英文奥数题
趣味智力题及答案英语
趣味智力题及答案英语趣味智力题大全及答案大全英语1、Why is the letter E so important?答案：Because letter E is the beginning of everything解析：为什么字母E特别重要由于字母E是“全部事情”(everything)的开始(字母)2、Why are the letter G and letter S in gloves close to each other?答案：Because there is love between them.解析：在单词gloves中，为什么字母G和字母S那么亲近?由于是g love s .她们中间有一个love3. b4. y5、How can you make a rope shorter without cutting or winding it?答案：Take a longer rope and compare with it解析：不剪，不卷，不折，怎么才能使一根绳子变短呢?拿一根长一点的绳子和它做比较就行了!6、What does everybody do at the same time?答案：grow old解析：全部人在同样的时间都在做的一件事是什么?有的.人会答：呼吸，心跳，但是都不严密只有“变老”是没有任何方法可以阻挡的。
7、Who works only one day in a year but never gets fired?答案：Santa Claus解析：这题关键要理解词组get fired的意思谁一年只工作一天，但是却永久不会被解雇呢?当然是“圣诞老人”。
8、How can you make 6 out of three 7s、sister.答案：7-7/7=6解析：其实这是一道简约的数学题，关键是要把题目看懂怎样才能从三个7中得到6.9、How can you make 1000 out of eight 8s第九题其实是同样的道理，怎么从8个8中得出1000答案：8+8+8+88+888=100010、A police officer had a brother ,but the brother had nobrother.How could that be?解析：这题主要是要打破常规思维，看见police officer,我们都会往男人去想，但是，其实为了男女同等，英文当中已经用police officer来代替polieman和policewoman了。
小学英语奥数题
小学英语奥数题1、Six trees are equally spaced along one side of a straight road. The distance from the first tree to the fourth is 60 feet. What is the distance in feet between the first and last trees?2、At the end of 2005 Walter was half as old as his grandmother. The sum of the years in which they were born is 3860. How old will Walter be at the end of 2006?3、The digits 1,2,3,4 and 9 are each used once to form the greatest possible even five-digit number. What is the digit in the tens place?4、Every edge of a cube is colored either red or green. In order to have at least one red edge on every face of the cube, find the minimum number of edges that must be colored red.5、When 31513 and 34369 are such divided by a certain 3-digit number, the remainders are equal. Find this remainder.6、Three signal lights were set to flash every certain specified time. The first light flashes every 12 seconds, the second flashed every 30 seconds and the third one every 66 seconds, the signal lights flash simultaneously at 8:30 am. At what time will the signal lights next flash。
趣味英语题20100828
一、常识题这类考题一般都应从生活常识的角度去分析和考虑，并不需要作高深的探究，但也不能被牵着鼻子走。
例如：How long does it take to boil two eggs if it takes four and a half minutes to boil one egg?不少考生的答案是：Nine minutes. 理由是煮一个鸡蛋要4.5分钟，2个鸡蛋则需9分钟。
其实，2个鸡蛋一起煮也是4.5分钟。
像这种简单的生活常识题，考生不能让人牵着鼻子走，否则因把简单的题目看得过于复杂而丢分就太可惜了。
又如：How many angles （角）are left if one angle is cut from a square （方形的）table? 此题答案是：Five (angles). 但有同学回答：Three (angles). 根据常识一张方形桌子，切去一个角，应该有5个角，而不是4 - 1 = 3。
二、语言知识题这类题常常是考查考生的语言知识和应变能力，有时要从句子本身或字形上分析，有时还涉及会意。
如：What is the distance betwe en the first letter and the last letter of “smiles”? 此题答案是：One mile. 问的是smiles这个单词中，第一个字母和最后一个字母之间的距离有多长，即除去第一个和最后一个字母s，剩下的是mile。
又如：What appears once in every minute, twice in every moment, but not once in thousand years? 此题答案为：The letter “m”. 问的是在minute中出现一次，moment中出现两次，而在thousand years中却一次也不出现的是什么？考生如果仅从字面意思上理解，就会误以为题目是问：每分钟出现一次、每一会儿出现两次，数千年却不出现的是什么？那问题就复杂了，答案也无从探究。
小升初英文奥数题系列
小升初英文奥数题系列一第一期英文奥数推出之后，家长和同学们都反映非常好，说受益匪浅。
因此我们推出系列二。
但同时也有些问题，很多家长反映孩子觉得比较难。
首先希望家长多多辅导孩子，其实奥数知识同学们是都会的，但是有些英文单词不认识，希望家长帮助孩子共同提高。
另外，我们只是希望那些比较优秀的同学和学有余力的同学来做，没有必要所有同学都要会做。
1、Did you know? In the decimal number system (base 10) ten different digits, 0 to 9, are used to write all the numbers. In the binary number system (base 2) two different digits are used, i.e. 0 and 1.Which one of the following numbers is not a valid number in theoctal number system (base 8)?A) 128 B) 127 C) 126 D) 125 E) 1242、The number of diagonals that can be drawn in a regular polygon with twenty sides (cosign) is_____.3、If a and b are integers, 10⊗3=1,152⊗7=3, and then 379⊗6 is equal to_____.4、Two numbers are in the ratio 2 : 3. When 4 is added to each number the ratio changes to 5 : 7.The sum of the two original numbers is____.5、The greatest number of Mondays which can occur in 45 consecutivedays is____6、Saul plays a video game in which he scores 4 for a hit and lost 6 for a miss. After 20 rounds his score is 30. The number of times he has missed is____.7、Three girls A, B and C run in a 100 m race. When A finishes, B is 10 m behind A and when B finishes C is 20 m behind B. How far in metres was C from A when A finished?(Let’s assume all the athletes run at a constant speed)8、The areas of the faces of a rectangulabox are 84 cm2 , 70 cm2and 30 cm2.The volume of the box in cm3 is____.9、You have 3 weights: 1 kg, 3 kg and 9 kg as well as an equal arm balance, as shown. How many different weight objects can you weigh with these three? [Remember the weights may be placed on either side]参考答案：1、A 考察我们学过的简单的进制问题，显然8进制中没有8出现2、170 找规律，公式为n×（n-3）÷23、1 定义新运算，就是求379÷6的余数。
外国语学校小升初入学考试中英文奥数题库1
外国语学校小升初英语奥数训练题第一部分1、三个素数的倒数之和是，则这三个素数中最大的是多少？1. The sum of the reciprocals of three prime numbers is, so what is the greatest one among the three prime numbers?2、有一个分数，它的分子加2，可以约简为；它的分母减2，可以约简为。
这个分数是多少？2. There is a fraction. If its numerator adds 2, it can be reduced to be; if its denominatorsubtracts 2, it can be reduced to be. So what is this fraction?3、一个数分别除以，，，所得的商都是自然数。
这个数最小是多少？3. A number is divided by, and respectively and thequotients are all natural numbers. So whatis the minimum value of this number?4、一片竹林，去年不开花的竹子比开花的2倍还多55棵，今年又多了100棵开花，这时开花的竹子恰好是不开花的4倍，这片竹林有多少棵竹子？4. There is a bamboo forest. Last year, the non-blooming bamboos were two times and 55 more than the blooming bamboos. With another 100 bamboos blooming this year, the blooming bamboos are four times as many as the non-blooming bamboos. So, how many bamboos are there in this forest?（红色的地方我有点不确定，葛老师您看看应该怎么翻）5、从中去掉两个分数，余下的分数之和为1。
小升初英文奥数题
英文奥数1．maths(English)1）Find the sum of all the prime numbers Between 20 and 30.2）Divide the sum of the first five odd numbers by the sum of the first two prime numbers.3）Subtract 2 from the sum of the first five even numbers. The difference is then divided by the first odd number following 5. What is the quotient4）Find the value of[4+8]÷[5×4-6]and give the answer correct to 2 decimal places 1）All the prime numbers between 20 and 30 are 23 and 29，The sum of all the prime numbers Between 20 and 30 = 23 + 29 = 522）The first five odd numbers are 1, 3, 5, 7and 9The sum of the first five odd numbers = 1 + 3 + 5 + 7 + 9= 253）The first two prime numbers are 2 and 3The sum of the first two prime numbers = 2 + 3= 54）The first five even numbers are 2, 4, 6, 8 and 10The sum of the first five even numbers= 2 + 4 + 6 + 8 + 10= 305）The difference between the sum of the first five even numbers and 2= 30 - 2= 286）The first odd number following 5 is 7The quotient = 28 / 7= 47）The value of [4+8]÷[5×4-6]= 12 ÷ ( 20 - 6 ) = 12 ÷ 14= ( Corr. to 2 decimal places )2．Six trees are equally spaced along one side of a straight road. the distance from the first tree to the fourth is 60 feet . what is the distancein feet between. The first and last trees3．At the end of 2005 walter was half as old as has snm of the years in which they were born is 3860 .how old will walter be at the end of 20064．The digits 1,2,3,4 and 9 are each used once to form the greatest possible even five-digit number. What is the digit in the tens place*************************************************1、In 2004, 16 June falls on a Wednesday. On what day of the week wi ll 16 June fall in 20102、In a magic square the sum of the numbers in each row, in eachdiagonal and in each column are equal. In this magic square the value of x is:3、If half of a number is 30, then three-quarters of that number is_ ___.4、The sum of the digits of the following product999×5555．Three positive integers have a sum of 28. The greatest possible p roduct that these integers canhave is_____.6、Jack was trying to tessellate regular pentagons. He managed thefollowing figure.The size of angle .a. is______.7、If the area of the shaded region of the regular hexagon in the diagrambelow is 36 cm2, the area of the whole hexagon in cm2 is_____. 8、In what follows, □and Δare different numbers.When 503 is divided by □the remainder is 20.When 503 is divided by Δthe remainder is 20.When 493 is divided by □x Δthe remainder is_____.9、A lady, her brother, her son and her daughter (all related by bi rth) playedvolleyball. The worst player's twin (who is one of the four pla yers) and thebest player are of opposite sex.The worst player and the best player are of the same age.Who cannot be the worst player(s)A) brother only B) daughter onlyC) son and daughter only D) lady and daughter onlyE) lady only10、If you continue the given number pattern, in what row and in wha tposition in that row will the number 320 be1 -------------- row 12 3 -------------- row 24 5 6 -------------- row 37 8 9 10 -------------- row 4The answers are given in the order of row ; position.参考答案：1、Wednesday2、123、454、27（求数位上上的数字之和）5、28=9+9+10，因此答案为8106、36度7、1088、503-20=483 483=3×7×23＝21×23，因此□x Δ＝483，因此此题余数是10.9、D 10、25，20##**##***************###############·········****#####附一道考试真题：At10:30, the angle less than 180°Formet by hour hand and minuteHand of normal clock is***( 英汉词典：angle角；less than小于；to from形成；hour hand 时针；Minute hand 分针；normal正常，正规； clock 钟，时钟）***上面是一道关于时钟问题的英文奥数题，猜猜是什么意思附参考答案：如果做过相关的时钟问题，借助于所给的解释可以理解到：求10：30分时，时针与分针所形成的小于180的夹角是多少答案：135°。
英文三年级数学应用题
英文三年级数学应用题# 英文三年级数学应用题Mathematics is a subject that can be both challenging and fun, especially when it comes to applying mathematical concepts to real-life situations. Here are some math word problemssuitable for third-grade students to practice their problem-solving skills and reinforce their understanding of basic mathematical operations.Problem 1: The Lemonade StandEmma and her friends decided to set up a lemonade stand. They sold 24 cups of lemonade on the first day and 32 cups on the second day. If each cup of lemonade costs $1.50, how much money did they make in total?Problem 2: The School TripA class of 36 students is going on a school trip. The cost of the bus ride is $120, and the entrance fee to the museum is$5 per student. What is the total cost for the class trip?Problem 3: The Birthday PartySarah is having a birthday party and wants to invite her friends. She has 15 friends and wants to give each of them 3 balloons. How many balloons does she need to buy?Problem 4: The Gardening ClubThe gardening club has 8 members. They are planting flowersin the school garden. Each member plants 7 rows of flowers.If each row has 12 flowers, how many flowers are planted in total?Problem 5: The Book FairAt the book fair, each book costs $5. If a student buys 4 books, how much will they spend? If they have $20, how many books can they buy with that amount?Problem 6: The Cookie BakingA recipe for cookies makes 18 cookies. If you need to bake enough cookies for 54 people, and each person gets 3 cookies, how many batches of cookies do you need to bake?Problem 7: The Sports EquipmentA sports shop has 12 soccer balls and each costs $8. If a customer buys 5 soccer balls, how much will they spend? How much money will they save if the shop offers a discount of 10% on the total bill?Problem 8: The Pet StoreAt a pet store, there are 9 kittens and 7 puppies. If 4kittens and 3 puppies are adopted, how many pets are left in the store?Problem 9: The Art ClassIn an art class, there are 20 students. Each student needs 2 sheets of paper and 3 paintbrushes for their project. Howmany sheets of paper and paintbrushes are needed in total?Problem 10: The Toy StoreA toy store has a sale where you buy 3 toys and get 1 free.If a child buys 9 toys, how many toys will they have in total?Problem 11: The Fruit BasketA fruit basket contains 5 apples, 3 oranges, and 2 bananas.If the same number of each fruit is added to the basket, how many of each fruit will there be then?Problem 12: The Cake DecorationA baker is decorating a cake with 24 sprinkles. If she uses 6 sprinkles on each side of the cake, how many sides does the cake have?Problem 13: The Bicycle RaceIn a bicycle race, there are 18 participants. If each participant finishes the race and receives a medal, and there are 2 spare medals, how many medals are there in total?Problem 14: The Library BooksA library has 40 books. If 12 books are borrowed and 8 books are returned, what is the new number of books in the library?Problem 15: The Classroom PlantsA classroom has 6 tables, and each table has 4 pots for plants. If each pot needs 2 seeds to start a new plant, how many seeds are needed in total for all the pots?These problems are designed to help third-grade students practice addition, subtraction, multiplication, and basic division in the context of everyday situations. By solvingthese problems, students can develop their mathematical reasoning and apply their skills to real-world scenarios.。
五年级英语小学六年级奥数题及答案
一．连词组句。
1．groups, all, chairs, are, in, desks, the, and(.)2．In, your, class, there, any, are, computers(?)3．Corner, have, we, twelve, computers, the, in, computer(.) 4．Corner, science, your, do, what, do, you, in(?)5．Cartoons, at, we, read, storybooks, and, look(.)二、句型转换(一) 根据答句写问句。
1．Yes, we often read in the library.2. We often play the violin in the club.3. We call it a foal.4. They often do project work here.（二）根据题意转换句型。
1. We often play the piano in the room.2. We read storybooks the library.3. There are some girls in our class.(一般疑问句)4．We have six classes a day. (否定句)5．Do the boys often play football? (肯定回答)6. Lucy works in a hospital. （一般疑问句）三．选择题：1．How many English you have a week?A. classesB. classC. a class2. Are there sheep on the farm?A. anyB. aC. an3. The photos shows how they learn.A. weB. usC. our4. They us around their school now.A. showingB. are showingC. shows5. __music classes do you have?A. Is there aB. How manyC. Are6. We often play games __groups.A. inB. atC. to7. They __on the paper.A. do printB. do printingC. print8. There is __art corner in our school.A. aB. anC. some9. They often do __in the class.A. listenB. listeningC. listens10. I often read __.A. storiesbookB. storybooksC. storybook11. Is there __apple on the table?A. anyB. anC. a12. Mark __from Britain.A. comeB. comingC. comes选择题：1.We play __rugby in our school.A. aB. theC. / 2．Those books are __the girls.A. forB. toC. on3. The boys __football.A. playingB. playsC. are playing4. We often do printing __the paper.A. inB. onC. for5. Is that __blue jacket?A. KateB. Kate’sC. Kates6. Are there __apples?A. anB. aC. any7. What __she do in the club?A. doB. doesC. is8. Are there __computers in the classroom?A. someB. anyC. a9. We play games __groups.A. onB. atC. in10. There __any bread on the table.A. isn’tB. isC. are11. __do you play volleyball?A. WhatB. WhereC. Who12. Are there any mugs on the table?Yes, __.A.There is notB. they areC. there are13. We often go __science field trips.A. inB. atC. on14.Our __are about flowers.A. studyB. studysC. studies15. How __you and your trip?A. isB. aboutC. do 16．Let’s __games after school.A. playingB. playC. to play17. There __any water in the bottle.A. isB. isn’tC. aren’t18. I study birds __the forest.A. onB. inC. to19. We like __hockey.A. playingB. playC. to play20. They are __TV now.A. watchingB. watchesC. watch21. Let __help you.A. isB. usC. does22. All of __are interested __music.A. they…atB. them…inC. their …in23. He often takes care __the boy.A. ofB. forC. to24. Your book is here. Put __away.A. itB. itsC. it’s25. Here __the money.A. amB. isC.are26. __is this bag?A. How manyB. How muchC. How nice27. What can I do __you?A. toB. atC. for28. __they have a football? Yes, they do.A. CanB. AreC. Do29. __are you doing? I’m singing.A. WhereB. WhatC. How30. Don’t speak! Listen to __.A. myB. meC. mine31. I like the __of the dress.A. moneyB. colourC. green32. I want a new a bag __my birthday.A. forB. inC. to33. Let’s meet __the gate.A.toB. atC. in34. How __orange?A. anyB. manyC. about35. The coat is beautiful. Try __on.A. themB. itC. its36. The sports shoes __too big for me.A. isB. areC. can37. Can you show __that one?A. IB. meC. mine38. How much __the shoes?A. isB. areC. about38.I want a pair __sandals.A. toB. ofC. about39. Hello, may I __to Mr Zhang?A. sayingB. speakC. tell40. I’m ill. I have a __cold.A. bigB. smallC. bad41. Her father will take __to the hospital.A. sheB. herC. hers42. Take good care __us.A. ofB. offC. for连词组句：句型转换：1.I, have, this, a, book, can, at, one (?) 1.The new toothbrush is five yuan.2.Much, is, it, how(?) 2. Can I help you? (同义句)3.it, take, we, will(.) 3. Can I have a look at this lamp?(同义句)4.the, is, money, here(.) 4. I can drink a glass of water. (一般疑问句)5.want, do, anything, you, else(?) 5. They want a towel. (否定句)6.Walkman, a, I, can, have(?) 6. What can I do for you? (同义句)7.English, want, I, to, to, listen(.) 7. I’ll take it. (同义句)8.Like, not, I, the, do, colour(.) 8. Please pass me the pen. （同义句）9.Pass, please, you, blue, can, me, the, one(?) 9. It’s eighty yuan.b, to, buy, I, want, a(.) 10. He can dance. (一般疑问句)11.Is, trip, Peter, week, on, going, a, next, field(.) 11. I like them very much.12.A, want, of, I, pair, sandals(.) 12. I want to buy a walkman. (同义句)13.For, too, are, big, they, me(.) 13. What can I do for you? （同义句）14.Try, on, them, can, I(?) 14. There is a book on the desk. （一般疑问句）15.That, show, me, pair, you, can(?) 15. He works in a hospital.16.Worry, do, your, lessons, not, about(.) 16. My uncle is a driver.17.Doctor, you, should, go, see, to, a(.) 17. How much is your new dress?(复数)18.Will, hospital, my, to, me, take, mum, the(.) 18. There are three people in my family.句型转换：19．Everyone in my class likes studying English.(一般疑问句。
北大附小三年级暑假奥数题
1.裁缝有一段16米长的呢子，每天剪去2米，第几天剪去最后一段？2.一根木料在24秒内被切成了4段，用同样的速度切成5段，需要多少秒？3.三年级同学120人排成4路纵队，也就是4个人一排，排成了许多排，现在知道每相邻两排之间相隔1米，这支队伍长多少米？4. 时钟4点钟敲4下，12秒钟敲完，那么6点钟敲6下，几秒钟敲完？5. 某人要到一座高层楼的第8层办事，不巧停电，电梯停开，如从1层走到4层需要48秒，请问以同样的速度走到八层，还需要多少秒？1．有一条公路长900米，在公路的一侧从头到尾每隔10米栽一根电线杆，可栽多少根电线杆？2．马路的一边每相隔9米栽有一棵柳树.张军乘汽车5分钟共看到501棵树.问汽车每小时走多少千米？3．某校五年级学生排成一个方阵，最外一层的人数为60人.问方阵外层每边有多少人？这个方阵共有五年级学生多少人？4．晶晶用围棋子摆成一个三层空心方阵，最外一层每边有围棋子14个.晶晶摆这个方阵共用围棋子多少个？5．一个圆形花坛，周长是180米.每隔6米种一棵芍药花，每相邻的两棵芍药花之间均匀地栽两棵月季花.问可栽多少棵芍药？多少棵月季？两棵月季之间的株距是多少米？1．一个街心花园如下图所示.它由四个大小相等的等边三角形组成.已知从每个小三角形的顶点开始，到下一个顶点均匀栽有9棵花.问大三角形边上栽有多少棵花？整个花园中共栽多少棵花？2．四个小动物排座位，一开始，小鼠坐在第1号位子上，小猴坐在第2号，小兔坐在第3号，小猫坐在第4号.以后它们不停地交换位子，第一次上下两排交换.第二次是在第一次交换后左右两列交换，第三次再上下两排交换，第四次再左右两列交换…这样一直换下去.问：第十次交换位子后，小兔坐在第几号位子上？（参看下图）3．将A、B、C、D、E、F 六个字母分别写在正方体的六个面上，从下面三种不同摆法中判断这个正方体中，哪些字母分别写在相对的面上。
4．观察下面的数列，找出其中的规律，并根据规律，在括号中填上合适的数.①2，5，8，11，（），17，20。
小升初英语奥数题
小升初英文奥数题11、Did you know In the decimal number system base 10 ten different digits, 0 to 9, are used to write all the numbers. In the binary number system base 2 two different digits are used, i.e. 0 and 1. Which one of the following numbers is not a valid number in the octal number system base 8你知道吗在十进制数字系统中十个不同的数字,用从0到9就可以写出所有的数字了；在二进制数字系统中使用了二个不同的数字0和1.请问在八进制数字系统中,下列的哪一个数字是不存在的A 128B 127C 126D 125E 1242、The number of diagonals that can be drawn in a regular polygon with twenty sides icosagon is_____.一个规则的20多边形总共可以画出多少条对角线3、If a and b are integers, 103=1, 1507=3, and then 3796 is equal to_____.如果a和b都是整数,那么3796是等于多少4、Two numbers are in the ratio 2 : 3. When 4 is added to each number the ratio changes to 5 : 7.The sum of the two original numbers is____.假设二个数字的比值是2:3,当每个数字加上4以后,比值就变成了5:7.那么这二个数字之和是多少5、The greatest number of Mondays which can occur in 45 consecutive days is____在日历中,续45天中最多可以出现多少个星期一6、Saul plays a video game in which he scores 4 for a hit and lost 6 for a miss. After 20 rounds his score is 30. The number of times he has missed is____.索尔在玩一个电子游戏,击中一次得4分,打偏则扣6分.20个回合后他的得分是30分,那么他打偏了多少次7、Three girls A, B and C run in a 100 m race. When A finishes, B is 10 m behindA and whenB finishesC is 20 m behind B. How far in metres was C from A when A finishedLet’s assume all the athletes run at a constant speed有A,B,C三个女孩进行100米赛跑.当A到达终点时,B落后A 10米,当B到达终点时,C落后B 20米.那么当A到达终点时,C落后A多少米假设所有的人都是匀速的8、The areas of the faces of a rectangular box are 84 cm2 , 70 cm2and 30 cm2.The volume of the box in cm3 is____.一个长方形盒子的表面积依次为84cm2,70 cm2 ,30 cm2,那么这个盒子的体积是多少9、You have 3 weights: 1 kg, 3 kg and 9 kg as well as an equal arm balance, as shown. How many different weight objects can you weigh with these threeRemember the weights may be placed on either side你有三个重物：1kg,3kg,9kg,还有一个等臂天平.那么你可以用这三个重物称多少个不同物体记得这些重物可以放在天平的任意一边小升初英文奥数题21. An ant covers adistance of 90 metrers in 3 hours .The average speed of the ant in decimetres per minute is____.一只小蚂蚁用了3个小时爬行了90米,那么它的平均速度是每分钟多少分米2.The Ancient Romans used the follwing different numerals in their number system as follows:Ⅰ=1 C=100 V=500 Ⅹ=10 M=1000 ,etc L=50 . They used these numerals to make up numbers as follows:Ⅰ=1,Ⅱ=2,Ⅲ=3,Ⅳ=4,Ⅴ=5,Ⅵ=6,Ⅶ=7,Ⅷ=8,Ⅸ=9,Ⅹ=10.So ,for example,XCIX is 99.What is the value of XLVI古罗马人用下面的符号来计数：…….,比如XCIX是99,XLVI等于多少3.In a certain town some people were affected bya'flu' epidemic .In the first month20% of the population contracted the ful whilst 80% were healthy.In the following month 20% of the sick people recovred and the 20% of the healthy people contractedthe disease . What fraction of the population is healthy at the end of the healthy month在某个小镇里一些人感染了流行性感冒.在第一个月里有20%的分感染了,但80%的人是健康的.在接下来一个月里,有20%的病人恢复了健康,但是20%的健康的人也感染了.那么现在总共健康的人数的百分比是多少4.Mpho,Barry ,Sipho,Erica and Fatima are sitting on a park bench .Mpho is not sittingon the far right. Barry is not sitting on the far left. Sipho is not sitting at either end .Erica si sitting to the right of Barry ,but not necessarily next tohim .Fatima is not sitting next to Sipho.Sipho is not sitting next to Barry .Who is sitting at the far rightMpho,Barry ,Sipho,Erica 和 Fatima坐在公园的长椅上.Mpho坐在不远的右边,Barry坐在不远的左边,Sipho不坐在长椅的任何一头,Eric坐在Barry的右边,但是并不需要挨着他,Fatima也不挨着Sipho,Sipho也不挨着Barry,那么谁坐在最右边5.Of the 28 T shirts in a drawer ,six are red ,five are blue ,and the rest arewhite.IfBob selects T-shirhs at random whilst packing for a holiday ,what is the least numberhe must remove from the drawer to be sure that he has three T-shirts of the same colour 抽屉里有28件T-恤衫,6个红色的,5个蓝色的,剩下的都是白色的,如果他随意取出T-恤衫去度假,要确保他有3件是同一颜色的,请问他从抽屉中拿出的T-恤衫的最少数量是多少件6.In an alien language ,jalez borg farn means "good maths skills".Nurf klar borg means"maths in harmony" and darko klar farn means"good in gold" .What is "harmony gold "in this language在外星人语言中,jalez borg farn 意思是 "good maths skills".………7.Five children ,Amelia ,Bongani ,Charles,Devine and Edwina ,were in the classroom when one of them broke a window. The teacher asked each of them to make a statement about the event,knowing that three of them always lie and two always tell the truth.Their statements were as follows :Amelia :"Charles did not break it ,nor did Devine." Bongani:"I didn't break it ,nor did Devine ." Charles:"I didn 't break it ,but Edwina did ." Devine;"Amelia or Edwina broke it ."Edwina:"Charles broke it."Who broke the window五个熊孩子,Amelia ,Bongani ,Charles,Devine and Edwina,在教室里玩疯了,突然有一个打破了窗户.老师先让每个人坦白一下事件,其实老师心里知道有三个孩子总是撒谎,二个爱讲实话.下面是他们的辩词：Amelia：不是Charles干的,也不是Devine干的.Bongani：我没有打破窗户,Devine也没有.Charles：我没有,是Edwina 干得.Devine：是Amelia或者Edwina 干得.Edwina：是Charles干得.那么到底是谁打破了窗户8. Did you know A palindrome你知道吗回文数字是一种顺读和倒读都一样的数.比如35453. 2002年就是一个这样的例子.那么2002和它前一个回文数字年的差别是多少小升初英文奥数题31、In 2004, 16 June falls on a Wednesday. On what day of the week will 16 June fall in2010在2004年,六月16号是星期三,那么在2010年六月16是星期几呢如果一个数的一半是30,那么这个数的四分之三是多少3、The sum of the digits of the following product 999×555 is999×555的积的数位上的数字之和是多少4、Three positive integers have a sum of 28. The greatest possible product that these integers can have is_____.三个正整数之和是28,那么这三个正整数之积最大可能是多少5、In what follows, □ and Δ are different numbers.When 503 is divided by □ the remainder is 20.When 503 is divided by Δ the remainder is 20.When 493 is divided by □ x Δ the remainder is_____.有如下几个例子,□和Δ都代表不同的数字,□除以503的余数是20. Δ除以503的余数也是20,那么493除以□×Δ之积的余数是多少6、A lady, her brother, her son and her daughter all related by birth played volleyball. The worst player's twin who is one of the four players and the best player are of opposite sex.The worst player and the best player are of the same age.Who cannot be the worst players一位女士,她的弟弟,她的儿子,她的女儿,都打排球.妈妈和她的弟弟是双胞胎,她自己的女儿和儿子也是双胞胎.最差的球员的双胞胎之一和最好的球员性别是相反的,最差的球员和最好的年龄是一样的.那么谁不可能是最差的A brother onlyB daughter onlyC son and daughter onlyD lady and daughter only7、If you continue the given number pattern, in what row and in what position in that row will the number 320 be如果你继续把下面这组数字模式写下去,那么在多少排、在这一排的什么位置将会是数字3201 -------------- row 12 3 -------------- row 24 5 6 -------------- row 37 8 9 10 -------------- row 4The answers are given in the order of row ; position.10.Two ants start at point A and walk at the same pace .One ant walks around a 3 cm by 3 cm square whilst the other walks around a 6cm by 3 cm rectangle .What is the minimun distance ,in centimetres,any one must cover before they meet again二只可怜的小蚂蚁都以同样的速度从A点出发.其中一只爬行了3cm×3cm的正方形,另一只爬行了6cm×3cm的长方形.那么如果二只蚂蚁再次相遇,每只必须爬行的最小距离的多少参考答案11、A 考察我们学过的简单的进制问题,显然8进制中没有8出现2、170 找规律,公式为n×n-3÷23、1 定义新运算,就是求379÷6的余数.4、40,16和245、76、57、28米,根据距离比求出速度比,三者的速度比为1：9/10：18/258、420 分解质因数9、13种21、单位换算,注意单词,90×10÷3×602、463、68%4、Erica5、7,抽屉原则6、nurf darko.注意一一对应,borg=maths,farn=good,jalez=skills,klar=in,nurf=harmony,darko=gold.7、Charles8、100131、Wednesday2、453、27求数位上上的数字之和4、28=9+9+10,因此答案为8105、503-20=483 483=3×7×23＝21×23,因此□ x Δ＝483,因此此题余数是10.6、D 7、25,2010、108△ABO的面积为16,等腰△DOC的面积占长方形面积的18％,那么阴影△AOC的面积是多少2、已知△ABC中,AB=AC=16, △ABC面积是64,P是BC上任意一点,P到AB,AC的距离分别是X、Y,那么X+Y=______3、从1到999这999个自然数中有______个数的各位数字之和能被4整除.4、如图乘法竖式中,"学而思杯"代表0 ~ 9中的一个数字,相同的汉字代表相同的数字,不同的汉字代表不同的数字,那么"学而思杯"分别代表的数字是_______5、学学和思思结伴骑车去图书馆看书,第一天他们从学校直接去图书馆；第二天他们先去公园再去图书馆；第三天公园修路不能通行．则这三天从学校到图书馆的最短路线分别有_______种不同的走法.1、10个不同非0自然数的和为1001,则这10个数的最大公约数的最大值_____2、"学而思杯思而学"是一个七位回文数字,其中相同的汉字代表相同的数字,不同的汉字代表不同的数字．已知这个七位数第1位能被2整除,前2位组成的2位数能为3整除,前3位组成的3位数数能被4整除,…… ,前7位数组成的七位数能被8整除．那么"学而思杯思而学"=_______ ．3、如图,△ABC是等腰直角三角形,DEFG是正方形,线段AB与CD相交于K点．已知正方形DEFG的面积48,AK: KB=1: 3,则△BKD的面积是_________4、甲、乙两队各出5名队员按事先排好的顺序出场参加象棋擂台赛,双方先由1号队员比赛,负者被淘汰,胜者再与负方2号队员比赛,……直至有一方队员全被淘汰为止,另一方获得胜利．各个队员的胜负排列便形成一种比赛过程．已知每次比赛都没有和局,问所有可能的比赛过程有多少种参考答案：一、填空题每题5分,共25分1、17702、153、24192004、35、14,63,147二、填空题每题7分,共35分1、3.52、83、2484、32015、16,8,81、132、42858243、124、25。
1975~2002匈牙利奥林匹克数学竞赛(英文版)
匈牙利数学奥林匹克试题(1975-2002)75th Kürschák Competition 1975Problem1.Transform the equation ab2(1/(a + c)2 + 1/(a - c)2) = (a - b) into a simpler form, given that a > c ≥ 0, b > 0.Problem2.Prove or disprove: given any quadrilateral inscribed in a convex polygon, we can find a rhombus inscribed in the polygon with side not less than the shortest side of the quadrilateral Problem3.Let x0 = 5, x n+1 = x n + 1/x n. Prove that 45 < x1000 < 45.1.76th Kürschák Competition 1976Problem1 ABCD is a parallelogram. P is a point outside the parallelogram such that angles PAB and PCB have the same value but opposite orientation. Show that angle APB = angle DPC. Problem2 A lottery ticket is a choice of 5 distinct numbers from 1, 2, 3, ... , 90. Suppose that 55 distinct lottery tickets are such that any two of them have a common number. Prove that one can find four numbers such that every ticket contains at least one of the fourProblem3 Prove that if the quadratic x2 + ax + b is always positive (for all real x) then it can be written as the quotient of two polynomials whose coefficients are all positive.77th Kürschák Competition 1977Problem1 Show that there are no integers n such that n4 + 4n is a prime greater than 5.Problem2ABC is a triangle with orthocenter H.The median from A meets the circumcircle again at A1,and A2 is the reflection of A1 in the midpoint of BC. Thepoints B2 and C2 are defined similarly. Show that H, A2,B2 and C2 lie on a circle.Problem3 Three schools each have n students. Eachstudent knows a total of n+1 students at the other twoschools. Show that there must be three students, onefrom each school, who know each other.78th Kürschák Competition 1978Problem1 a and b are rationals. Show that if ax2 + by2 =1 has a rational solution (in x and y), then it must have infinitely many.Problem2 The vertices of a convex n-gon are colored so that adjacent vertices have different colors. Prove that if n is odd, then the polygon can be divided into triangles with non-intersecting diagonals such that no diagonal has its endpoints the same color.Problem3 A triangle has inradius r and circumradius R. Its longest altitude has length H. Show that if the triangle does not have an obtuse angle, then H ≥ r + R. When does equality hold?79th Kürschák Competition 1979Problem1 The base of a convex pyramid has an odd number of edges. The lateral edges of the pyramid are all equal, and the angles between neighbouring faces are all equal. Show that the base must be a regular polygon.Problem 2f is a real-valued function defined on the reals such that f(x) ≤ x and f(x+y) ≤ f(x) + f(y) for all x, y. Prove that f(x) = x for all x.Solution Suppose f(0) < 0. Then f(y) ≤ f(0) + f(y) < f(y). Contradiction. So f(0) ≥ 0 and hence f(0) = 0. Now suppose f(x) < x. Then f(0) ≤ f(x) + f(-x) < x + f(-x) ≤ x - x = 0. Contradiction. Hencef(x) = x.Problem3 An n x n array of letters is such that no two rows are the same. Show that it must be possible to omit a column, so that the remaining table has no two rows the same.80th Kürschák Competition 1980Problem1 Every point in space is colored with one of 5 colors. Prove that there are four coplanar points with different colors.Problem2 n > 1 is an odd integer. Show that there are positive integers a and b such that 4/n = 1/a + 1/b iff n has a prime divisor of the form 4k-1.Problem3 There are two groups of tennis players, one of 1000 players and the other of 1001 players. The players can ranked according to their ability. A higher ranking player always beats a lower ranking player (and the ranking never changes). We know the ranking within each group. Show how it is possible in 11 games to find the player who is 1001st out of 2001.81st Kürschák Competition 1981Problem1 Given any 5 points A, B, P, Q, R (in the plane) show that AB + PQ + QR + RP <= AP + AQ + AR + BP + BQ + BRProblem2 n > 2 is even. The squares of an n x n chessboard are painted with n2/2 colors so that there are exactly two squares of each color. Prove that one can always place n rooks on squares of different colors so that no two are in the same row or column.Problem3 Divide the positive integer n by the numbers 1, 2, 3, ... , n and denote the sum of the remainders by r(n). Prove that for infinitely many n we have r(n) = r(n+1).82nd Kürschák Competition 1982Problem1 A cube has all 4 vertices of one face at lattice points and integral side-length. Prove that the other vertices are also lattice points.Problem2 Show that for any integer k > 2, there are infinitely many positive integers n such that the lowest common multiple of n, n+1, ... , n+k-1 is greater than the lowest common multiple of n+1, n+2, ... , n+k.Problem3 The integers are colored with 100 colors, so that all the colors are used and given any integers a < b and A < B such that b - a = B - A, with a and A the same color and b and B the same color, we have that the whole intervals [a, b] and [A, B] are identically colored. Prove that -1982 and 1982 are different colors.83rd Kürschák Competition 1983Problem1 Show that the only rational solution to x3 + 3y3 + 9z3 - 9xyz = 0 is x = y = z = 0. Problem2 The polynomial x n + a1x n-1 + ... + a n-1x + 1 has non-negative coefficients and n real roots. Show that its value at 2 is at least 3n.Problem3 The n+1 points P1, P2, ... , P n, Q lie in the plane and no 3 are collinear. Given any two distinct points P i and P j, there is a third point P k such that Q lies inside the triangle P i P j P k. Prove that n must be odd.84th Kürschák Competition 1984Problem1 If we write out the first four rows of the Pascal triangle and add up the columns we get: 11 11 2 11 3 3 11 1 4 3 4 1 1If we write out the first 1024 rows of the triangle and add up the columns, how many of theresulting 2047 totals will be odd?Problem2 A1B1A2, B1A2B2, A2B2A3, B2A3B3, ... , A13B13A14, B13A14B14, A14B14A1, B14A1B1 are thin triangular plates with all their edges equal length, joined along their common edges. Can the network of plates be folded (along the edges A i B i) so that all 28 plates lie in the same plane? (They are allowed to overlap).Problem3 A and B are positive integers. We are given a collection of n integers, not all of which are different. We wish to derive a collection of n distinct integers. The allowed move is to take any two integers in the collection which are the same (m and m) and to replace them by m + A and m - B. Show that we can always derive a collection of n distinct integers by a finite sequence of moves.85th Kürschák Competition 1985Problem1 The convex polygon P0P1 ... P n is divided into triangles by drawing non-intersecting diagonals. Show that the triangles can be labeled with the numbers 1, 2, ... , n-1 so that the triangle labeled i contains the vertex P i (for each i).Problem2 For each prime dividing a positive integer n, take the largest power of the prime not exceeding n and form the sum of these prime powers. For example, if n = 100, the sum is 26 + 52 = 89. Show that there are infinitely many n for which the sum exceeds n.Problem3 Vertex A of the triangle ABC is reflected in theopposite side to give A'. The points B' and C' are definedsimilarly. Show that the area of A'B'C' is less than 5 times thearea of ABC.86th Kürschák Competition 1986Problem1 Prove that three half-lines from a given pointcontain three face diagonalsof a cuboid iff the half-linesmake with each other threeacute angles whose sum is180o.Problem2 Given n > 2, find the largest h and the smallest H such that h < x1/(x1 + x2) + x2/(x2 + x3) + ... + x n/(x n + x1) < H for all positive real x1, x2, ... , x n.Problem3 k numbers are chosen at random from the set {1, 2, ... , 100}. For what values of k is the probability ½ that the sum of the chosen numbers is even?87th Kürschák Competition 1987Problem1 Find all quadruples (a, b, c, d) of distinct positive integers satisfying a + b = cd and c + d = ab.Problem2 Does there exist an infinite set of points in space such that at least one, but only finitely many, points of the set belong to each plane?Problem3 A club has 3n+1 members. Every two members play just one of tennis, chess and table-tennis with each other. Each member has n tennis partners, n chess partners and n table-tennis partners. Show that there must be three members of the club, A, B and C such that A and B play chess together, B and C play tennis together and C and A play table-tennis together.88th Kürschák Competition 1988Problem1 P is a point inside a convex quadrilateral ABCD such that the areas of the triangles PAB, PBC, PCD and PDA are all equal. Show that one of its diagonals must bisect the area of thequadrilateral.Problem2 What is the largest possible number of triples a < b < c that can be chosen from 1, 2, 3, ... , n such that for any two triples a < b < c and a' < b' < c' at most one of the equations a = a', b = b', c = c' holds?Problem3 PQRS is a convex quadrilateral whose vertices are lattice points. The diagonals of the quadrilateral intersect at E. Prove that if the sum of the angles at P and Q is less than 180o then the triangle PQE contains a lattice point apart from P and Q either on its boundary or in its interior.89th Kürschák Competition 1989Problem1 Given two non-parallel lines e and f anda circle C which does not meet either line. Constructthe line parallel to f such that the length of its segmentinside C divided by the length of its segment from Cto e (and outside C) is as large as possible.Problem2 Let S(n) denote the sum of the decimaldigits of the positive integer n. Find the set of allpositive integers m such that s(km) = s(m) for k = 1,2, ... , m.Problem3 Walking in the plane, we are allowed to move from (x, y) to one of the four points (x, y ± 2x), (x ± 2y, y). Prove that if we start at (1, √2), then we cannot return there after finitely many moves.90th Kürschák Competition 1990Problem1 Show that for p an odd prime and n a positive integer, there is at most one divisor d of n2p such that d + n2 is a square.Problem2 I is the incenter of the triangle ABC and A' is the center of the excircle opposite A. The bisector of angle BIC meets the side BC at A". The points B', C', B", C" are defined similarly. Show that the lines A'A", B'B", C'C" are concurrent.Problem3 A coin has probability p of heads and probability 1-p of tails. The outcome of each toss is independent of the others. Show that it is possibleto choose p and k, so that if we toss the coin k times we can assign the 2k possible outcomes amongst 100 children, so that each has the same 1/100 chance of winning. [A child wins if one of its outcomes occurs.]91st Kürschák Competition 1991Problem1 a >= 1, b >= 1 and c > 0 are reals and n is a positive integer. Show that ( (ab + c)n - c) <= a n ( (b + c)n - c).Problem2 ABC is a face of a convex irregular triangularprism(the triangular faces are not necessarily congruent orparallel). The diagonals of the quadrilateral face opposite Ameet at A'. The points B' and C' are defined similarly. Showthat the lines AA', BB' and CC' are concurrent.Problem3 There are 998 red points in the plane, no threecollinear. What is the smallest k for which we can alwayschoose k blue points such that each triangle with red vertices has a blue point inside?92nd Kürschák Competition 1992Problem1 Given n positive integers a i, define S k = Σ a i k, A = S2/S1, and C = ( S3/n)1/3. For each of n = 2, 3 which of the following is true: (1) A >= C; (2) A <= C; or (3) A may be > C or < C,depending on the choice of a i?Problem2 Let f1(k) be the sum of the (base 10) digits of k. Define f n(k) = f1(f n-1(k) ). Find f1992(21991).Problem3 A finite number of points are given in the plane, no three collinear. Show that it is possible to color the points with two colors so that it is impossible to draw a line in the plane with exactly three points of the same color on one side of the line.93rd Kürschák Competition 1993Problem1 a and b are positive integers. Show that there are at most a finite number of integers n such that an2 + b and a(n + 1)2 + b are both squares.Problem2 The triangle ABC is not isosceles. The incircle touches BC, CA, AB at K, L, M respectively. N is the midpoint of LM. The line through B parallel to LM meets the line KL at D, and the line through C parallel to LM meets the line MK at E. Show that D, E and N are collinear. Problem3 Find the minimum value of x2n + 2 x2n-1 + 3 x2n-2 + ... + 2n x + (2n+1) for real x.94th Kürschák Competition 1994Problem1 Let r > 1 denote the ratio of two adjacent sides of a parallelogram. Determine how the largest possible value of the acute angle included by the diagonals depends on r.Problem2 Prove that after removing any n-3 diagonals of a convex n-gon, it is always possible to choose n-3 non-intersecting diagonals amongst those remaining, but that n-2 diagonals can be removed so that it is not possible to find n-3 non-intersecting diagonals amongst those remaining. Problem3 For k = 1, 2, ... , n, H k is a disjoint union of k intervals of the real line. Show that one can find [(n + 1)/2] disjoint intervals which belong to different H k.95th Kürschák Competition 1995Problem1 A rectangle has its vertices at lattice points and its sides parallel to the axes. Its smaller side has length k. It is divided into triangles whose vertices are all lattice points, such that each triangle has area ½. Prove that the number of the triangles which are right-angled is at least 2k. Problem2 A polynomial in n variables has the property that if each variable is given one of the values 1 and -1, then the result is positive whenever the number of variables set to -1 is even and negative when it is odd. Prove that the degree of the polynomial is at least n.Problem3 A, B, C, D are points in the plane, no three collinear. The lines AB and CD meet at E, and the lines BC and DA meet at F. Prove that the three circles with diameters AC, BD and EF either have a common point or are pairwise disjoint.96th Kürschák Competition 1996Problem1 The diagonals of a trapezium are perpendicular. Prove that the product of the two lateral sides is not less than the product of the two parallel sides.Problem2 Two delegations A and B, with the same number of delegates, arrived at a conference. Some of the delegates knew each other already. Prove that there is a non-empty subset A' of A such that either each member in B knew an odd number of members from A', or each member of B knew an even number of members from A'.Problem3 2kn+1 diagonals are drawn in a convex n-gon. Prove that among them there is a broken line having 2k+1 segments which does not go through any point more than once. Moreover, this is not necessarily true if kn diagonals are drawn.97th Kürschák Competition 1997Problem1 Let S be the set of points with coordinates (m, n), where 0 <= m, n < p. Show that we can find p points in S with no three collinearProblem2 A triangle ABC has incenter I and circumcenter O. The orthocenter of the three points at which the incircle touches its sides is X. Show that I, O and X are collinear.Problem3 Show that the edges of a planar graph can be colored with three colors so that there is no monochromatic circuit.98th Kürschák Competition 1998Problem1 Can you find an infinite set of positive integers such that each pair has a common divisor (greater than 1), no integer (greater than 1) divides all members of the set, and no member of the set divides any other member?Problem2 Show that there is a polynomial with integer coefficients whose values at 1, 2, ... , n are different powers of 2.Problem3 For which n > 2 can we find n points in the plane, no three collinear, so that for each triangle of the points which are in the convex hull, exactly one of the points belongs to its interior. 99th Kürschák Competition 1999Problem1 Let e(k) be the number of positive even divisors of k, and let o(k) be the number of positive odd divisors of k. Show that the difference between e(1) + e(2) + ... + e(n) and o(1) + o(2) + ... + o(n) does not exceed n.Problem2 ABC is an arbitrary triangle. Construct an interior point P such that if A' is the foot of the perpendicular from P to BC, and similarly for B' and C', then the centroid of A'B'C' is P. Problem3 Prove that every set of integers with more than 2k members has a subset B with k+2 members such that any two non-empty subsets of B with the same number of members have different sums.100th Kürschák Competition 2000Problem1 The square 0 ≤ x ≤ n, 0 ≤ y ≤ n has (n+1)2 lattice points. How many ways can each of these points be colored red or blue, so that each unit square has exactly two red vertices? Problem2 ABC is any non-equilateral triangle. P is any point in the plane different from the vertices. Let the line PA meet the circumcircle again at A'. Define B' and C' similarly. Show that there are exactly two points P for which the triangle A'B'C' is equilateral and that the line joining them passes through the circumcenter.Problem3 k is a non-negative integer and the integers a1, a2, ... , a n give at least 2k different remainders on division by n+k. Prove that among the a i there are some whose sum is divisible by n+k.101st Kürschák Competition 2001Problem 1Given any 3n-1 points in the plane, no three collinear, show that it is possible to find 2n whose convex hull is not a triangle.Solution The difficulty is that extra points can reduce thenumber of points in the convex hull. Consider for example theconfiguration above. If we take at least one point from each arc,then the convex hull is a triangle. So we can pick at most 2npoints to get a convex hull which is not a triangle.On the other hand, if we have N > 4 points with convex hull nota triangle, then it is easy to remove a point and still have theconvex hull not a triangle. If there is an interior point then wecan remove that. If there are no interior points, then the convex hull has N points and we can remove any point to get a convex hull of N-1 > 3 points.So if the result is false, then if we take any N ≥ 2n of thepoints the convex hull must be a triangle. Suppose the convexhull of the 3n-1 points is A1, B1, C1. If we remove A1, then theconvex hull is a triangle. Now B1 must be one of the verticesof this triangle, for if it belonged to a triangle XYZ of otherpoints, then it could not be part of the convex hull of thewhole set. Similarly for C1. So the convex hull after removingA1 must be A2B1C1 for some A2. We can now remove A2 and the convex hull must be A3B1C1 for some A3, and so on. Finally, we remove A n-1 to get A n B1C1 as the convex hull of the remaining 2n points.Similarly, we can define B2, B3, ... , B n, so that the convex hull after removing B1, B2, ... B i is A1B i+1C1, and we can define C2, C3, ... , C n, so that the convex hull after removing C1, C2, ... , C i is A1B1C i+1.But now we have 3n points A i, B j, C k chosen from 3n-1, so two must be the same. The A i are all distinct, and similarly the B j and the C k. So wlog we have A i = B j for some i,j. Now if we remove all the As, all the Bs and C1 from the original set we are left with at least n-1 points (because we are removing at most 2n distinct points) and these must belong to the interior of the blue triangle A i B1C1 and the yellow triangle A1B j C1 = A1A i C1. But the interiors are disjoint, so we have a contradiction.Problem 2k > 2 is an integer and n > kC3 (where aCb is the usual binomial coefficient a!/(b! (a-b)!) ). Show that given 3n distinct real numbers a i, b i, c i (where i = 1, 2, ... , n), there must be at least k+1 distinct numbers in the set {a i + b i, b i + c i, c i + a i | i = 1, 2, ... , n}. Show that the statement is not always true for n = kC3.Solution Suppose there are at most k distinct numbers. Then there are at most kC3 and hence <n distinct sets of 3 numbers chosen from them. So for some i ≠ j we must have {a i + b i, b i + c i, c i + a i} = {a j + b j, b j + c j, c j + a j}. But the set {a i, b i, c i} is uniquely determined by {a i + b i, b i + c i, c i + a i}, so {a i, b i, c i} = {a j, b j, c j}. Contradiction.Suppose we take S to be the set {30, 31, 32, ... , 3k-1}. Take all n = kC3 subsets of 3 elements and for each such subset A i take {a i, b i, c i} so that {a i + b i, b i + c i, c i + a i} = A i. Obviously a i, b i, c i are distinct, so we have to show that if (3a + 3b - 3c)/2 = (3r + 3s - 3t)/2, where a, b, c are distinct and r, s, t are distinct, then {a,b,c} = {r,s,t}. We have 3a + 3b + 3t = 3r + 3s + 3c. There are two cases. If a = t, then since the representation base 3 is unique, we must have one of r, s, c equal to b and the other two equal to a. Since c ≠ a, and c ≠ b that is impossible. So a, b, t must all be distinct and hence {a, b, t} = {r, s, c}. Since c ≠ a or b, we must have c = t and hence {a, b} = {r, s} and so {a, b, c} = {r, s, t} as required.For example, if k = 4, then n = 4 and we can take: a1, b1, c1 = -5/2, 7/2, 11/2 ;a2, b2, c2 = -23/2, 25/2, 29/2 ;a3, b3, c3 = -17/2, 19/2, 35/2 ;a4, b4, c4 = -15/2, 21/2, 33/2 .Problem 3The vertices of the triangle ABC are lattice points and there is no smaller triangle similar to ABC with its vertices at lattice points. Show that the circumcenter of ABC is not a lattice point.Solution Let the points A, B, C have coordinates A (0,0), B(a,b), C(c,d). Suppose the circumcenter D (x,y) is a lattice point. Then AD2 = AB2, so x2 + y2 = (x-a)2 + (y-b)2. Hence a2 + b2 is even. Hence a + b and a - b are even. Similarly, c + d and c - d are even. So the points X = ((a+b)/2, (a-b)/2) and Y = ((c+d)/2, (c-d)/2) are lattice points.But AX2 = (a+b)2/4 + (a-b)2/4 = (a2+b2)/2 = AB2/2. Similarly AY2 = AC2/2. XY2 = (a+b-c-d)2/4 + (a-b+c-d)2/4 = ((b-c)2 + (a-d)2)/2 = BC2/2. So AXY is similar to ABC and smaller. Contradiction. 102th Kürschák Competition 2002Problem1 ABC is an acute-angled non-isosceles triangle. H is the orthocenter, I is the incenter and O is the circumcenter. Show that if one of the vertices lies on the circle through H, I and O, then at least two vertices lie on it.Problem2 The Fibonacci numbers are defined by F1 = F2 = 1, F n = F n-1 + F n-2. Suppose that a rational a/b belongs to the open interval (F n/F n-1, F n+1/F n). Prove that b ≥ F n+1.Problem3 S is a convex 3n gon. Show that we can choose a set of triangles, such that the edges of each triangle are sides or diagonals of S, and every side or diagonal of S belongs to just one triangle.。
六年级英语复习必看的五大题型
六年级英语复习必看题型⼀：单词和语法
1、简单的词组利于：look at等在⼩升初英语试卷中经常出现;
2、情态动词⽤法如must之类是⼩升初英语的必考内容;
3、⾮谓语动词也是⼩升初英语的必考内容;
4、虚拟语⽓是⼩升初英语的必考内容。
六年级英语复习必看题型⼆：完形填空
完形填空会全⾯考察孩⼦的英语知识体系，词汇、语法、词组搭配等。
六年级英语复习必看题型三：阅读题
⼀般⼩升初英语试卷会有四道阅读题，四篇阅读题中，最后⼀篇⾮常难，有的甚⾄达到了⾼考英语⽔平。
但是，做阅读提有⼀些技巧，是可以强化的。
六年级英语复习必看题型四：英⽂奥数题
英⽂奥数题是孩⼦认为最难的题⽬，因为会出现很多孩⼦从来没有见过的⽣词，⽽且还要孩⼦有⼀些数学基础。
⼀般数学成绩好的孩⼦，英语不咋地，那么这类题就成为了他们的难题。
六年级英语复习必看题型五：摘要写作题
这类题型对于⼤多数孩⼦来说很简单，只要孩⼦了解⼀些基本的做题技巧，都能顺利。
孩⼦在考试之前，可以学习新概念英语教材，以为⼩升初英语写作题是从新概念英语教材中的摘要写作部分借鉴⽽来的。
英语奥数题10道
小外的考试中，每年都会有几道英文数学题，近两年更多的重点中学，在选拔的时候都考察到了英文数学题。
下面就是几道这种类型的题目，我们将分批次让大家提前感受小卷的必考题型之一——英文数学题。
做这种题目的时候，其实你不需要完全读懂题目的意思，而是应该根据自己所学的只是猜测题目需要求什么。
所涉及到的知识都是比较简单的奥数知识。
1、In 2004, 16 June falls on a Wednesday. On what day of the week will 16 June fal l in 2010?2、In a magic square the sum of the numbers in each row, in each diagonal and in e ach column are equal. In this magic square the value of x is:3、If half of a number is 30, then three-quarters of that number is____.4、The sum of the digits of the following product999×5555、Three positive integers have a sum of 28. The greatest possible product that these i ntegers can have is_____.6、Jack was trying to tessellate regular pentagons. He managed thefollowing figure.The size of angle .a. is______.7、If the area of the shaded region of the regular hexagon in the diagrambelow is 36 cm2, the area of the whole hexagon in cm2 is_____.8、In what follows, □ and Δ are different numbers.When 503 is divided by □ the remainder is20.When 503 is divided by Δ the remainder is 20.When 493 is divided by □ x Δ the remainder is_____.9、A lady, her brother, her son and her daughter (all related by birth) playedvolleyball. The worst player's twin (who is one of the four players) and thebest player are of opposite ***.The worst player and the best player are of the same age.Who cannot be the worst player(s)?A) brother only B) daughter onlyC) son and daughter onlyD) lady and daughter onlyE) lady only10、If you continue the given number pattern, in what row and in whatposition in that row will the number 320 be?1 -------------- row 12 3 -------------- row 24 5 6 -------------- row 37 8 9 10 -------------- row 4The answers are given in the order of row ; position.答案：1、Wednesday2、123、454、27(求数位上上的数字之和)5、28=9+9+10，因此答案为8106、36度7、1088、503-20=483 483=3×7×23=21×23，因此□ x Δ=483，因此此题余数是10.9、D10、25，20。
字母与数字奥数题
字母与数字奥数题一、字母奥数题1：1、题目：在一个等式中，ABCD × 9 = EFGHI，其中A、B、C、D、E、F、G、H和I分别代表0-9中的一个数字。
求A、B、C、D、E、F、G、H和I分别代表什么数字。
2、解答：观察这个等式，我们可以发现9是一个关键数字。
因为A乘以9得到I，而1乘以9得到9，2乘以9得到8，依此类推。
我们可以看到如果A不是1，那么乘积将会超过9。
因此，A必须是1。
因为E不可能是0，否则乘积将是0而不是10或以上。
因此，E 必须是一个大于0的数字。
由于A是1，所以E是9。
因此，我们可以推断出B是0，因为如果B是任何其他数字，乘积将大于9。
接下来，我们知道H乘以9得到8，所以H必须是8。
现在我们已经知道A=1，B=0，E=9和H=8。
接下来，我们可以推断出C是2，因为如果C 是任何其他数字，乘积将大于8且不是3位数。
接下来我们可以看到乘积为923，所以D必须是3。
现在我们已经知道A=1，B=0，C=2，E=9和H=8。
接下来我们可以推断出F是4，因为如果F是任何其他数字，乘积将大于8且不是4位数。
现在我们已经知道A=1，B=0，C=2，E=9，H=8和F=4。
接下来我们可以推断出G是5，因为如果G是任何其他数字，乘积将大于8且不是5位数。
现在我们已经知道A=1，B=0，C=2，E=9，H=8，F=4和G=5。
最后我们可以推断出I是6，因为如果I是任何其他数字，乘积将大于8且不是6位数。
现在我们已经知道A=1，B=0，C=2，D=3，E=9，H=8。
二、字母与数字奥数题2：1、题目：在字母竖式ABCD * E = FGHI中，A、B、C、D、E、F、G、H和I分别代表0-9中的一个数字。
求A、B、C、D、E、F、G、H 和I分别代表什么数字。
2、解答：首先观察这个字母竖式，我们可以发现E乘以D的位置是CD，因此E和D不能是0，而且E乘以D的结果是个位数，所以E也不能是1。
英文奥数题
英文奥数题1. An ant covers a distance of 90 metres in 3 hours. The average speed of the ant in decimetres per minute is____.2. The Ancient Romans used the following different numerals in thei number system:They used these numerals to make up numbers as follows:So, for example, XCIX is 99.What is the value of XLVI?3. In a certain town some people were affected by a ’flu’ epidemic. In the first month 20% of the populationcontracted the flu whilst 80% were healthy.I n the following month 20% of the sick people recovered and 20% ofthe healthy people contracted the disease. What fraction of the population is healthy at the end of the secondmonth?4. Mpho, Barry, Sipho, Erica and Fatima are sitting on a park bench.Mpho is not sitting on the far right. Barry is not sitting on the far left.Sipho is not sitting at either end. Erica is sitting to the right ofBarry,but not necessarily next to him. Fatima is not sitting next to Sipho.Sipho is not sitting next to Barry.Who is sitting at the far right?5. Of the 28 T?shirts in a drawer, six are red, five are blue, and the rest arewhite. If Bob selects T?shirts at random whilst packing for a holiday, what is the least number he must remove from the drawer to be sure thathe has three T?shirts of the same colour?6. In an alien language, jalez borg farn means “good maths skills”. Nurf klar borg means“maths in harmony”and darko klar farn means “good in gold”.What is “harmony gold” in this language?7. Five children, Amelia, Bongani, Charles, Devine and Edwina, were in the classroom when one of them broke awindow. The teacher asked each of them to make a statement about the event, knowing that three of them alwayslie and two always tell the truth. Their statements were as follows:Amelia: “Charles did not break it, nor didDevine.”Bongani: “I didn’t break it, nor did Devine.”Charles: “I didn’t break it, but Edwina did.”Devine: “Amelia or Edwina broke it.”Edwina: “Charles broke it.”Who broke the window?8. Did you know? A palindrome is a number which reads the same forwards as backwards e.g. 35453. Next year2002 is an example of a palin dromic number. What is the difference between 2002 and the number of the previouspalindromic year?9、If a Åb = (2×a)+(3×b) then the value of 2 Å(3 Å5) is____.9. Two ants start at point A and walk at the same pace. One ant walks around a 3 cm by 3 cm square whilst theother walks around a 6 cm by 3 cmrectangle. What is the minimum distance, in centimetres, any one must coverbefore they meet again?参考答案：1. 单位换算，注意单词，90×10÷(3×60)=52. 463. 0.684、Erica5、抽屉原则76、注意一一对应，borg= maths, farn=good, jalez=skills. Klar=in, Nurf=harmony, darko=gold. 答案是Nurf darko.7、Charles8、10019、67第9题图：试题二1. Did you know? In the decimal number system (base 10) ten different digits, 0 to 9, are used to write allthe numbers. In the binary number system (base 2) two different digits are used, i.e. 0 and 1.Which one of the following numbers is not a valid number in theoctal number system (base 8)?A) 128 B) 127 C) 126 D) 125 E) 1242. The number of diagonals that can be drawn in a regular polygon withtwenty sides (icosagon) is_____.3. If a and b are integers, 10Ä3=1,152Ä7=3, and then 379Ä6 is equal to_____.4. Two numbers are in the ratio 2 : 3. When 4 is added to each number the ratio changes to 5 : 7.The sum of thetwo original numbers is____.5. The greatest number of Mondays which can occur in 45 consecutivedays is____6. Saul plays a video game in which he scores 4 for a hit and lost 6 for a miss. After 20 rounds his score is30. The number of times he has missed is____.7. Three girls A, B and C run in a 100 m race. When A finishes, B is 10 mbehind A and when B finishes C is 20 m behind B. How far in metres was C from A when Afinished?(Let’s assumeall the athletes run at a constant speed)8. The areas of the faces of a rectangular box are 84 cm2 , 70 cm2and 30 cm2.The volume of the box in cm3is____.9. You have 3 weights: 1 kg, 3 kg and 9 kg as well as an equal arm balance, as shown. How many different weightobjects can you weigh with these three?[Remember the weights may be placed on either side]参考答案：1. A 考察我们学过的简单的进制问题，显然8进制中没有8出现2. 170 找规律，公式为n×(n-3)÷23. 1 定义新运算，就是求379÷6的余数。
二年级英语奥数题 100道(含答案)
二年级英语奥数题 100道(含答案)第一部分：语言运用1. —How many ____ do you see?—I see four.- birds- dogs- pens ✅- books2. —Excuse me, is this your _____?—No, it isn't.- book- ruler ✅- backpack- pencil3. ______ is my favorite color.- Blue ✅- Red- Yellow- Green...第二部分：数学题1. If you have 23 candies and you give 10 candies to your friend, how many candies do you have left?- 10- 13 ✅- 23- 332. What is the value of the digit 5 in the number 859?- 5- 50- 500 ✅- 50003. What is 33 + 48?- 81 ✅- 84- 88- 90...第三部分：写作题请根据下面的图片写一篇作文，介绍你最喜欢的季节。
（文章至少50个字）Sample Answer我最喜欢的季节是春天。
春天里，天气非常好，温度适宜，阳光明媚。
万物开始复苏，花儿开了，草儿绿了。
在春天里，我可以与朋友一起去公园里野餐、玩耍，也可以在户外尽情畅玩。
春天是开始新生活的季节，我热爱春天！结语以上是二年级的英语奥数题，希望你们都做得很好哦！。
imo英文试题及答案
imo英文试题及答案IMO英文试题及答案1. 单选题：选择下列句子中语法正确的一项。
A. She doesn't like to play football.B. She don't like to play football.C. She doesn't likes to play football.D. She don't likes to play football.答案：A2. 填空题：根据语境填写合适的单词。
The teacher asked the students to ________ their names on the blackboard.A. writeB. wroteC. writesD. writing答案：A3. 阅读理解：阅读以下短文，然后回答问题。
Once upon a time, there was a little girl named Lucy. She loved to play in the garden with her dog, Max. One day, Max disappeared. Lucy was very sad and searched everywhere for him. Finally, she found Max in the neighbor's yard. He was playing with a kitten. Lucy was happy to see her dog again.问题：What did Lucy find in the neighbor's yard?A. A kittenB. Her dog, MaxC. A gardenD. A toy答案：B4. 完形填空：阅读短文，从每题所给的选项中选择一个最佳答案填入空白处。
Tim was a hardworking student. He always finished his homework on time. One day, he was feeling sick, but he still went to school. He was worried that if he didn't go to school, he would miss the important lesson. During the class, he felt even worse. Finally, he decided to go to the nurse's office. The nurse examined him and told him to go home and rest.46. Tim was feeling ________.A. happyB. sickC. boredD. excited47. What did Tim do when he felt sick?A. He stayed at home.B. He went to school.C. He played with friends.D. He went to the doctor.答案：46. B47. B5. 翻译题：将下列句子从英文翻译成中文。
英文奥数题
1、What is the square of the rectangle, which its perimeter is 30cm and the length is 2 times as itswidth.（和倍问题）一个长方形，周长（perimeter）是30厘米，长（length）是宽（width）的2倍，求这个长方形的面积。
2、Jia has 80yuan less than Yi .The money of Yi is 5 times more than Jia .Do you know the numberof their money?（和倍问题）甲的钱比乙的钱少80元，乙的钱比甲的钱多5倍。
甲、乙两人各有多少元？3、A number has 6 digits, it can be divided exactly by 9.Its primacy digit is 5 and all the digits aredifferent, so what’s the minimum number?（整除问题）digit：数字primacy：最早的，首个minimum：最小的3、The area of the quadrangular field is 52 hectares, the two catercorners divided it into 4small triangles. The areas of the two small triangles is 6 hectares and 7 hectares, so how about the area of the bigger triangles?（风筝定理）4、The circumference of parallelogram ABCD is 75 centimeters. When we take BC as base, theheight is 14cm and when we take CD as base, the height is 16cm.Find out the area ofparallelogram ABCD.（求面积）词汇表：面积：area 中点：midpoint四边形的：quadrangular 田地：field公顷：hectare 对角线：catercorners分成：divide into 三角形：triangle梯形：trapezoid 平行四边形：parallelogram周长：circumference 厘米：centimeters底：base 高：height5、A ship that leaks has been into some water and water ran into the boat in uniform speed. If 10 people dip water, it needs three hours to finish and if five people to do, it needs 8 hours. If we ask for people for 2 hours to finish. How many people we need? （牛吃草问题）牧场：pasture漏：leak船：boat匀速：uniform speed淘水：dip water完成：finish要求：ask for6、Calculate the value of the following formula and the answer to retain integer.33．3332-3.1415926÷0.618估算：estimating比较：compare计算：calculate值：value保留整数：retain integer整数：integral7、The sum of five consecutive odd numbers is1985, what’s the smallest one?（高斯求和）8、A natural number minuses 45 is a perfect square number, and it pluses 54 is also a perfect square number. Find out this natural number.（平方差公式）9、Four groups of young pioneers picked up seeds. The mean of A ,B,C is 24kg.The mean ofB,C,D is 26kg. The number of group D is 28kg. Do you know how many had group.A pick up? （平均数）10、There are two baskets of apples A and B.A is 19 kilograms more than B. How many kilograms We need to take out from A basket to B basket to make the apples in B basket three kilograms more than A?（和倍问题）11、A car spent six hours from A to B .It quiked 8 km per hour when it returned .As a result,it took an hour less. What’s the distance between A and B ?（行程问题）12、The ratio of A and B is 8:5.A and B are both reducing by 34,then A is two times than of B. Find out the two numbers.（比例应用题）比：ratio比例：proportion减少：reduce by杯子：glass商场：market13、The waterway is 100 kilometers from A to B。
PETS“英文奥数题”的刺激
PETS“英文奥数题”的刺激PETS“英文奥数题”的刺激PETS“英文奥数题”的刺激英文奥数题的刺激这个我怎么背？这个我怎么会？这个我怎么考啊？面对pets三级教程，正在北大附中上初中的胡月，冲着母亲烦躁地几乎吼叫起来。
在母亲张女士眼中，胡月是个争气的孩子，在小学毕业前便考过了pets二级，对于女儿的烦躁她也早就习以为常。
没背，怎知背不下？没学，怎知学不会？没考，又怎知考不过呢？张女士耐心地打着太极。
小升初时，凭借着手中的pets二级证、三好学生证、书法获奖证、滑冰、跳舞、弹琴胡月首轮即获北大附中承办的北达资源中学免考资格。
最终，在每年交纳14000元钱后，胡月得偿所愿，并成为亲朋眼中的骄子。
如今，刚上初二的胡月又踏上了pets三级考试(相当于本科毕业英语水平)征程。
不重视英语不行。
崔方的家长告诉记者，崔方2004年在参加人大附中小升初考试时，碰到了一道英文奥数题。
又是英文又是奥数。
这让崔方的家长印象深刻。
如今，崔方舞蹈过了4级，单簧管过了9级，外语也已经考过了pets二级。
去年，章东在育新小学上六年级时开始学习pets，并考过了一级b，今年来新东方继续攻战二级。
我们学pets真的是被迫无奈。
章东的母亲说。
因为没有pets二级证书，所以章东去年在第一轮筛选中就被人大附中淘汰出局，用章妈妈的话说，尽管儿子学习很优秀，但我们连参加人家入学考试的资格都没有。
只好上了一所普通中学。
培训费比择校费便宜培训费虽贵，总比择校费便宜得多。
小升初时，一级就算特长生，二级就可以免试直升特长班了。
9月10日，一名在考场外等候的家长一语道破pets火爆的原因之一，孩子如果考级成功，能省下几万块钱择校费；就算考不上，英语多学点也不会有坏处。
我不想去学，太累了，连玩的时间也没有了，可我妈已经交了500元的培训费，我没办法了！朝阳区针织路小学二年级只有9岁的学生元元，谈起pets后便低下了头。
记者在一家培训学校看到，pets考试有两种培训班，一种是10个月的pets课程，一种是为期一周左右的pets辅导课。
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英文奥数题
把正确答案填到横线上。
1. Subtract 10 from 40 is _________
2. What is the total of 13 and 5 ? ________
3. How much less is 12 than 20? _________
4. What is ½ of 8? _________
5. What is next in the pattern?12 10 8 6 ?___________
6. How much more is 15 than 9? ___________
7. Add 3, 4, and 1 ___________
8. Take 12 from 18 ___________
9. Share 10 between 2 ___________
10. Subtract 10 from 13 ____________
11. Divide 9 by 3 ____________
12.Which is smaller 36 or 63? ___________
13.Put these into order – smallest to biggest 21 12 9 19 10 _______________________________
14.What is 14 – 11? _______________
15.Subtract 10 from 13 _______________
16.What fraction is not shaded? ________________
17.How many minutes in an hour? _______________
18.What is 3 + 11? _______________
19.Continue the pattern 21 – 19 – 17 – 15 - ? ________________
20.M T W Th F Sa Su
1 2 3 4 5
6 7 8 9 10 11 12 What day will it be on the 20th? _________________
21.Double 5 ____________
22.How many pennies in a pound? ________________
23.What fraction is not blue? __________________
24.What is one more and less of 17? _______________
25.How many days are in a week? _______________
26. 4. John is 10 – Ann is 2 years younger , and Don is 1 year older than she is. How old is Don? ______________
27.Which are odd? 26 21 25 24 20 ________________
28.Which are even? 18 19 11 28 _____________
29.Draw an arrow facing West _________________
30.Draw a rectangle ________________
31.What is the difference between 8 and 15? ______________
32.How many sides has a hexagon? _____________
33.How many sides has an octagon? ________________
34.How much less is 14 than 19? _______________
35.How much more than 3 is 20? _______________
36.I have no corners and 2 faces? _______________
37. Which is missing?________
38. I have 3 corners and 3 sides _____________
39. How seconds in ½ a minute? _____________
40I am 3D shape with 2 edges, 2 circular faces and one other. Draw me __________________
41Draw a line of symmetry on this shape
42. Which numbers are missing? ____________
43. Jane is half as old as me. I am 22 ，how old is Jane? _________________
答案：1~5 30 18 8 4 4 6~10 6 8 6 5 3 11~15 3 36 (9,10,12,19,21) 3 3 16~20 1/4 60 14 13 Monday 21~25 10 100 1/3 16 and 18 7
26~30 9 (21 25) (18 28) or 31~35 7 6 8 5 17
36~40 31 30 41~43 (56 and 65) 11。