)Section A (8 marks)86. 我们花了多少时间来思考我们生活中的许多“要是……该多好啊”？87. 要是我没有稍微晚了一步而是第一个想到那个主意该有多好。
2008全国赛获奖名单
附件：
2008年全国职业院校技能大赛获奖名单
一、中职组计算机应用技术
1．动画片制作
—5 —
2．园区网互联及网站建设
—6 —
—7 —
—8 —
3．企业网络搭建及应用
一等奖
—9 —
—10 —
—11 —
—12 —
二、中职组电工电子
1．电子产品装配与调试
—13 —
2．机电一体化设备组装与调试
三、中职组烹饪1．中餐热菜
2．中餐面点
3．中餐冷拼
4．果蔬雕
四、中职组数控技术1．数控车
2．数控铣
五、中职组汽车运用与维修1．汽车维修基本技能
2．汽车二级维护
六、中职组服装设计制作与模特表演
1．女式夏季裙套装现场电脑样板制作与缝制
2．中职学生模特表演
七、中职组美容美发1．新娘化妆
2．有缝推剪吹风造型。
2008全国数学建模竞赛(A题)
基于非线性规划的数码相机定位摘要本文从数码相机的成像原理出发，综合图像处理和空间坐标变换，运用了非线性规化等方法，避免矛盾方程组，解决了所提出的问题。
对于问题一，运用Matlab 对图像处理，采用阈值两极化方法提取其边界并对图像进行相应的分析，根据线性畸变下切线的性质，用切线法定出了相应的特征点，在此过程中引入扰动因子，对切线法进行优化，建立了非线性规划模型，并基于LINGO 进行参数估计来解决物和像的对应几何关系，减小了误差。
对于问题二，利用问题一的优化模型进行求解，得到特征点在相机像平面坐标系中模型的精确度进行讨论，发现图中特征点与其对应的最长轴距离为0.735767毫米，像中C A ''、C B ''的连线夹角为rad 0092.0，说明计算结果有良好的精确度；通过调整扰动因子α，得到6组不同的特征点的坐标，以及特征点与其对应长轴之间的距离，并进行方差分析，方差值波动性不强，说明计算结果有比较好的稳定性。
对于问题四，利用坐标旋转、矩阵变换和平移向量，将问题转化成对平移向量的求解，沿用问题一中的非线性规划思想，对问题进行求解。
关键词：边界检测 矛盾方程 非线性规划 参数估计 旋转矩阵一、问题重述数码相机定位在交通监管（电子警察）、航天科技、机器人控制等方面有广泛的应用。
数码相机定位最常用的定位方法是双目定位，即用两部相机来定位。
于是对双目定位，精确地确定两部相机的相对位置就是关键，这一过程称为系统标定。
标定的一种做法是：在一块平板上画若干个点， 同时用这两部相机照相，分别得到这些点在它们像平面上的像点，利用这两组像点的几何关系就可以得到这两部相机的相对位置。
然而，无论在物平面或像平面上我们都无法直接得到没有几何尺寸的“点”。
实际的做法是在物平面上画若干个圆（称为靶标），它们的圆心就是几何的点。
而它们的像一般会变形，所以必须从靶标上的这些圆的像中把圆心的像精确地找到，标定就可实现。
2008年全国生物联赛试卷答案与解析
2008年全国生物联赛试卷答案与解析1.根细胞有丝分裂发生在分生区，随后进行分化，且植物的根细胞几乎不存在无丝分裂2.内皮层不仅存在于根中，如松针3.了解各种植物叶形态的区别，水稻叶卷曲主要因为上表皮泡状细胞的作用4.被子植物生殖细胞产生的基本过程中，只有大小孢子母细胞—大小孢子过程为减数分裂，其余过程均为有丝分裂5.充分水饱和的细胞水势为0，放入任意浓度的溶液中，溶液的水势均小于0，所以水由水势高处向水势低处流动6.植物体吸水的两种主要方式为蒸腾拉力和根压，落叶树的叶片在早春尚未展开，其蒸腾作用很弱，所以主要的溪水方式为根压。
7.涝灾使根长期处于无氧环境中，发酵作用产生的大量酒精对植物产生生理胁迫8.光合作用中PGAld的终产物淀粉在叶绿体中合成，而蔗糖则在细胞质中合成9.花为植物的生殖器官，在花旗中花为代谢库，所以其呼吸作用在花期中侧重于生殖器官10.植物的年龄影响植物的发育过程，许多多年生的植物在未达到成熟年龄时不会由侧重营养生长转向侧重生殖生长11.影响植物光周期的色素是光敏色素，其中红光能光敏色素pr型向Pfr型转变，其作用对中断暗期最为有效，而远红光使Pfr转为Pr，作用相反，其余两种色光对光周期作用不大。
12.完全为记忆性内容，可理解为有机物只能通过韧皮部运输13.离层的产生原因为植物细胞间的分离，细胞壁胞间层的主要成分为果胶和纤维素。
14.冷、冻害来临时，植物体内的谷aa，可溶性糖，脱落酸等物质增加，提高渗透压，以抵抗低温脱水对细胞造成的伤害15.蔷薇科的四亚科间区别及其特征应为植物分类学重点记忆内容，此部分内容可参考有关教材16.二原型，侧根发生于木质部与韧皮部之间，三、四原型，侧根发生对着初生木质部，多元型侧根发生对着初生韧皮部17.豆科三亚科也为重点记忆内容，上升覆瓦状排列对应假蝶形花冠，下降覆瓦状排列对应碟形花冠，按照花图式记忆18.四个选项中的蕨类植物，分别属于石松亚门，水韭亚门，真蕨亚门，木贼亚门，其中只有木贼亚门具有节和节间的分化，应熟悉蕨类各亚门间特征19.此题有争议，内鼻孔在肺鱼中已经出现，答案应不唯一，侧线在两栖类中依然存在，而高等爬行类出现槽生齿20.门静脉将血液输送到肝脏中进行解毒，正确。
2008年全国中学生语文能力竞赛(决赛)试题参考答案 高一年级组
一、听力（一）1.× 2.√ 3. √ 4.× 5. √ 6. √7.×8.×9.因为“我”怕录音会留下证据（答此点给1分），怕西多罗夫找“我”算账（答此点给1分）。
10.“我”是一个胆小怕事、没有义气，不值得信赖、交心的人。
11.是西多罗夫打来的（答此点给1分），因为今天是四月一日——愚人节，他想和“我”开个玩笑。
（答此点给1分）（二）1.因为他不敢相信自己的眼睛，能这么巧的遇见两位大明星。
2.因为他看见了两位明星，有些激动（答此点给1分）；想索要签名，又没有签名本。
（答此点给1分）3.为人谦和，平易近人，处事低调，幽默风趣。
（每对两点，给1分）二、益智1. C2. 移步视钦3. 武松4.能屈能伸三、积累与应用1. 雄关漫道真如铁会当凌绝顶（有错字不给分）2. 这幅漫画的内容是：一棵大树被拦腰折断，压坏了一所房子，而房子的主人却巧借大树的断枝悠然地荡起了秋千。
（答对内容给1分）寓意为：面对灾难，要保持乐观的心态。
（答对寓意给2分）3. 发奋识遍天下字，立志读尽人间书。
4. 1聪明，能干，美丽，厉害。
（2分）②能说会道，虚伪阴狠，善使手段。
（2分）③善良温顺，与世无争，独善其身。
（2分）5.逻辑是回答对不对的问题，也就是看是否符合客观事实（答案合理给1分）；语法是回答通不通的问题，也就是看是否符合语法规则（答案合理给1分）；修辞是回答好不好的问题，也就是看是否能运用一些表现方法，使语言表达得准确、鲜明而生动。
（答案合理给1分）6.示例：散发着激情和热力秋，是金黄的诗收获着成熟和希望（有错字不给分）7.第一处：将“经过”调至“工作人员”之前，或者将后面一句改为“为赴台大熊猫选定了响亮的名字”。
（语病为谓语残缺）（答案正确给1分）第二处：“内涵”与“意蕴”保留一个即可。
（语病为语意重复）（答案正确给1分）第三处：“积聚”换成“凝聚”。
（语病为谓语残缺）（答案正确给1分）8.出版商能出奇制胜，大发其财，主要是利用了总统的回答和人们的心理来大做文章。
2008年全国大学生数学建模竞赛吉林赛区竞赛成绩
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2008高教社杯全国大学生数学建模竞赛承诺书我们仔细阅读了中国大学生数学建模竞赛的竞赛规则.我们完全明白，在竞赛开始后参赛队员不能以任何方式（包括电话、电子邮件、网上咨询等）与队外的任何人（包括指导教师）研究、讨论与赛题有关的问题。
我们知道，抄袭别人的成果是违反竞赛规则的, 如果引用别人的成果或其他公开的资料（包括网上查到的资料），必须按照规定的参考文献的表述方式在正文引用处和参考文献中明确列出。
我们郑重承诺，严格遵守竞赛规则，以保证竞赛的公正、公平性。
如有违反竞赛规则的行为，我们将受到严肃处理。
我们参赛选择的题号是（从A/B/C/D中选择一项填写）： A我们的参赛报名号为（如果赛区设置报名号的话）：1800所属学校（请填写完整的全名）：华南师范大学参赛队员(打印并签名) ：1. 蔡明辉2. 滕菲3. 林轩腾指导教师或指导教师组负责人(打印并签名)：杨坦日期： 2008 年 9 月22 日赛区评阅编号（由赛区组委会评阅前进行编号）：数码相机定位摘要数码相机在当今社会上正在得到广泛的应用。
本题采用系统标定过程，提出有效的方法，对两部处于不同位置的相机间的相对位置给予准确描述。
首先我们在忽略畸变作用的前提下，利用小孔成像建立一个简单的模型得到靶标上圆心在像平面上的坐标，并对畸变作用的影响对模型进行修正。
为求得具体像坐标，我们分别使用了标注连接分量法和图像配准法，求得两组结果均比较接近且精确地获得五个圆心在像平面上的坐标。
为分析模型的稳定性和精确度，我们通过与光轴夹角与模型精度的变化曲线进行讨论。
对于两台相机A和B，利用第一部分理论求得世界坐标系到像坐标的映射，构造逆映射，从而把两相机坐标映射到世界坐标系，求得两相机的相对位置。
关键字：系统标定正交旋转矩阵平移矩阵透视投影映射畸变像素密度1.问题的重述双目定位就是利用两台数码相机摄制物体的相片以确定物体表面某些特征点的位置的一种常用的定位方法。
对于物体上一个特征点，用两部固定于不同位置的相机摄得物体的像，可以获得该点在两部相机像平面上的坐标。
只要知道两部相机精确的相对位置，可采用几何的方法得到该特征点在固定一部相机的坐标系中的坐标，即确定了特征点的位置。
题目进行如下标定：在一块平板上画五个圆，即标靶（如图1），同时用两部相机照相，分别得到这几个圆在像平面上的像，利用这两组像的几何关系就可以得到这两部相机的相对位置。
图 1 靶标示意图图2 靶标的像我们的任务是：（1）建立数学模型和算法以确定靶标上圆的圆心在该相机像平面的像坐标；（2）对由图1、图2分别给出的靶标及其像，计算靶标上圆的圆心在像平面上的像坐标；（3）设计一种方法检验我们的模型，并对方法的精度和稳定性进行讨论；（4）建立用此靶标给出两部固定相机相对位置的数学模型和方法。
2.基本假设1.假设相机的光学系统可以由一凸透镜唯一的替代，靶标特征点的光学性质不随时间变化；2.假设相机的光学成像系统仅限于几何光学范畴，不考虑光的波动性；3.假设相机透镜对入射光线具有100%的透射率，被看做一块各向同性的无色散均匀介质，在透镜的边界处不考虑不同介质间的边界效应。
3.符号说明(,,,)x y z o：三维世界坐标系；w w w wx y z o：三维相机坐标系；(,,,)X Y z O：三维像平面基准坐标系；(,,,)'z：光学中心到像平面的距离；ψθφ、、：相机坐标系与世界坐标系对应坐标轴的夹角；X Y：靶标在小孔成像模型下得到的图像坐标；(,)u uX Y：由透镜畸变引起的实际图像的坐标；(,)d dR：像平面基准坐标系下径向半径的平方；dk：凸透镜径向畸变系数；γ：检测模型过程中，发光点与坐标中心的连线和光轴的夹角；4.问题分析与简化4.1问题分析4.1.1问题(1)首先利用简化的小孔成像模型，靶标上的特征点和它们在像平面上的坐标分别被看做光学系统的输入和输出信息。
假定靶标在三维相机坐标系内的位置坐标后，可求解靶标的物像坐标的一般表达式。
在实际的透镜光学系统中，尤其在拍摄四方形的物体时，其周围图像常出现卷翘或膨鼓的现象。
因而在对小孔成像模型的修正中应考虑镜头畸化的影响。
在常出现的畸化类别中，透镜径向畸化对于图像的扭曲作用最大。
在考虑畸化作用的影响下，我们可以得到更为贴近实际像坐标分布的表达式4.1.2问题(2)在图像的标定的众多做法中，有一种是直接在平板上画若干个点，通过两台相机同时照相得以标定。
可是实际操作中无论在物平面或是像平面上我们都无法直接得到没有几何尺寸的“点”。
此时的做法是精确地找出画在平板山的圆的圆心在像平面上的坐标，这也是本题的重要所在。
4.1.3问题(3)对于在问题(1)种建立的小孔成像模型，可以将靶标上的特征点作为光学系统的输入信息，同时在像平面上所体现的像点分布为光学系统的输出信息。
因而，对于模型的检验可看做对于此光学系统输入输出效果的评估。
为使得试验模型被保持为小孔成像模型，同时透镜系统被看成单个薄凸透镜，我们要求输入特征点的位置与光学中心的距离大于透镜半径至少一个量级。
4.1.4问题(4)正如人用双眼可以识别立体的物品，在空间中要使用机器进行智能识别需要两台相机同时照相。
可是在机器内部要对获取的信息进行整合，还需要知道两部相机之间的相对位置。
事实上，每部相机的照相系统都是建立在一个特殊的坐标系下进行讨论的，而最终要整合所有的信息，则需要首先建立一个公共的坐标系，这也是本题我们一开始就建立的世界坐标系。
4.2问题的简化①由于数码相机的镜头由许多形状大小都不相同的镜片组成，同时镜片上镀有用于滤光等用途的光学薄膜，因而，实际的数码相机光学系统极为复杂。
从成像原理角度出发，我们将复杂的光学元件系统简化成一凸透镜，根据模型精确度的要求不同，在优化模型中考虑透镜的纵向畸变给靶标物像带来的影响。
②本模型仅在几何光学范围中考虑像面上所得的靶标像。
不考虑光的波动性对于像面成像图样的影响。
5. 模型的建立5.1问题(1)的模型5.1.1采用小孔成像模型分析将相机的光学元件系统简化为薄透镜，如图一所示，光学中心位置对应于坐标系原点，靶标及其像分列于x y -平面两侧，作为透射的输入与输出信息。
对于各个靶标特征点，由于在题中仅需被确定各圆心的空间位置坐标，靶标上各圆心点来代替各圆的透射过程。
透射光路与小孔成像模型的光路一致，各靶标特征点和它的像在坐标系中可与坐标系原点、z 轴投影点形成相似的两个三角形。
以C 点为例，设靶标特征点C 的空间坐标为(,,)C C C x y z ，其像的空间坐标为'(,,)C C X Y z ，根据相似三角形间对应边的比例关系，我们有：' (1)C C C C C x y zX Y z==图3 小孔成像模式下的透射示意图由于像平面与x y -平面平行，摄像中，作为系统信息给出。
因而靶标圆心的像坐标为：''''// (2)C C C C C C X z x z Y z y z z z ⎧=⎪=⎨⎪=⎩至此，靶坐标与像坐标在小孔成像模型下，构成如上的关系式。
通过相片定位各像点在像平面上的位置，可以进而求得靶标的空间坐标。
5.1.2模型的修正实际相机中的透射在像平面上得到的相片存在径向和切向等类型的畸变，其中影响较大的径向畸变是由于镜头中透镜的表面曲率存在误差引起的。
在考虑畸变因素的情况下，理想图像中的图像点沿着径向，在离中心点位置越远的地方，产生越大的形变量。
为求取畸变状态下相机像平面的像坐标。
相机坐标系如图3所示，坐标原点o 在相机的光心，z 轴与光轴重合。
图像坐标系(z )X Y O 、、、的X Y 、轴分别平行于摄像机坐标系的x y 、轴。
根据投影和透视变换理论和矩阵转换知识。
得到靶标上特征点在世界坐标系下和相机坐标系下的坐标变换关系[1,5]：(3)w w w x x y R y T z z ⎡⎤⎡⎤⎢⎥⎢⎥=+⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦其中，R T 和分别为从世界坐标系到相机坐标系的旋转矩阵变换和平移变换。
它们可分别表示为：123456789cos cos cos sin sin sin cos cos sin cos sin sin sin cos sin sin sin cos cos sin sin cos cos sin sin cos sin cos cos = R r r r r r r r r r ψθψθφψφψθφψφψθψθφψφψθφψφθθφθφ--⎡⎤⎢⎥=++⎢⎥⎢⎥-⎣⎦⎡⎤⎢⎥⎢⎥⎢⎥⎣⎦[ ]x y z T T T T T =在小孔成像模型下，等同于5.1.1中的做法，可得：''/ (4)/ (5)u u X z x z Y z y z ==在此基础上，引入畸变模型，得到的实际图像坐标(,)d d X Y 和小孔成像模型下所得坐标关系为：2121(1) (6)(1) (7)d d u d d uX kR X Y kR Y --=+=+其中222dd d R X Y =+。
经算得 '12322789'45622789 (8)11 (9)11u w w w x d d d w w w z w w w yu d d d w w w zX r x r y r z T z X kR kR r x r y r z T r x r y r z T Y zY kR kR r x r y r z T +++==⨯++++++++==⨯+++++其中，x T 和y T 可根据公式(3)和公式(6)，(7)公式得出。
通过解(8)，(9)两个方程组，可以确定'z 、k 、z T 。
同时，可求出修正后真实的靶标像坐标'(,,)d d X Y z 。
5.2问题(2)的模型依然如图3所示，根据已知，相机的像距是1577个像素单位，同时根据体重所给出的长度像素换算条件，即：1mm 约为3.78个像素单位，可知相机的像距为417.2mm 。
考虑到像平面平行于x y -平面，得知：417.20A B C D E z z z z z mm =====-。
靶标的像为一张1024768⨯的图片，首先我们通过imread 命令将其读入matlab 中作为一个矩阵7681024⨯的矩阵7681024A ⨯输出。
下面用两种不同的方法寻找靶标上A 、B 、C 、D 、E 五个圆的圆心在像平面上的横纵像坐标，并进行比较。
方法一（标注连接分量法）：为了将五个靶标的像标记为五个连通区域，我们先将图4中黑点、白点分别标记为0和1。
其次运用标注连接分量法[2]将五个连通区域分别标记为1、2、3、4、5。
事实上，像平面上的五个像的尺寸并非特别大（直径仅约15mm ），我们可以将其近似看成比较对称的椭圆群。