美国数学题2
VCE考试数学2007试题2
Reading time: 11.45 am to 12.00 noon (15 minutes) Writing time: 12.00 noon to 2.00 pm (2 hours)
QUESTION AND ANSWER BOOK
Structure of book
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SECTION 1 – continued TURN OVER
2007 MATHMETH(CAS) EXAM 2
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Question 7 The random variable X has a normal distribution with mean 11 and standard deviation 0.25. If the random variable Z has the standard normal distribution, then the probability that X is less than 10.5 is equal to A. Pr (Z > 2) B. Pr (Z < –1.5) C. Pr (Z < 1) D. Pr (Z ≥ 1.5) E. Pr (Z < – 4) Question 8 Which one of the following is not true about the function f : R → R, f (x) = |2x + 4|? A. The graph of f is continuous everywhere. B. The graph of f ′ is continuous everywhere. C. f (x) ≥ 0 for all values of x D. f ′ (x) = 2 for all x > 0 E. f ′ (x) = –2 for all x < –2 Question 9
SAT-II-Math(美国SAT数学几何题目和答案)
10.Geometry1.In a circle with center at O, arc ST measures 110°. What is the measure of angle STO?2.If the angles of a triangle are in the ratio of 2:3:5, what is the measure of the smallest angle?3.In the figure x.4.The diagonal of a rectangle is 26 cm and its height is 10 cm. Find the area of the rectangle in squarecentimeters.5. A ship travels 60 mi north, 90 mi west, and then 60 mi north again. How many miles is it fromits starting point?6.Find the length in inches of a tangent drawn to a circle with a 10 in. radius from a point 26 in.from the center of the circle.7.If the largest possible circular disc is cut from a rectangular piece of tin 8 in. by 12 in., what isthe area of the waste tin in square inches in terms of π?8. A circle is inscribed in a square. What is the ratio of the area of the square to that of the circle interms of π?9.At 4:20 PM., how many degrees has the hour hand of a clock moved since noon?10.Find the volume of a cube in cubic centimeters if the total surface area of its faces is 150 sq cm.11.A cylindrical can has a circular base with a diameter of 14 in. and a height of 9 in. Approxi-12.What is the volume in cubic inches of an open box made by snipping squares 2 in. by 2 in. fromthe corners of a sheet of metal 8 in. by 11 in. and then folding up the sides?13.Water 6 in. high in a fish tank 15 in. long by 8 in. wide is poured into a tank 20 in. long by 12 in.wide. What height in inches does it reach in the larger tank?14.A 6 ft pole is casting a 5 ft shadow at the same time that a flagpole is casting a 22 ft shadow. Howmany feet high is the flagpole?15.In the figure, PQRST is a regular pentagon inscribed in the circle. andthe pentagon, forming an angle of x° as shown. What does x equal?16.A treasure is buried 10 ft from tree T and 12 ft from a straight fence F. If T is 20 ft from F, in howmany places may the treasure be buried?17.If a given statement is true, which of the following statements must also be true?(A)the converse of the statement(B)the inverse of the statement(C)the contrapositive of the statement(D)the negative of the statement(E)none of these18.Given point P on a line. In a given plane containing the line, what is the total number of pointsthat are at a distance of 4 units from P and also at a distance of 3 units from the given line? 19.Point Q is 20 cm from plane P in space. What is the locus of points 8 cm from P and 12 cm frompoint Q?20.The sum of the measures of the interior angles of a convex polygon is 720°. What is the sum of themeasures of the interior angles of a second convex polygon that has two more sides than the first?10.GEOMETRY1.The correct answer is (35º).Let PT be a diameter of Circle O.2.The correct answer is (36º). Let 2x = smallest angle in degrees, 3x and 5x = other two anglesin degrees.3.The correct answer is (120º).m (alernate interior angles), and mAngle x is an exterior angle of ∆DEF and isequal to the sum of the measures of the two remote interior angles.x= 50º + 70º = 120º5.The correct answer is (150).The tangent is perpendicular to the radiusat the point of tangency.In right triangle OTP7.The correct answer is (96 - 16π).8.The correct answer isLet radius of circle be 1.Side of square is 1 + 1 =2.9.The correct answer is (130º).From noon to 4:00 PM., the hour hand has moved from 12 to 4: of360º = 120º. In the next 20 minutes it moves of the distance from 4:00 to 5:00: of30º = 10º.120º + 10º = 130º10.The correct answer is (125).11.The correct answer is (6).12.The correct answer is (56).13.The correct answer is (3).14.The correct answer isCross-multiply.15.The correct answer is (108°).16.The correct answer is (2).The locus of points 10 ft. from T is acircle of radius 10. The locus of points12 ft from F consists of two parallellines 12 ft from F on each side. Thecircle and one parallel line intersect intwo points.17.The correct answer is (C). The only statement that has the same truth value as the given statement isthe contrapositive of the original statement. This is the converse of the inverse of the original statement.18.The correct answer is (4).The locus of points 4 units from P is a circleof radius 4 with center at P. The locusof points 3 units from the given line consistsof two parallel lines 3 units from theline. The two parallel lines intersect thecircle in 4 points.19.The correct answer is (1).The locus of points 8 cm from P consistsof two planes parallel to P and 8 cm fromit. The locus of points 12 cm from Q is asphere of radius 12 and center at Q. Thesphere intersects one of the parallel planesin one point (point of tangency).20.The correct answer is (1080°). The sum of the interior angles of an n-sided polygon is (n – 2) 180°= 720°. Divide by 180.The second polygon has 6 + 2 = 8 sides.Sum of the measures of its interior angles = 180(8 – 2)= 180(6) = 1080°。
数学思维【初中】：2018-2019年美国“大联盟”思维探索七和八、九年级试卷(2份,含参考答案)
2018-2019年度美国“大联盟”(Math League)思维探索活动第一阶段(七年级)(活动日期：2018年11月25日，答题时间：90分钟，总分：200分)学生诚信协议：答题期间，我确定没有就所涉及的问题或结论，与任何人、用任何方式交流或讨论，我确定我所填写的答案均为我个人独立完成的成果，否则愿接受本次成绩无效的处罚。
请在装订线内签名表示你同意遵守以上规定。
考前注意事项：1. 本试卷是七年级试卷，请确保和你的参赛年级一致；2. 本试卷共4页(正反面都有试题)，请检查是否有空白页，页数是否齐全；3. 请确保你已经拿到以下材料：本试卷(共4页，正反面都有试题)、答题卡、答题卡使用说明、英文词汇手册、草稿纸。
试卷、答题卡、答题卡使用说明、草稿纸均不能带走，请留在原地。
4. 本试卷题目很多也很难，期待一名学生所有题目全部答对是不现实的，能够答对一半题目的学生就应该受到表扬和鼓励。
选择题：每小题5分，答对加5分，答错不扣分，共200分，答案请填涂在答题卡上。
1.(4 × 6 × 8 × 10) ÷ (6 × 8 × 10) =A) 3 B) 4 C) 12 D) 3 × 6 × 8 × 102.(2 ÷ 3) rounded to the nearest hundredth isA) 0.33 B) 0.66 C) 0.67 D) 0.703.Baby Amy is one day older than Baby Barry. The product of theirages measured in days could beA) 33 B) 132 C) 245 D) 2464.(The largest even divisor of 200) ÷ (the largest odd divisor of 200) =A) 4 B) 8 C) 20 D) 2005.An equilateral triangle with integer side-lengths has a perimeter that is numerically equalto the area of a square. Which of the following could be the length of a side of the square?A)12 B)10 C) 8 D) 46.I have only nickels, dimes, and quarters to pay for my dinner, which costs $12.60. Thesmallest number of coins I can use to pay isA) 51 B) 52 C) 54 D) 557.The smallest prime factor of 2019 isA) 1 B) 3 C) 19 D) 6738.The product of four consecutive integers must be divisible by each of the following exceptA) 4 B) 6 C) 10 D) 129.There are ? hours in 4 weeks.A) 48 B) 96 C) 336 D) 67210.If I divide my favorite number by its reciprocal, the quotient is 10 times as large as myfavorite number. My favorite number isA)110B)15C)12D) 1011.The height of the smoke from my barbecue is 100 000 cm, which is thesame as ? km.A) 1 B) 10C) 100 D) 100012.If the degree measures of the angles of a triangle are in a 4:5:6 ratio, whatis the difference between the measures of the largest and the smallest angles?A) 12°B) 24°C) 30°D) 36°13.The population of a town started at 1000, then went up 10%, then down 20%, then backup 10%. The population of the town ended atA) 968 B) 972 C) 1000 D) 102414.In my orchard, there are 60 more apples than oranges, and 5times as many apples as oranges. How many apples are there?A) 50 B) 75 C) 100 D) 12515.A polygon in which every pair of angles is supplementary must be aA) triangle B) squareC) rectangle D) hexagon16.Which of the following is smallest in value?A) 2600B) 3500C) 4400D) 530017.(2100 × 450) ÷ 2 =A) 275B) 2100C) 2149D) 219918.What is the remainder when 3333 is divided by 10?A) 1 B) 3 C) 7 D) 919.On a series of tests, Gus got 100 once, 90 twice, and 80 five times. What was his averagescore for all of the tests?A) 80 B) 85 C) 90 D) 9220.The product of the thousands and tenths digits of 1234.5678 isA) 5 B) 10 C) 35 D) 40第1页，共4页第2页，共4页21. The probability of heads then tails then heads on 3 tosses of a coin isA) 0.125B) 0.25C) 0.375D) 0.522. On January 1 last year, Rui got a jar of jellybeans. On each day he ate the same number ofjellybeans. He counted 560 on January 31 before eating any and he counted 380 on March 17 before eating any. There were ? jellybeans in the jar when Rui got it.A) 600 B) 650 C) 680D) 74023. Jake used 120 boxes of tissues in 3 days! There are 144 tissues per box. That’s ? tissues per minute!A) 2B) 3C) 4D) 524. The number 5184 has ? positive odd divisors.A) 1B) 2C) 4D) 525. The sum of 5 consecutive even integers could beA) 120B) 125C) 164D) 21226. Jacques, who paints only smiley faces, signs and numbers each of his paintings. If he started with Smiley #1 and has painted through Smiley #111, how many times has he used the digit 1 in his numbering?A) 12B) 22C) 24D) 3627. How many whole numbers have squares that are between 2 and 200?A) 12B) 13C) 24D) 2628. A baker cuts circular cookies out of a flat rectangle of cookie dough. If the rectangle is2 m by 1 m, and the cookies have radius 10 cm, at most how many cookies can the baker cut from the sheet of dough?A) 50B) 63C) 64D) 20029. 0.02% of 20% of ? = 200% of 2000A) 1000 B) 100 000 C) 1 000 000D) 100 000 00030. A miner combines 1200 kg of ore that is on average 3% gold with 2400 kg of ore that is on average 6% gold. If the 100 kg containing the most gold of the 3600 kg is 40% gold, the remaining ore will be ? gold.A) 2%B) 3%C) 4%D) 5%31. Including face diagonals, the total number of diagonals of a cube isA) 12B) 14C) 16D) 2432. How many odd 3-digit integers greater than 500 are composed of 3 different non-zero digits?A) 154B) 175C) 185D) 20033. If I square all whole-number factors of 36 and multiply the resulting numbers, the product will be equal toA) 362B) 364C) 368D) 36934. When the four members of the Beaverton family carry a log, each has a 0.02 probability of tripping, and each probability is independent of the others. What is theprobability that they will carry the log without any of them tripping?A) 1 – (0.02)4 B) (0.98)4 C) (0.02)4D) 1 – (0.98)435. What is the largest prime factor of the product of all even numbers from 2 through 200?A) 47B) 97C) 199D) 201936. What is the sum of the solutions to |10 – 4x | = 5?A) 1.25 B) 3.75C) 5 D) 1037. If 2x × 42x × 83x = 2y , then y =A) 2x 3B) 6xC) 6x 3D) 14x38. Each time Alan falls asleep, he sleeps for exactly 8m minutes and then is awake for the next 4m minutes. If he falls asleep for the 1st time at 11 P.M. and wakes from his 6th time asleep at 4:06 A.M., then m = A) 4.25 B) 4.5 C) 5.125D) 6.37539. If x is a positive integer, the remainder when 2018x is divided by 10 could NOT beA) 4B) 6C) 8D) 040. If a + b = 8 and114a b+=, then ab = A) 2 B) 6 C) 12 D) 32第3页，共4页 第4页，共4页2018-2019年度美国“大联盟”(Math League)思维探索活动第一阶段(八、九年级)(活动日期：2018年11月25日，答题时间：90分钟，总分：200分)学生诚信协议：答题期间，我确定没有就所涉及的问题或结论，与任何人、用任何方式交流或讨论， 我确定我所填写的答案均为我个人独立完成的成果，否则愿接受本次成绩无效的处罚。
世界数学十大未解难题
世界数学十大未解难题（其中“一至七”为七大“千僖难题”；附录“希尔伯特23个问题里尚未解决的问题”）一：P（多项式算法）问题对NP（非多项式算法）问题在一个周六的晚上，你参加了一个盛大的晚会。
由于感到局促不安，你想知道这一大厅中是否有你已经认识的人。
你的主人向你提议说，你一定认识那位正在甜点盘附近角落的女士罗丝。
不费一秒钟，你就能向那里扫视，并且发现你的主人是正确的。
然而，如果没有这样的暗示，你就必须环顾整个大厅，一个个地审视每一个人，看是否有你认识的人。
生成问题的一个解通常比验证一个给定的解时间花费要多得多。
这是这种一般现象的一个例子。
与此类似的是，如果某人告诉你，数13，717，421可以写成两个较小的数的乘积，你可能不知道是否应该相信他，但是如果他告诉你它可以因子分解为3607乘上3803，那么你就可以用一个袖珍计算器容易验证这是对的。
不管我们编写程序是否灵巧，判定一个答案是可以很快利用内部知识来验证，还是没有这样的提示而需要花费大量时间来求解，被看作逻辑和计算机科学中最突出的问题之一。
它是斯蒂文·考克（StephenCook）于1971年陈述的。
二：霍奇(Hodge)猜想二十世纪的数学家们发现了研究复杂对象的形状的强有力的办法。
基本想法是问在怎样的程度上，我们可以把给定对象的形状通过把维数不断增加的简单几何营造块粘合在一起来形成。
这种技巧是变得如此有用，使得它可以用许多不同的方式来推广；最终导至一些强有力的工具，使数学家在对他们研究中所遇到的形形色色的对象进行分类时取得巨大的进展。
不幸的是，在这一推广中，程序的几何出发点变得模糊起来。
在某种意义下，必须加上某些没有任何几何解释的部件。
霍奇猜想断言，对于所谓射影代数簇这种特别完美的空间类型来说，称作霍奇闭链的部件实际上是称作代数闭链的几何部件的(有理线性)组合。
三：庞加莱(Poincare)猜想如果我们伸缩围绕一个苹果表面的橡皮带，那么我们可以既不扯断它，也不让它离开表面，使它慢慢移动收缩为一个点。
《美国数学二年级上学期数学期中试卷全攻略》
试卷 第1页 共9页《美国数学二年级上学期数学期中试卷全攻略》姓名：__________ 班级：__________ 学号：__________一、填空题 （共20题）1．12 比（ ）多 4。
2．一个数是 18，另一个数是它的一半，这两个数的和是（ ）。
3．小明有 8 颗糖，小红的糖数是小明的 3 倍，小红有（ ）颗糖。
4．15 连续减去 3，减（ ）次结果是 0。
5．有 16 个苹果，平均分给 4 个小朋友，每个小朋友分（ ）个。
6．20 以内的质数有（ ）。
7．比 10 大比 15 小的数有（ ）。
8．一个三角形有（ ）条边。
9．5+5+5+5+5=（ ）×（ ）。
10．18÷3=（ ），表示把 18 平均分成（ ）份，每份是（ ）。
11．钟面上有（ ）个大格，（ ）个小格。
12．1 时 =（ ）分。
13．一个两位数，个位上是 6，十位上是 1，这个数是（ ）。
14．最小的三位数是（ ），最大的两位数是（ ），它们相差（ ）。
15．3 个 4 相加的和是（ ），3 和 4 相加的和是（ ）。
16．12 是 4 的（ ）倍。
17．（ ）里最大能填几？（ ）×4＜20。
18．把 24 个苹果平均分成 6 份，每份是（ ）个。
19．5×（ ）=40。
20．10 个一是（ ），10 个十是（ ）。
二、选择题 （共20题）21．15÷3=5，读作（ ）。
A ．15 除 3 等于 5 B ．15 除以 3 等于 5C ．3 除以 15 等于 522．有 12 个苹果，每 3 个放一盘，可以放（ ）盘。
A ．4 B ．3 C ．5试卷 第2页 共9页23．2×6 和 6×2 的结果（ ）。
A ． 不一样B ． 一样C ． 无法确定24．一个数的 5 倍是 20，这个数是（ ）。
A ．4B ．5C ．1525．18÷6=3，表示把 18 平均分成 6 份，每份是 3，也可以表示（ ）。
全国中学生生物学联赛试题及答案及美国数学竞赛8年级(AMC8)真题(附答案)(电脑版)
全国中学生生物学联赛试题注意事项：1．所有试题使用2B铅笔在机读卡上作答；2．试题按学科分类，单选和多选题混排，多选题答案完全正确才可得分3．纸质试卷72题，电子试卷48题，共计l20题；4．答题时间120分钟。
一、细胞生物学、生物化学、微生物学16题1．癌细胞与正常细胞的不同之处在于A．癌细胞不能合成DNA B．癌细胞被锁定在细胞周期中的S期C．癌细胞能持续分裂尽管彼此紧密相接D．癌细胞始终处于细胞周期中的分裂期2．人的肌肉组织分为快缩纤维和慢缩纤维两种，快缩纤维负责剧烈运动如举重，短跑，易产生酸痛感觉；慢缩纤维负责慢跑，游泳等有氧运动。
以下关于慢缩纤维和快缩纤维的描述，哪个是正确的A．快缩纤维含有的线粒体多，有氧呼吸能产生大量乳酸和ATP供能B．慢缩纤维含有的线粒体多，有氧呼吸不产生乳酸，产生的ATP也少C．快缩纤维含有的线粒体少，主要依靠糖酵解产生ATP供能，因此产生大量乳酸D．慢缩纤维含有的线粒体多，主要依靠糖酵解产生ATP供能3．在光合作用中参与电子传递的载体是A．叶绿素B．磷酸烯醇式丙酮酸C．NADH D．NADPH4．肽链生物合成时，信号肽A．是线粒体的定位信号B．将新生肽链导入内质网C．控制蛋白质分子的最终构象D．处于肽链的C末端5．原核细胞的特征包括A．无核糖体B．无转录后修饰C．无有氧呼吸D．无染色体6．以下糖类中属于寡糖的有(多选2分)A．甘露糖B．麦芽糖C．半乳糖D．蔗糖E．糖原7．以下关于蛋白质变性后所产生现象的描述不正确的是：A．溶解度降低B．疏水侧链基团被包裹在分子内部C．形成沉淀D．失去结晶能力8．真菌中的半知菌是指。
A．没有光合作用B．菌丝没有横隔C．没有发现有性生殖阶段D．不能运动。
9．关于维生素A的生理功能，下面的描述中哪个是错误的?A．抑制胆碱酯酶活性B．构成视觉细胞感光物质的成分C．参与上皮组织细胞膜糖蛋白合成D．严重缺乏时导致夜盲症10．磷酸戊糖途径的发生部位在A．细胞质B．线粒体C．叶绿体D．细胞膜11．在C4植物循环中，CO2进入叶肉细胞被固定的最初产物是A．甘油酸-3-磷酸B．苹果酸C．草酰乙酸D．丙酮酸12．在等电聚焦电泳过程中，随着蛋白质样品的迁移，电流的变化为A．越变越大，当样品到达其等电点位置时，电流达到最大值B．越变越小，当样品到达其等电点位置时，电流达到最小值，接近于零C．基本不变，为一恒定值D．不确定13．利用酵母菌进行乙醇发酵时若通入氧气，会导致菌株对葡萄糖利用速度降低，甚至停止生成乙醇，这种现象称为A．呼吸抑制效应B．巴斯德效应C．分子氧效应D．葡萄糖效应14．巴斯德设计的曲颈瓶实验，可以(单选1分)A．证明酒是酵母菌发酵获得B．否定自然发生学说的论点C．说明培养基灭菌后的作用D．验证某细菌是不能培养的15．营养缺陷型菌株是指(单选1分)A．不需要添加生长因子可在基础培养基上生长的菌株B．不需要添加生长因子可在丰富培养基上生长的菌株C．因突变需提供某种营养才能在基础培养基上生长的菌株D．因自发或诱发突变而导致的可抵抗环境不良因素的菌株16．以下哪类化合物属于微生物的次生代谢产物(多选2分)A．脂肪B．色素C．抗生素D．蛋白质E．毒素二、植物和动物的解剖、生理、组织和器官18题17．草履虫、水螅、乌贼、蟾蜍受到刺激后，均可从体内发出一些物质以攻击或防御敌害，在他们身体上，发出这些物质的结构是A．刺丝泡、刺细胞、墨囊、耳后腺B．刺丝泡、刺丝囊、外套腔、唾液腺C．表膜泡、刺丝囊、墨囊、唾液腺D．表膜泡、刺细胞、外套腔、耳后腺18．在动物卵裂时期，由于不同动物受精卵内卵黄多少及其在卵内分布的不同，卵裂方式也有很大差异，海胆、沙蚕、昆虫、乌贼的卵裂方式依次分别为(单选1分) A．完全均等卵裂(等裂)、表面卵裂、螺旋形卵裂、盘裂B．螺旋形卵裂、完全均等卵裂(等裂)、盘裂、表面卵裂C．螺旋形卵裂、完全均等卵裂(等裂)、表面卵裂、盘裂D．完全均等卵裂(等裂)、螺旋形卵裂、表面卵裂、盘裂19．不同动物类群具有独特的特征，现存棘皮动物、海绵动物、哺乳动物、鸟类所特有的特征依次为A．水管系、水沟系、下颌为单一齿骨、羽毛B．后口、水沟系、胎生、飞翔C．后口、骨针、胎生、羽毛D．水管系、骨针、下颌为单一齿骨、飞翔20．节肢动物类群很多，不同类群的排泄器官亦有差异，节肢动物门甲壳纲动物的排泄器官有(多选l分)A．基节腺B．触角腺C．颚腺D．马氏管21．家鸽的一侧体动脉弓退化，雌家鸽的一侧卵巢和输卵管也退化了，退化的这些器官是(单选1分)A．左体动脉弓和右侧的卵巢、输卵管B．左体动脉弓和左侧的卵巢、输卵管C．右体动脉弓和左侧的卵巢、输卵管D．右体动脉弓和右侧的卵巢、输卵管22．在海滨潮间带经常可以见到石鳖和沙蚕，以下不属于它们共同特征的是A．以裂体腔法形成真体腔B．后肾型排泄系统C．具有担轮幼虫期D．开管式循环系统23．以下哪项不是文昌鱼的特征A．具有脊索，背神经管，鳃裂B．有分节的肌肉，有哈氏窝C．有头，有心脏D．有特化的口器24．一家饭店涉嫌出售野生鸟类，检查人员在检查时发现了一种鸟类的足，三趾向前一趾向后，后趾与前面三趾在同一平面上，趾长，基部有蹼相连，这种鸟类是A．鹈形目B．鹳形目C．雁形目D．鹤形目25．以下哪组元素在植物体内参与氧化还原反应(单选2分)A．钼镍铜铁B．铁铜镁钼C．钙镁镍铜D．锰镍镁铜26．盐胁迫条件下，较耐盐的禾本科植物大麦可以通过将盐分局域于以下部位来缓解盐分对植物生长造成的危害(多选l分)A．根系B．幼叶C．叶鞘D．老叶27．关于植物的种子，下列描述正确的是(多选2分)A．种子由胚珠发育而来B．种子表面均有种孔、种脐和种脊的结构C．种子中的胚乳多来源于受精后的中央细胞，也有来自于雌配子体的细胞D．胚是休眠的幼小孢子体E．无胚乳种子在发育过程中没有胚乳形成28．有关被子植物花的叙述，下列哪一个是错误的(单选2分)A．花是适应于繁殖功能的变态短枝B．花托、花萼和花冠被认为是叶的变态C．雄蕊和雌蕊也被认为是叶的变态D．花托、花被、雄蕊和雌蕊均有茎的顶端分生组织产生29．玉米干旱缺水时叶片的内卷主要是失水造成的A．叶肉细胞B．叶表皮的毛状体C．位于上表皮的泡状(运动)细胞D．位于下表皮的泡状(运动)细胞30．有关C4植物，以下说法中正确的是(多选2分)A．叶解剖结构中可观察到“花环结构”B．光合作用CO2的初步固定和同化在不同细胞中进行C．光合作用CO2的初步固定和同化在同一细胞中进行D．在一定范围的强光、高温条件下光合效率高31．心肌细胞有效不应期的长短主要取决于A．静息电位水平B．0期去极化的速度C．阈电位水平D．平台期的长短32．血液中CO2分压升高使呼吸运动加强的最主要途径是(单选2分)A．直接刺激脑桥的呼吸相关神经元B．直接刺激延髓呼吸中枢的神经元C．刺激中枢化学感受器D．刺激颈动脉体和主动脉体感受器33．当去甲肾上腺素与β受体结合时，下列哪一种肌肉收缩或收缩加强(单选1分) A．心室肌B．子宫平滑肌C．小肠平滑肌D．血管平滑肌E．支气管平滑肌34．下列哪种因素可引起人尿量的明显增加的(多选2分)A．献血200ml后B．饮用清水1000ml后C．静脉注射神经垂体激素D．饮用生理盐水100ml后三、动物行为学、生态学15题35．如果一项研究，专注于了解不同生态因子对生物的影响，及生物对它们的耐受，那么这个研究属于哪一层次上的研究A．个体生态学B．种群生态学C．群落生态学D．生态系统生态学36．关于高等动物种群中性别比例，下面论述中错误的是A．大多数种群倾向于使出生性比趋近于l：1 B．老年组往往雌性多于雄性C．出生的时候，往往雄性多于雌性D．种群性比与世代长度直接相关37．社会性寄生是指A．寄生在动物社会中是普遍现象B．寄生只发生在特定社会等级的动物中C．社会性昆虫中发生的寄生行为D．强迫寄主动物为其提供食物或其他利益38．关于外来物种，以下论述错误的是A．所有的外来物种都是入侵种，都是有害的B．外来物种可以依靠风、鸟、昆虫等自然因素入侵C．有些外来物种是人类有意引入的D．入侵物种可能对生态系统造成长久的破坏39．适合度是指A．动物单一行为的适应性B．动物调整自己的行为以适合于生活在当时的环境C．动物的总体繁殖成功性D．最适合动物生活习性、满足营养需求的食物40．以下哪种情况不属于动物的行为节律A．候鸟随季节的迁徙B．哺乳动物晨昏活动习性C．细菌生长速度随营养物浓度起落而快慢变化D．招潮蟹的活动随潮汐变化而变化41．动物的生长和发育是需要一定温度的，下列哪个说法是正确的(单选2分) A．外界温度的高低直接决定了动物机体的体温，进而影响其生长发育B．当外界温度低于某一温度时，昆虫就停止生长发育，这一温度阈值称为发育起点温度C．动物的发育速度总是随环境温度的增高而加快的D．昆虫发育的有效积温是发育历期乘以发育期的平均温度，然后求和42．下列有关水生群落演替的说法中哪个是错误的A．水生群落的演替一般会依次经历裸底期、浮水植物期、沉水植物期、挺水植物期、湿生草本植物期等阶段B．在这一演替过程中池底逐渐变浅，最终向陆地变化C．挺水植物根系往往较发达，可以使水底迅速增高D．浮水植物的叶子漂浮在水面，影响到水下光照，不利于沉水植物生长43．关于固定行为型，下述论述正确的是(多选2分)A．固定行为型被特定的外部刺激所释放B．每一个物种都有物种特异的固定行为型C．固定行为型一旦释放就会持续到底D．固定行为型是一种先天行为44．在动物行为学研究中，严格定义行为类型是研究工作的基础。
AMC10美国数学竞赛讲义全
AMC10美国数学竞赛讲义全AMC 中的数论问题1：Remember the prime between 1 to 100：2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 91 2：Perfect number:Let is the prime number.if21p - is also the prime number. then1(21)2p p --is the perfect number. For example:6,28,496.3: Let ,0n abc a =≠ is three digital integer .if 333n a b c =++Then the number n is called Daffodils number . There are only four numbers: 153 370 371 407 Let ,0n abcd a =≠ is four digital integer .if 4444d n a b c +=++Then the number n is called Roses number . There are only three numbers: 1634 8208 94744：The Fundamental Theorem of ArithmeticEvery natural number can be written as a product of primes uniquely up to order.5：Suppose that a and b are integers with b =0. Then there exists unique integers q a d such that 0 ≤ < |b| a d a = bq + .6：(1)G eatest Commo D v so : Let gcd (a, b) = max {d ∈ Z: d | a a d d | b}. For any integers a and b, we havegcd(a, b) = gcd(b, a) = gcd(±a, ±b) = gcd(a, b ? a) = gcd(a, b + a). For example: gcd(150, 60) = gcd(60, 30) = gcd(30, 0) = 30 (2)Least commo mult le:Let lcm(a,b)=m {d∈Z: a | d a d b | d }. (3)We have that: ab= gcd(a, b) lcm(a,b) 7：Congruence modulo nIf ,0a b mq m -=≠,then we call a congruence b modulo m and we rewritemod a b m ≡.(1)Assume a b c d m Z 0 m 0 .If a b mod m c d mod m then we havemod a c b d m ±≡± , mod ac bd m ≡ , mod k k a b m ≡(2) The equat o ax ≡ b (mod m) has a solut o f a d o ly f gcd(a, m) d v des b.8：How to find the unit digit of some special integers (1)How many zero at the end of !nFor example, when 100n =, Let N be the number zero at the end of 100!then10010010020424525125N=++=+=(2) ,,a n Z ∈Find the unit digit n a . For example, when 100,3n a ==9：Palindrome, such as 83438, is a number that remains the same when its digits are reversed.There are some number not only palindrome but 2 2 ，222 ， (1)Some special palindrome n that 2n is also palindrome. For example :222221111121111123211111123432111111111112345678987654321=====(2)How to create a palindrome? Almost integer plus the number of its reverseddigits and repeat it again and again. Then we get a palindrome. For example: 87781651655617267266271353135335314884+=+=+=+=But whether any integer has this Property has yet to prove(3) The palindrome equation means that equation from left to right and right to left it all set up. For example ：1242242112231132211121241388888831421211====Let ab and cde are two digital and three digital integers. If the digits satisfy the,,9a c b e d c e d ?=?=+≤, then ab cde edc ba ?=?.10: Features of an integer divisible by some prime number If n is even ，then 2|n⼀个整数n 的所有位数上的数字之和是3（或者9）的倍数，则n 被3（或者9）整除⼀个整数n 的尾数是零，则n 被5整除⼀个整数n 的后三位与截取后三位的数值的差被7、11、13整除，则n 被7、11、13整除⼀个整数n 的最后两位数被4整除，则n 被4整除⼀个整数n 的最后三位数被8整除，则n 被8整除⼀个整数n 的奇数位之和与偶数位之和的差被11整除，则n 被11整除 11. The number Theoretic functions If 312123t rr rrt n p p p p =(1) {}12()#0:|(1)(1)(1)t n a a n r r r χ=>=+++(2) 12222111222|()(1)(1)(1)t r r r t t t a nn a p pp p p p p p p δ==+++++++++∑(3) {}11221111122()#:,gcd(,)1()()()t t r r rr rr t t n a N a n a n p p p p p p φ---=∈≤==---For example: 2(12)(23)(21)(11)6χχ=?=++= 22(12)(23)(122)(13)28δδ=?=+++= 22(12)(23)(22)(31)4φφ=?=--= Exercise1. The sums of three whole numbers taken in pairs are 12, 17, and 19. What is the middle number?(A) 4 (B) 5 (C) 6 (D) 7(E) 83. For the positive integer n, let denote the sum of all the positive divisors of n with the exception of n itself. For example,<4>=1+2=3 and <12>=1+2+3+4+6=16. What is <<<6>>>?(A) 6 (B) 12 (C) 24 (D) 32 (E) 36 8. What is the sum of all integer solutions to 21<(x-2)<25? (A) 10 (B) 12(C) 15(D) 19(E) 5(A) 6 (B) 7 (C) 8 (D) 9 (E) 10(A) 1(B) 2(C) 3(D) 4(E) 515．The figures 123,,F F F and 4F shown are the first in a sequence of figures. For3n ≥, n F is constructed from -1n F by surrounding it with a square and placing one more diamond on each side of the new square than -1n F had on each side of its outside square. For example, figure 3F has 13 diamonds. How many diamonds are there in figure 20F ?18. Positive integers a, b, and c are randomly and independently selected with replacement from the set {1, 2, 3,…, 2010}. What is the probability that abc ab a ++ is divisible by 3? (A)13(B)2981(C)3181(D)1127(E)132724. Let ,a b and c be positive integers with >>a b c such that 222-b -c +=2011a ab and222+3b +3c -3-2-2=-1997a ab ac bc . What is a ?(A) 249 (B) 250 (C) 251 (D) 252 (E)2535. In multiplying two positive integers a and b, Ron reversed the digits of the two-digit number a. His erroneous product was 161. What is the correct value of the product of a and b?(A) 116 (B) 161 (C) 204 (D) 214 (E) 224 23. What is the hundreds digit of 20112011?(A) 1 (B) 4 (C) 5 (D) 6 (E) 99. A palindrome, such as 83438, is a number that remains the same when its digits are reversed. The numbers x and x+32 are three-digit and four-digit palindromes, respectively. What is the sum of the digits of x?(A) 20 (B) 21 (C) 22 (D) 23 (E) 2421. The polynomial 322010x ax bx -+- has three positive integer zeros. What is the smallest possible value of a?(A) 78 (B) 88 (C) 98 (D) 108 (E) 11824. The number obtained from the last two nonzero digits of 90! Is equal to n. What is n?(A) 12 (B) 32 (C) 48 (D) 52 (E) 6825. Jim starts with a positive integer n and creates a sequence of numbers. Each successive number is obtained by subtracting the largest possible integer square less than or equal to the current number until zero is reached. For example, if Jim starts with n=55, then his sequence contain 5 numbers:55 55-72= 6 6-22= 2 2-12= 1 1-12= 0Let N be the smallest umbe fo wh ch J m’s seque ce has umbe s. What is the units digit of N?(A) 1 (B) 3 (C) 5 (D) 7 (E) 9 21．What is the remainder when 01220093+3+3++3is divided by 8?(A) 0 (B) 1 (C) 2 (D) 4 (E) 65．What is the sum of the digits of the square of 111,111,111? (A) 18 (B) 27(C) 45(D) 63(E) 81ABC 100643a S 2(A) 6 (B) 7(C) 8(D) 9(E) 1024. Let 2200820082k =+. What is the units digit of 222k +? (A) 0 (B) 1(C) 4(D) 6(E) 8AMC about algebraic problems⼀、Linear relations(1) Slope y-intercept form: y kx b =+ (k is the slope, b is the y-intercept) (2)Standard form: 0Ax By C ++= (3)Slope and one point 0000(,),()()P x y k slope y y k x x -=- (4) Two points 1122(,),(,)P x y P x y12121212y y y y y y x x x x x x ---==--- (5)x,y-intercept form: (,0),(0,),(0,0)1x yP a Q b a b a b≠≠+= ⼆、the relations of the two lines 111222:0,:0l A x B y C l A x B y C ++=++=(1) 1l ∥2l 122112210,0A B A B C B C B ?-=-≠ (1) 1l ⊥2l 12120A A B B ?-=三、Special multiplication rules:222223322332212211222112222()()()2()()()()()()(2)()((1)(1))(1)()n n n n n n n n n n n n n n a b a b a b a b a ab b a b a b a ab b a b a b a ab b a b a b a a b ab b n a b a b a a b ab b n is odd n a b c ab bc ac a b -----------=-+±=±+-=-+++=+-+-=-++++≥+=+-++-+->++=++?-22()()0b c c a a b c+-+-=?==四、quadratic equations and PolynomialThe quadratic equations 2(0)y ax bx c a =++≠ has two roots 12,x x then we has1212b c x x x x a a+=-=More generally, if the polynomial 121210nn n n n x a x a x a x a ---+++++= has nroots 123,,,,n x x x x ,then we have:1231122312123(1)n n n n n nx x x x a x x x x x x a x x x x a -++++=-++==-开⽅的开⽅、估计开⽅数的⼤⼩绝对值⽅程Arithmetic Sequence123(1)(2)(3)()n m a a n d a n d a n d a n m d =+-=+-=+-==+-121321()()()()2222n n n m n m n n a a n a a n a a n a a s ---+++++=====1(1)2n n n ds na -=+If n=2k, then we have 1()n k k s k a a +=+ If n=2k+1, then we have 1n k s na += Geometric sequence123123n n n n m n m a a q a q a q a q ----=====1(1)1,1n n a q q s q-≠=-Some special sequence , , 2, 3, 5, ,… 9,99,999,9999，… 1，11，111，1111，… Exercise4 .When Ringo places his marbles into bags with 6 marbles per bag, he has 4 marbles left over. When Paul does the same with his marbles, he has 3 marbles left over. Ringo and Paul pool their marbles and place them into as many bags as possible, with 6 marbles per bag. How many marbles will be left over?7 For a science project, Sammy observed a chipmunk and a squirrel stashing acorns in holes. The chipmunk hid 3 acorns in each of the holes it dug. The squirrel hid 4 acorns in each of the holes it dug. They each hid the same number of acorns, although the squirrel needed 4 fewer holes. How many acorns did the chipmunk hide?21. Four distinct points are arranged on a plane so that the segments connecting them have lengths ,,,,, and . What is the ratio of to ?13. An iterative average of the numbers 1, 2, 3, 4, and 5 is computed the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, and then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. What is the difference between the largest and smallest possible values that can be obtained using this procedure?16. Three runners start running simultaneously from the same point on a 500-meter circular track. They each run clockwise around the course maintaining constant speeds of 4.4, 4.8, and 5.0 meters per second. The runners stop once they are all together again somewhere on the circular course. How many seconds do the runners run?24. Let ,a b and c be positive integers with >>a b c such that 222-b -c +=2011a ab and222+3b +3c -3-2-2=-1997a ab ac bc . What is a ?(A) 249 (B) 250(C) 251(D) 252(E) 2531. What is246135135246++++-++++? (A) -1 (B) 536(C) 712(D)14760(E)43310. Consider the set of numbers {1, 10, 102, 103(010)}. The ratio of the largest element of the set to the sum of the other ten elements of the set is closest to which integer?(A) 1 (B) 9 (C) 10 (D) 11 (E) 101=(A) -64 (B) -24 (C) -9 (D) 24 (E) 5764. Let X and Y be the following sums of arithmetic sequences: X= 10 + 12 + + …+ 00. Y= 2 + + + …+ 02. What is the value of Y X -?(A) 92 (B) 98 (C) 100 (D) 102 (E) 1127. Which of the following equations does NOT have a solution?(A) 2(7)0x +=(B) -350x += 20=80= (E) -340x -=(A)(B)(C)2(D) (E) 613. What is the sum of all the solutions of 2602x x x =--?(A) 32 (B) 60 (C) 92 (D) 120 (E) 12414. The average of the numbers 1, 2, 3… 9 , 99, and x is 100x. What is x?(A)49101(B)50101(C)12 (D)51101(E)509911. The length of the interval of solutions of the inequality 23a x b ≤+≤ is 10.What is b-a?(A) 6 (B) 10 (C) 15 (D) 20 (E) 3013. Angelina drove at an average rate of 80 kph and then stopped 20 minutes for gas. After the stop, she drove at an average rate of 100 kph. Altogether she drove 250 km in a total trip time of 3 hours including the stop. Which equation could be used to solve for the time t in hours that she drove before her stop? (A) 880100()2503t t +-=(B) 80250t = (C) 100250t =(D) 90250t =(E) 880()1002503t t -+=21. The polynomial 32-2010x ax bx +- has three positive integer zeros. What is the smallest possible value of a?(A) 78 (B) 88 (C) 98 (D) 108 (E) 118 15．When a bucket is two-thirds full of water, the bucket and water weigh kilograms. When the bucket is one-half full of water the total weight is kilograms. In terms of and , what is the total weight in kilograms when the bucket is full of water?13．Suppose thatand. Which of the following is equal tofor every pair of integers16．Let ,,, and be real numbers with,, and. What is the sum of all possible values of5. Which of the following is equal to the product?81216442008............481242004n n +? (A) 251(B) 502(C) 1004(D) 2008 (E) 40167. The fraction 20082200622007220052(3)(3)(3)(3)-- simplifies to which of the following? (A) 1 (B) 9/4 (C) 3 (D) 9/2 (E) 913. Doug can paint a room in 5 hours. Dave can paint the same room in 7 hours. Doug and Dave paint the room together and take a one-hour break for lunch. Let t be the total time, in hours, required for them to complete the job working together, including lunch. Which of the following equations is satisfied by t ?(A) 11()(1)157t ++=(B) 11()1157t ++= (C) 11()157t +=(D) 11()(1)157t +-=(E) (57)1t +=15. Yesterday Han drove 1 hour longer than Ian at an average speed 5 miles per hour faster than Ian. Jan drove 2 hours longer than Ian at an average speed 10 miles per hour faster than Ian. Han drove 70 miles more than Ian. How many more miles did Jan drive than Ian?(A) 120 (B) 130 (C) 140 (D) 150 (E) 160AMC 中的⼏何问题⼀、三⾓形有关知识点1.三⾓形的简单性质与⼏个⾯积公式①三⾓形任何两边之和⼤于第三边；②三⾓形任何两边之差⼩于第三边；③三⾓形三个内⾓的和等于 0°；④三⾓形三个外⾓的和等于3 0°；⑤三⾓形⼀个外⾓等于和它不相邻的两个内⾓的和；⑥三⾓形⼀个外⾓⼤于任何⼀个和它不相邻的内⾓。
2021 -美国袋鼠数学一二年级竞赛练习题
SAMPLE QUESTION FOR 3 POINTS6. The pink tower is taller than the red tower but shorter than the green tower. The silver tower is taller than the green tower. Which tower is the tallest?(A) pink tower (B) green tower (C) red tower (D) silver tower(E) We don’t know.SAMPLE QUESTION FOR 4 POINTS15. The picture shows the five houses of five friends and their school. The school is the largest building in the picture. To go to school, Doris and Ali walk past Leo’s house. Eva walks past Chole’s house. Which is Eva’s house?(A) (B) (C) (D) (E) SAMPLE QUESTION FOR 5 POINTS18. Every time the witch has 3 apples, she turns them into 1 banana. Every time she has3 bananas, she turns them into 1 apple. What will she end up with if she starts with4 apples and5 bananas?(A) (B) (C) (D) (E)SAMPLE QUESTION FOR 3 POINTS6. The pink tower is taller than the red tower but shorter than the green tower. The silver tower is taller than the green tower. Which tower is the tallest?(A) pink tower (B) green tower (C) red tower (D) silver tower(E) We don’t know.SAMPLE QUESTION FOR 4 POINTS15. The picture shows the five houses of five friends and their school. The school is the largest building in the picture. To go to school, Doris and Ali walk past Leo’s house. Eva walks past Chole’s house. Which is Eva’s house?(A) (B)(C) (D) (E) SAMPLE QUESTION FOR 5 POINTS18. Every time the witch has 3 apples, she turns them into 1 banana. Every time she has3 bananas, she turns them into 1 apple. What will she end up with if she starts with4 apples and5 bananas?(A)(B) (C) (D) (E)SAMPLE QUESTION FOR 3 POINTS2. How many fish will have their heads pointing towards the ring when we straighten the line?(A) 3 (B) 5 (C) 6 (D) 7 (E) 8SAMPLE QUESTION FOR 4 POINTS15. On a tall building there are 4 fire escape ladders, as shown. The heightsof 3 ladders are at their tops. What is the height of the shortest ladder?(A) 12 (B) 14 (C) 16 (D) 20 (E) 22SAMPLE QUESTION FOR 5 POINTS23. Elena wants to write the numbers from 1 to 9 in the squares shown. The arrows always point from a smaller number to a larger one. She has already written 5 and 7. Which number should she write instead of the question mark?(A) 2 (B) 3 (C) 4 (D) 6 (E) 8SAMPLE QUESTION FOR 3 POINTS2. How many fish will have their heads pointing towards the ring when we straighten the line?(A) 3 (B) 5 (C) 6 (D) 7 (E) 8SAMPLE QUESTION FOR 4 POINTS15. On a tall building there are 4 fire escape ladders, as shown. The heightsof 3 ladders are at their tops. What is the height of the shortest ladder?(A) 12 (B) 14 (C) 16 (D) 20 (E) 22SAMPLE QUESTION FOR 5 POINTS23. Elena wants to write the numbers from 1 to 9 in the squares shown. The arrows always point from a smaller number to a larger one. She has already written 5 and 7. Which number should she write instead of the question mark?(A) 2 (B) 3 (C) 4 (D) 6 (E) 8SAMPLE QUESTION FOR 3 POINTS10. There is a square with line segments drawn inside it. The line segments are drawn either from the vertices or the midpoints of other line segments. We colored 18 of the large square. Which one is our coloring?(A) (B) (C) (D) (E)SAMPLE QUESTION FOR 4 POINTS13. Rosa wants to start at the arrow, follow the line, and get out at the other arrow. Which piece, if placed in the middle, cannot produce this? Note: The piece can be rotated.(A) (B) (C) (D) (E)SAMPLE QUESTION FOR 5 POINTS29. 10 elves and trolls each were given a token with a different number from 1 to 10 written on it. They were each asked what number was on their token and all answered with a number from 1 to 10. The sum of the answers was 36. Each troll told a lie and each elf told the truth. What is the smallest number of trolls there could be in the group?(A) 1 (B) 3 (C) 4 (D) 5 (E) 7SAMPLE QUESTION FOR 3 POINTS10. There is a square with line segments drawn inside it. The line segments are drawn either from the vertices or the midpoints of other line segments. We colored 18 of the large square. Which one is our coloring?(A) (B) (C) (D)(E)SAMPLE QUESTION FOR 4 POINTS13. Rosa wants to start at the arrow, follow the line, and get out at the other arrow. Which piece, if placed in the middle, cannot produce this? Note: The piece can be rotated.(A) (B) (C) (D) (E)SAMPLE QUESTION FOR 5 POINTS29. 10 elves and trolls each were given a token with a different number from 1 to 10 written on it. They were each asked what number was on their token and all answered with a number from 1 to 10. The sum of the answers was 36. Each troll told a lie and each elf told the truth. What is the smallest number of trolls there could be in the group?(A) 1 (B) 3 (C) 4 (D) 5 (E) 7SAMPLE QUESTION FOR 3 POINTS5. When the five pieces shown fit together correctly, the result is a rectangle with a calculationwritten on it. What is the result of this calculation?(A) –100 (B) –8 (C) –1 (D) 199 (E) 208SAMPLE QUESTION FOR 4 POINTS13. The area of the large square is 16 cm 2 and the area of each small square is 1 cm 2. What is the total area of the black flower?(A) 3 cm 2(B) 72 cm 2(C) 4 cm 2(D) 112 cm 2(E) 6 cm 2SAMPLE QUESTION FOR 5 POINTS26. 2021 colored kangaroos are arranged in a row and are numbered from 1 to 2021. Eachkangaroo is colored red, gray, or blue. Among any three consecutive kangaroos, there are always kangaroos of all three colors. Bruce guesses the colors of five kangaroos. These are his guesses: Kangaroo 2 is gray; Kangaroo 20 is blue; Kangaroo 202 is red; Kangaroo 1002 is blue; Kangaroo 2021 is gray. Only one of his guesses is wrong. What is the number of the kangaroo whose color he guessed incorrectly?(A) 2 (B) 20 (C) 202 (D) 1002 (E) 2021SAMPLE QUESTION FOR 3 POINTS5. When the five pieces shown fit together correctly, the result is a rectangle with a calculation written on it. What is the result of this calculation?(A) –100 (B) –8 (C) –1 (D) 199 (E) 208SAMPLE QUESTION FOR 4 POINTS13. The area of the large square is 16 cm2 and the area of each smallsquare is 1 cm2. What is the total area of the black flower?(A) 3 cm2(B) 72 cm2(C) 4 cm2(D) 112 cm2(E) 6 cm2SAMPLE QUESTION FOR 5 POINTS26. 2021 colored kangaroos are arranged in a row and are numbered from 1 to 2021. Each kangaroo is colored red, gray, or blue. Among any three consecutive kangaroos, there are always kangaroos of all three colors. Bruce guesses the colors of five kangaroos. These are his guesses: Kangaroo 2 is gray; Kangaroo 20 is blue; Kangaroo 202 is red; Kangaroo 1002 is blue; Kangaroo 2021 is gray. Only one of his guesses is wrong. What is the number of the kangaroo whose color he guessed incorrectly?(A) 2 (B) 20 (C) 202 (D) 1002 (E) 2021SAMPLE QUESTION FOR 3 POINTS1. Each year, the third Thursday in March is named Kangaroo Day. The dates of Kangaroo Day for the next few years are shown below, with one error. Which date is wrong?(A) March 17, 2022 (B) March 16, 2023 (C) March 14, 2024(D) March 20, 2025 (E) March 19, 2026SAMPLE QUESTION FOR 4 POINTS13. The numbers from 1 to 6 are placed in the circles at the intersections of three rings.The position of number 6 is shown. The sums of the numbers on each ring are the same. What number is placed in the circle with the question mark?(A) 1 (B) 2 (C) 3 (D) 4 (E) 5SAMPLE QUESTION FOR 5 POINTS25. The smaller square in the picture has an area of 16 and the gray triangle has an area of 1. What is the area of the larger square?(A) 17 (B) 18 (C) 19 (D) 20 (E) 21SAMPLE QUESTION FOR 3 POINTS1. Each year, the third Thursday in March is named Kangaroo Day. The dates of Kangaroo Day for the next few years are shown below, with one error. Which date is wrong?(A) March 17, 2022 (B) March 16, 2023 (C) March 14, 2024(D) March 20, 2025 (E) March 19, 2026SAMPLE QUESTION FOR 4 POINTS13. The numbers from 1 to 6 are placed in the circles at the intersections of three rings.The position of number 6 is shown. The sums of the numbers on each ring are the same. What number is placed in the circle with the question mark?(A) 1 (B) 2 (C) 3 (D) 4 (E) 5SAMPLE QUESTION FOR 5 POINTS25. The smaller square in the picture has an area of 16 and the gray triangle has an area of 1. What is the area of the larger square?(A) 17 (B) 18 (C) 19 (D) 20 (E) 21LEVELS 11 AND 12SAMPLE QUESTION FOR 3 POINTS1. Paula’s weather app shows a diagram of the predicted weather and maximum temperatures for the next seven days, as shown. Which of the following represents the corresponding graph of maximum temperatures?(A) (B) (C)(D) (E)SAMPLE QUESTION FOR 4 POINTS19. A naughty puppy grabs the end of a roll of toilet paper and walks away at a constant speed. Which of the functions below best describes the thickness y of the roll as a function of the unrolled part x?(A) (B) (C)(D) (E)SAMPLE QUESTION FOR 5 POINTS30. A certain game is won when one player gets 3 points ahead. Two players A and B are playingthe game and at a particular point, A is 1 point ahead. Each player has an equal probability of winning each point. What is the probability that A wins the game?(A) 12(B) 23(C) 34(D) 45(E) 56LEVELS 11 AND 12 ANSWERSSAMPLE QUESTION FOR 3 POINTS1. Paula’s weather app shows a diagram of the predicted weather and maximum temperatures for the next seven days, as shown. Which of the following represents the corresponding graph of maximum temperatures?(A) (B) (C)(D) (E)SAMPLE QUESTION FOR 4 POINTS19. A naughty puppy grabs the end of a roll of toilet paper and walks away at a constant speed. Which of the functions below best describes the thickness y of the roll as a function of the unrolled part x?(A) (B) (C)(D) (E)SAMPLE QUESTION FOR 5 POINTS30. A certain game is won when one player gets 3 points ahead. Two players A and B are playingthe game and at a particular point, A is 1 point ahead. Each player has an equal probability of winning each point. What is the probability that A wins the game?(A) 12(B)23(C) 34(D) 45(E) 56。
2007-美国袋鼠数学一二年级竞赛练习题
LEVELS 1 AND 2SAMPLE QUESTION FOR 3 POINTSWhich toy comes before the 5th toy?A) B) C) D) E)SAMPLE QUESTION FOR 4 POINTSWhich letter is missing from each of the words below? SCHOL BOK PRBLEM QUESTINA) A B) E C) O D) I E) USAMPLE QUESTION FOR 5 POINTSWhich number should replace the question mark in the pyramid?A) 10 B) 14 C) 22 D) 24 E) 34LEVELS 1 AND 2 ANSWERSSAMPLE QUESTION FOR 3 POINTSWhich toy comes before the 5th toy?A) B) C) D)E)SAMPLE QUESTION FOR 4 POINTSWhich letter is missing from each of the words below? SCHOL BOK PRBLEM QUESTINA) A B) E C) O D) I E) USAMPLE QUESTION FOR 5 POINTSWhich number should replace the question mark in the pyramid?A) 10 B) 14 C) 22 D) 24E) 34LEVELS 3 AND 4SAMPLE QUESTION FOR 3 POINTSNot taking any steps backwards, Anna travelled toward the car using a path shown in the picture, and picked up numbers she encountered along her way. Which set of the numbers below could she pick up?A) 1, 2, 4 B) 2, 3, 4 C) 2, 3, 5 D) 1, 5, 6 E) 1, 2, 5SAMPLE QUESTION FOR 4 POINTSThe square shown in the picture must be filled in such a way that each of thedigits 1, 2, and 3 appears in each row and in each column once and onlyonce. If Harry started to fill in the square as shown, what number can hewrite in the square marked with the question mark?A) 1 B) 2 C) 3 D) 1 or 2 E) 1, 2 or 3SAMPLE QUESTION FOR 5 POINTSHow many digits have to be written in order to write down every number from 1 to 100 inclusive?A) 100 B) 150 C) 190 D) 192 E) 200LEVELS 3 AND 4 ANSWERSSAMPLE QUESTION FOR 3 POINTSNot taking any steps backwards, Anna travelled toward the car using a path shown in the picture, and picked up numbers she encountered along her way. Which set of the numbers below could she pick up?A) 1, 2, 4 B) 2, 3, 4 C) 2, 3, 5D) 1, 5, 6 E) 1, 2, 5SAMPLE QUESTION FOR 4 POINTSThe square shown in the picture must be filled in such a way that each of thedigits 1, 2, and 3 appears in each row and in each column once and onlyonce. If Harry started to fill in the square as shown, what number can hewrite in the square marked with the question mark?A) 1 B) 2 C) 3D) 1 or 2 E) 1, 2 or 3SAMPLE QUESTION FOR 5 POINTSHow many digits have to be written in order to write down every number from 1 to 100 inclusive?A) 100 B) 150 C) 190 D) 192E) 200LEVELS 5 AND 6SAMPLE QUESTION FOR 3 POINTSEvaluate 2007 ÷ (2 + 0 + 0 + 7) – 2 × 0 × 0 × 7A) 1 B) 9 C) 214 D) 223 E) 2007SAMPLE QUESTION FOR 4 POINTSAlex, Ben, Carl, and Daniel each participates in a different sport: karate, soccer, volleyball, and judo. Alex does not like sports played with a ball. Ben practices judo and often attends soccer games to watch his friend play. Which of the following statements is true?A) Alex plays volleyball.B) Ben plays soccer.C) Carl plays volleyball.D) Daniel does karate.E) Alex does judo.SAMPLE QUESTION FOR 5 POINTSTo the right of a certain two-digit number the same number has been written, creating a four-digit number. How many times is the new four-digit number greater than the original two-digit number?A) 100 B) 101 C) 1000 D) 1001 E) 10LEVELS 5 AND 6 ANSWERSSAMPLE QUESTION FOR 3 POINTSEvaluate 2007 ÷ (2 + 0 + 0 + 7) – 2 × 0 × 0 × 7A) 1 B) 9 C) 214 D) 223E) 2007SAMPLE QUESTION FOR 4 POINTSAlex, Ben, Carl, and Daniel each participates in a different sport: karate, soccer, volleyball, and judo. Alex does not like sports played with a ball. Ben practices judo and often attends soccer games to watch his friend play. Which of the following statements is true?A) Alex plays volleyball.B) Ben plays soccer.C) Carl plays volleyball.D) Daniel does karate.E) Alex does judo.SAMPLE QUESTION FOR 5 POINTSTo the right of a certain two-digit number the same number has been written, creating a four-digit number. How many times is the new four-digit number greater than the original two-digit number?A) 100 B) 101C) 1000 D) 1001 E) 10LEVELS 7 AND 8SAMPLE QUESTION FOR 3 POINTSRose bushes are planted in a line on both sides of a path. The distance between the bushes is 2 m. What is the largest number of bushes that can be planted if the path is 20 m long?A) 22 B) 20 C) 12 D) 11 E) 10SAMPLE QUESTION FOR 4 POINTSx is a strictly negative integer. Which of the expressions is the greatest?A) x + 1 B) 2x C) −2x D) 6x + 2 E) x − 2SAMPLE QUESTION FOR 5 POINTSA certain broken calculator does not display the digit 1. For example, if we type in the number 3131, only the number 33 is displayed, with no spaces. Mike typed a 6-digit number into that calculator, but only 2007 appeared on the display. How many different numbers could Mike have typed?A) 12 B) 13 C) 14 D) 15 E) 16LEVELS 7 AND 8 ANSWERSSAMPLE QUESTION FOR 3 POINTSRose bushes are planted in a line on both sides of a path. The distance between the bushes is 2 m. What is the largest number of bushes that can be planted if the path is 20 m long?A) 22B) 20 C) 12 D) 11 E) 10SAMPLE QUESTION FOR 4 POINTSx is a strictly negative integer. Which of the expressions is the greatest?A) x + 1 B) 2x C) −2x D) 6x + 2 E) x − 2SAMPLE QUESTION FOR 5 POINTSA certain broken calculator does not display the digit 1. For example, if we type in the number 3131, only the number 33 is displayed, with no spaces. Mike typed a 6-digit number into that calculator, but only 2007 appeared on the display. How many different numbers could Mike have typed?A) 12 B) 13 C) 14 D) 15E) 16LEVELS 9 AND 10SAMPLE QUESTION FOR 3 POINTSIn the picture, what is the sum of the number of dots on the faces of the dice which you cannot see?A) 15 B) 12 C) 7 D) 27 E) another answerSAMPLE QUESTION FOR 4 POINTSTo fill in the table, we need to write 0 or 1 in each cell in such a way that the sum of numbers of each row and of each column is equal to 2. What are x and y ? A) x = 1, y = 1 B) x = 1, y = 0 C) x = 0, y = 1D) x = 0, y = 0 E) It is impossible to determine.SAMPLE QUESTION FOR 5 POINTSA certain island is inhabited by liars and truth-tellers (the liars always lie and the truth-tellers always tell the truth). One day 12 islanders, both liars and truth-tellers, gathered together and issued a few statements. Two people said: “Exactly two people among us twelve are liars.” Four other people said: “Exactly four people among us twelve are liars.” The other six people said: “Exactly six people among us twelve are liars.” How many liars were there? A) 2 B) 4 C) 6 D) 8 E) 10LEVELS 9 AND 10 ANSWERSSAMPLE QUESTION FOR 3 POINTSIn the picture, what is the sum of the number of dots on the faces of the dice which you cannot see?A) 15 B) 12C) 7D) 27E) another answerSAMPLE QUESTION FOR 4 POINTSTo fill in the table, we need to write 0 or 1 in each cell in such a way that the sum of numbers of each row and of each column is equal to 2. What are x and y ? A) x = 1, y = 1 B) x = 1, y = 0C) x = 0, y = 1D) x = 0, y = 0 E) It is impossible to determine.SAMPLE QUESTION FOR 5 POINTSA certain island is inhabited by liars and truth-tellers (the liars always lie and the truth-tellers always tell the truth). One day 12 islanders, both liars and truth-tellers, gathered together and issued a few statements. Two people said: “Exactly two people among us twelve are liars.” Four other people said: “Exactly four people among us twelve are liars.” The other six people said: “Exactly six people among us twelve are liars.” How many liars were there? A) 2 B) 4C) 6D) 8E) 10LEVELS 11 AND 12SAMPLE QUESTION FOR 3 POINTSA billiard ball always bounces off the side of a billiard table at an angle of 45° as shown. If it continues on the path shown, which pocket will the ball fall into?A) A B) B C) C D) DE) The ball will not fall into any pocket.SAMPLE QUESTION FOR 4 POINTSThe square ABCD lies in a plane and its edge measures 1. Consider all squares that share at least two vertices with the square ABCD. What is the area of the region covered by all of such squares, not including ABCD?A) 5 B) 6 C) 7 D) 8 E) 9SAMPLE QUESTION FOR 5 POINTSWhat is the measure of the acute angle of a rhombus with side of length equal to the geometric mean of its diagonals?A) 15° B) 30° C) 45° D) 60° E) 75°LEVELS 11 AND 12 ANSWERSSAMPLE QUESTION FOR 3 POINTSA billiard ball always bounces off the side of a billiard table at an angle of 45° as shown. If it continues on the path shown, which pocket will the ball fall into?A) A B) B C) C D) DE) The ball will not fall into any pocket.SAMPLE QUESTION FOR 4 POINTSThe square ABCD lies in a plane and its edge measures 1. Consider all squares that share at least two vertices with the square ABCD. What is the area of the region covered by all of such squares, not including ABCD?A) 5 B) 6 C) 7D) 8 E) 9SAMPLE QUESTION FOR 5 POINTSWhat is the measure of the acute angle of a rhombus with side of length equal to the geometric mean of its diagonals?A) 15° B) 30°C) 45° D) 60° E) 75°。
关于汽车通过通不过的二年级数学题
汽车通过通不过的二年级数学题1. 引言在日常生活中，我们常常会遇到各种各样的数学问题，有些问题看似简单，实际上却充满了巧妙的逻辑和推理。
作为一个普通的驾驶者，我们可能很少会想到汽车与数学之间会有什么联系，然而实际上，在某些情况下，数学知识可能会对我们的驾驶产生直接的影响。
2. 背景在美国，有一道被誉为“通不过的二年级数学题”的经典数学问题。
这个问题是这样的：如果一辆汽车以每小时60英里的速度行驶，那么在以每小时60英里的速度行驶的情况下，这辆汽车需要多长时间才能行驶一英里的距离？这道题看似简单，但是其背后隐藏着许多有趣的数学知识。
3. 分析我们可以直接计算出这辆汽车以60英里每小时的速度行驶一英里需要的时间。
根据速度等于路程除以时间的公式，我们可以得到时间等于路程除以速度，那么行驶一英里需要的时间就是1英里除以60英里每小时，计算得到的结果是0.0166667小时。
然而，这个结果并不是最终的答案。
4. 深入探讨事实上，这个问题涉及到了时间与速度的关系，通过分析我们可以发现，时间的概念在这个问题中其实并不是简单的时间，而是涉及到了小时、分钟的转换。
我们需要将0.0166667小时转换成分钟，这样才能得到这辆汽车实际需要的时间。
继续计算下去，我们将0.0166667小时转换成分钟，得到的结果是1分钟，这意味着这辆汽车实际上只需要1分钟的时间就可以行驶一英里的距离。
5. 结论总结来说，这道“通不过的二年级数学题”看似简单，实则涉及到了时间、速度和单位转换等多个数学知识点，对于我们来说，这个问题不仅是一次简单的数学练习，更是一个思维的训练。
通过这样的问题，我们可以更好地理解数学知识在现实生活中的应用，也能够培养我们的逻辑思维能力和解决问题的能力。
即使是如此简单的问题，在日常生活中我们也应该多加关注并加以重视。
6. 应用除了在数学课堂上，这个问题也可以在驾驶学校中应用。
在学习驾驶的过程中，了解汽车速度和时间的关系是非常重要的，这不仅可以帮助学员计算汽车行驶的时间，还可以提高他们对于速度和时间的直观感知。
美国AMC10中文版试题及答案
2000到20XX年AMC10美国数学竞赛0 0P 0 A 0 B 0 C 0D 0 全美中学数学分级能力测验(AMC 10)2000年 第01届 美国AMC10 (2000年2月 日 时间75分钟)1. 国际数学奥林匹亚将于 在美国举办，假设I 、M 、O 分别表示不同的正整数，且满足I ⨯M ⨯O =2001，则试问I +M +O 之最大值为 。
(A) 23 (B) 55 (C) 99 (D) 111 (E) 6712. 2000(20002000)为 。
(A) 20002001 (B) 40002000 (C) 20004000 (D) 40000002000 (E) 200040000003. Jenny 每天早上都会吃掉她所剩下的聪明豆的20%，今知在第二天结束时，有32颗剩下，试问一开始聪明豆有 颗。
(A) 40 (B) 50 (C) 55 (D) 60 (E) 754. Candra 每月要付给网络公司固定的月租费及上网的拨接费，已知她12月的账单为12.48元，而她1月的账单为17.54元，若她1月的上网时间是12月的两倍，试问月租费是 元。
(A) 2.53 (B) 5.06 (C) 6.24 (D) 7.42 (E) 8.775. 如图M ，N 分别为PA 与PB 之中点，试问当P 在一条平行AB 的直 在线移动时，下列各数值有 项会变动。
(a) MN 长 (b) △P AB 之周长 (c) △P AB 之面积 (d) ABNM 之面积(A) 0项 (B) 1项 (C) 2项 (D) 3项 (E) 4项 6. 费氏数列是以两个1开始，接下来各项均为前两项之和，试问在费氏数列各项的个位数字中， 最后出现的阿拉伯数字为 。
(A) 0 (B) 4 (C) 6 (D) 7 (E) 97. 如图，矩形ABCD 中，AD =1，P 在AB 上，且DP 与DB 三等分∠ADC ，试问△BDP 之周长为 。
高中数学竞赛典型题目
数学竞赛典型题目（一）1.(美国数学竞赛)设n a a a ,,,21是整数列,并且他们的最大公因子是1.令S 是一个整数集,具有性质:（1）),,2,1(n i S a i （2）}),,2,1{,(n ji S a a ji,其中j i,可以相同（3）对于S y x,，若S yx，则Syx证明：S 为全体整数的集合。
2．(美国数学竞赛)c b a ,,是正实数，证明：3252525)()3)(3)(3(c b a ccbbaa3．(加拿大数学竞赛)T 为1002004的所有正约数的集合，求集合T 的子集S 中的最大可能的元素个数。
其中S 中没有两个元素，一个是另一个的倍数。
4．(英国数学竞赛)证明：存在一个整数n 满足下列条件：（1）n 的二进制表达式中恰好有2004个1和2004个0；（2）2004能整除n .5．(英国数学竞赛)在0和1之间，用十进制表示为21.0a a 的实数x 满足：在表达式中至多有2004个不同的区块形式，)20041(20031ka a a kkk ，证明：x 是有理数。
6．(亚太地区数学竞赛)求所有由正整数组成的有限非空数集S ，满足：如果S nm,，则Sn m n m),(7．(亚太地区数学竞赛)平面上有2004个点，并且无三点共线，S 为通过任何两点的直线的集合。
证明：点可以被染成两种颜色使得两点同色当且仅当S 中有奇数条直线分离这两点。
8．(亚太地区数学竞赛)证明：)()!1(*2N n nnn 是偶数。
9．(亚太地区数学竞赛)z y x ,,是正实数，证明：)(9)2)(2)(2(222zx yz xy zyx10．（越南数学竞赛）函数f 满足)0(2sin 2cos )(cot xx xx f ，令)11)(1()()(xx f x f x g ，求)(x g 在区间]1,1[的上最值。
11．（越南数学竞赛）定义17612)(,91524)(2323x xxx q x xxx p ，证明：（1）每个多项式都有三个不同的实根；（2）令A 为)(x p 的最大实根，B 为)(x q 的最大实根，证明：4322B A 12．（越南数学竞赛）令F 为所有满足R R f :且x x f f x f )]2([)3(对任意R x成立的函数f 的集合。
amc10美国数学竞赛真题
amc10美国数学竞赛真题amc10（美国数学竞赛）真题：
1. 微积分方面：
（1）求一元二次微分方程的有限解；
（2）求曲线y = f（x）在a点的切线斜率；
（3）求曲面z = f（x，y）在a点的切面方向；（4）决定泰勒展开式中n次项的系数；
（5）判断函数f（x）的极值；
（6）研究极限函数的存在性及无穷性；
（7）解决解析函数的定义域；
2. 数论方面：
（1）计算包含一定系数的多项式的余数；
（2）判断两个多项式的最大公因数；
（3）判断一个数的完全平方数；
（4）证明或否定一个数在某数域中的存在性；
3. 统计学：
（1）奇数和偶数的含义；
（2）样本容量的定义；
（3）求一组数据中最小值、最大值和中位数；（4）决定随机变量的条件概率；
（5）求一组数据的散布图；
4. 几何学：
（1）求平面两点间的距离；
（2）求直线的斜率；
（3）分析不相交平面的法向量；
（4）解决三角形的各个角度；
（5）计算圆的周长；
（6）两个圆的位置关系；
（7）求圆环的面积；
（8）求立体图形的体积。
5. 其它方面：
（1）分析逻辑表达式；
（2）求图论问题的最小生成树；
（3）计算算法问题的复杂度；
（4）计算概率论问题中的概率；
（5）规划组合优化问题的最优解；
（6）根据资料采纳正确的统计预测方法。
数学史上的24道经典名题
数学史上的24道经典名题1.不说话的学术报告1903年10月，在美国纽约的一次数学学术会议上，请XXX教授作学术报告。
他走到黑板前，没说话，用粉笔写出2^67-1，这个数是合数而不是质数。
接着他又写出两组数字，用竖式连乘，两种计算结果相同。
回到座位上，全体会员以暴风雨般的掌声表示祝贺。
证明了2自乘67次再减去1，这个数是合数，而不是两百年一直被人怀疑的质数。
有人问他论证这个问题，用了多长时间，他说：“三年内的全部星期天”。
请你很快回答出他至少用了多少天?2.国王的重赏传说，印度的舍罕国王打算重赏国际象棋的发明人——大臣XXX。
这位聪明的大臣跪在国王面敢说：“陛下，请你在这张棋盘的第一个小格内，赏给我一粒麦子，在第二个小格内给两粒，在第三个小格内给四粒，照这样下去，每一小格内都比前一小格加一倍。
陛下啊，把这样摆满棋盘上所有64格的麦粒，都赏给您的仆人吧?”国王说：“你的要求不高，会如愿以偿的”。
说着，他下令把一袋麦子拿到宝座前，计算麦粒的工作开始了。
……还没到第二十小格，袋子已经空了，一袋又一袋的麦子被扛到国王面前来。
但是，麦粒数一格接一格地增长得那样迅速，很快看出，即使拿出来全印度的粮食，国王也兑现不了他对象棋发明人许下的语言。
算算看，国王应给象棋发明人多少粒麦子?3.王子的数学题传说从前有一位王子，有一天，他把几位妹妹召集起来，出了一道数学题考她们。
题目是：我有金、银两个手饰箱，箱内分别装自若干件手饰，如果把金箱中25%的手饰送给第一个算对这个题目的人，把银箱中20%的手饰送给第二个算对这个题目的人。
然后我再从金箱中拿出5件送给第三个算对这个题目的人，再从银箱中拿出4件送给第四个算对这个题目的人，最后我金箱中剩下的比分掉的多10件手饰，银箱中剩下的与分掉的比是2∶1，请问谁能算出我的金箱、银箱中原来各有多少件手饰?4.公主出题古时候，传说捷克的公主XXX出过这样一道有趣的题：“一只篮子中有若干李子，取它的一半又一个给第一个人，再取其余一半又一个给第二人，又取最后所余的一半又三个给第三个人，那么篮内的李子就没有剩余，篮中原有李子多少个?”5.XXX猜测XXX是二百多年前德国的数学家。
2024年AIME2数学竞赛(英文试题中文答案)
2024年AIME2数学竞赛（英文）1、Among the 900 residents of Aimeville, there are 195 who own a diamond ring, 367 who own a set of golf clubs, and 562 who own a garden spade. In addition, each of the 900 residents owns a bag of candy hearts. There are 437 residents who own exactly two of these things, and 234 residents who own exactly three of these things. Find the number of residents of Aimeville who own all four of these things.2、A list of positive integers has the following properties:· The sum of the items in the list is 30.· The unique mode of the list is 9.· The median of the list is a positive integer that does not appear in the list itself.Find the sum of the squares of all the items in the list.3、Find the number of ways to place a digit in each cell of a 2×3 grid so that the sum of the two numbers formed by reading left to right is 999, and the sum of the three numbers formed by reading top to bottom is 99. The grid below is an example of such an arangement because 8+991=999 and 9+9+ 81=99.4、Let xx, yy, and zz be positive real numbers that satisfy the following system of equations:log2 (xx yyyy)=12log2 (yy xxyy)=13log2 (yy xxyy)=14Then the value of |log2 (xx4yy3zz2)| is mm nn, where mm and nn are relatively prime positive integers. Find mm+ nn.5、Let AAAAAAAAAAAA be a convex equilateral hexagon in which all pairs of opposite sides are parallel. The triangle whose sides are extensions of segments AAAA, AAAA, and AAAA has side lengths 200, 240, and 300. Find the side length of the hexagon.6、Alice chooses a set AA of positive integers. Then Bob lists all finite nonempty sets AA of positive integers with the property that the maximum element of AA belongs to AA. Bob's list has 2024 sets. Find the sum of the elements of AA.7、Let NN be the greatest four-digit positive integer with the property that whenever one of its digits is changed to 1, the resulting number is divisible by 7. Let QQ and RR be the quotient and the remainder, respectively, when NN is divided by 1000. Find QQ+RR.8、Torus TT is the surface produced by revolving a circle with radius 3 around an axis in the plane of the circle that is a distance 6 from the center of the circle.Let SS be a sphere with radius 11. When TT rests on the inside of SS, it is internally tangent to SS along a circle with radius rr ii, and when TT rests on the outside of SS, it is externally tangent to SS along a circle with radius rr0. The difference rr1−rr0 can be written as mm nn, where mm and nn are relatively prime positive integers. Find mm+nn.9、There is a collection of 25 indistinguishable black chips and 25 indistinguishable white chips. Find the number of ways to place some of these chips in 25 unit cells of a 5×5 grid so that· each cell contains at most one chip,· all chips in the same row and all chips in the same column have the same color, and· any additional chip placed on the grid would violate one or more of the previous two conditions.10、Let △AAAAAA have incenter II, circumcenter OO, inradius 6, and circumradius 13. Suppose that II AA⊥OOII. Find AAAA⋅AAAA.11、Find the number of triples of nonnegative integers (aa,bb,cc) satisfying aa+bb+cc=300 and aa2bb+aa2cc+bb2aa+bb2cc+cc2aa+cc2bb=600000012、Let OO(0,0), AA(12,0), and AA(0,√32) be points in the coordinate plane. Let AA be the family of segments PPQQ of unit length lying in the first quadrant with PP on the x-axis and QQ on the y-axis. There is a unique point AA on AAAA, distinct from AA and AA, that does not belong to any segment from AA other than AAAA. Then OOAA2=pp qq, where pp and qq are relatively prime positive integers. Find pp+qq.13、Let ωω≠1 be a 13th root of unity. Find the remainder when∏12kk=0(2−2ωωkk+ωω2kk)is divided by 1000.14、Let bb⩾2 be an integer. Call a positive integer nn bb-eautiful if it has exactly two digits when expressed in base bb, and these two digits sum to √nn. For example, 81 is 13-eautiful because 6313 and 6+ 3=√81. Find the least integer bb⩾2 for which there are more than ten bb-eautiful integers.15、Find the number of rectangles inside a fixed regular dodecagon (12-gon) where each side of the rectangle lies on a side or on a diagonal of the dodecagon. The diagram below shows three of those rectangles.1 、【答案】073;【解析】记四种都有的人数为aa．由容斥原理有1×437+2×234+3×aa=195+367+562解得aa=073【标注】2 、【答案】236;【解析】由30<4×9知至多3个9，又由9为唯一众数知至少2个9.由中位数不在数列中，必为偶数个数．若恰有3个9，只能为3，9，9，9，中位数为9，不合，舍去．因此恰为2个9，其余数互不相同．若为6个数，1+2+3+4+9+9=28只能1，2，3，6，9，9或1，2，4，5，9，9中位数均不为整只能为4个数且9为较大两数，5，7，9，9合题52+72+92+92=236【标注】3 、【答案】045;【解析】由横向两数和为999知对应数位和为9则纵向形成的数均为9的倍数9aa+9bb+9cc=99aa+bb+cc=11正整数解有AA102=045【标注】4 、【答案】033;【解析】 设log 2 xx =aa ，log 2 yy =bb ， log 2 zz =cc 则有 ⎩⎪⎨⎪⎧aa −bb −cc =12−aa +bb −cc =13−aa −bb +cc =14 解得 ⎩⎪⎨⎪⎧aa =−724bb =−924cc =−524原式 =|4aa +3bb +2cc |=25825+8=033【标注】5 、【答案】 80; 【解析】 设六边形边长为 xx△PPAAAA ∽△PPRRQQPPPP PPBB =PPPP PPRR =23⇒PPAA =2xx 3 △RRAAAA ∽△RRPPQQPPRR RRAA =PPPP PPRR =56⇒RRAA =5xx 62xx 3+xx +5xx 6=200⇒xx =80【标注】 6 、【答案】 55;【解析】 AA 中以AA 中的元素kk 为最大元素的集合个数为 2kk−1只需将2024拆成互不相同的2的幂之和2024=2048−24=211−(24+23)=210+29+28+27+26+25+23则AA={11,10,9,8,7,6,4}，元素和为55【标注】7 、【答案】699;【解析】设题中的数为aabbccaa，则由题意有7|1bbccaa+aa1ccaa+aabb1aa+aabbcc1即7|3aabbccaa+1111，又1111≡5（mod 7），可得aabbccaa≡3（mod 7）则aabbccaa−aabbcc1≡3（mod 7）⇒aa−1≡3（mod 7）⇒aa=4则aabbccaa−aabb1aa≡3（mod 7）⇒10(cc−1)≡3（mod 7）⇒cc=2,9则aabbccaa−aa1ccaa≡3（mod 7）⇒100(bb−1)≡3（mod 7）⇒bb=6则aabbccaa−1bbccaa≡3（mod 7）⇒1000(aa−1)≡3（mod 7）⇒aa=5取最大，则为5694，694+5=699【标注】8 、【答案】127;【解析】TT内切于SS时，如图OOAA=11，OO1AA=3，OO1AA=6，OOOO1=11−3=8 OOOO1OORR=OO1PP RRBB⇒rr ii=AAAA=334TT外切于SS时，如图OOAA=11，OO2AA=3，OO2AA=6，OOOO2=11+3=14DDDD OO2AA=OODD OOOO2⇒rr0=AAAA=337rr ii−rr0=334−337=992899+28=127【标注】9 、【答案】902;【解析】（1）若棋子全黑或全白：2种（2）若棋子不同色原问题于确定5行5列分别为黑或白形成对应由题意，原问题的一种填法唯一确定行列的黑/白而当行列的黑白均确定，只能是：黑行黑列交叉点填黑、白行白列交叉点填白、其余位置不填同样唯一确定填法不为全黑/全白的行列选法有(25−2)2=900种综上，2+900=902【标注】10 、【答案】468;【解析】由垂径AAII=AAII=RRDD2AAAA=AAII=AAAA由托勒密AAAA⋅AAAA+AAAA⋅AAAA=AAAA⋅AAAA可得AAAA+AAAA=2AAAA，即c+b=2a再由三角形面积SS△RRPPBB=aaaaaa4PP=(aa+aa+aa)rr2代入RR=13，r=6aaaaaa52=3aa⋅62⇒bbcc=468【标注】11 、【答案】601;【解析】aa2bb+aa2cc+bb2aa+bb2cc+cc2aa+cc2bb=aabb(aa+bb)+bbcc(bb+cc)+ccaa(cc+aa)=aabb(300−cc)+bbcc(300−aa)+ccaa(300−bb)=300(aabb+bbcc+ccaa)−3aabbcc=6000000则100(aabb+bbcc+ccaa)−aabbcc=2000000=1002(aa+bb+cc)−1003 1003−1002⋅(aa+bb+cc)+100(aabb+bbcc+ccaa)−aabbcc=0(100−aa)(100−bb)(100−cc)=0aa，bb，cc均为100:1种aa，bb，cc恰有1个100：3×(201−1)=600【标注】12 、【答案】23;【解析】AAAA：yy=√32−√3xx可设AA 中直线为yy =sin θθ−tan θθ⋅xx （θθ 为直线与xx 轴所夹锐角）则AAAA 为θθ=ππ3 情况AAAA 与yy =sin θθ−tan θθ⋅xx 联立有√32−√3xx =sin θθ−tan θθ⋅xx ，得 xx =√3−2sin θθ2√3−2tan θθ 则当θθ→ππ3 时的极限即符合题意中不在AA 中另一直线上的要求洛必达则xx aa =18，得AA (18,3√38)，OOAA 2=716，7+16=23．【标注】 13 、【答案】 321;【解析】 ��2−2ωωkk +ωω2kk �12kk=0=���1−ωωkk �2+1�12kk=0=��1−ωωkk −ii�12kk=0�1−ωωkk +ii�=�[(1−ii )−ωωkk ]12kk=0[(1+ii )−ωωkk ] 由ωω 为13 次单位根，xx 13−1的分解为 ��xx −ωωkk �12kk=0 则原式 =[(1−ii )13−1][(1+ii )13−1]=[−64(1−ii )−1][−64(1+ii )−1]=(64ii −65)(−64ii −65)=652+642=83218321≡321（mod 1000） 【标注】 14 、【答案】 211;【解析】 考虑bb 进制两位数 xxyy aa则它需满足(xx +yy )2=bbxx +yy ，其中1⩽xx ⩽bb −1，0⩽yy ⩽bb −1 且 为整数．以上方程需至少有10组解． 首先， (xx +yy )2=bbxx +yy ⩽bb (bb −1)+(bb −1)=bb 2−1<bb 2∴ xx +yy ⩽bb −1 其次， (xx +yy )2=bbxx +yy =xx +yy +(bb −1)xx 则 (xx +yy )(xx +yy −1)=(bb −1)xx注意2⩽xx+yy⩽bb−1，1⩽xx⩽bb−1则转化为在bb−1以内可找到至少十组xx+yy⩾xx使得上式成立，以下分析bb−1数论性质记bb−1=bb′=Πnn ii=1pp ii kk ii（pp ii为质数）由bb′|(xx+yy)(xx+yy−1)且(xx+yy,xx+yy−1)=1则xx+yy与xx+yy−1无公共质因子但包含bb′全部质因子．考虑将bb′质因子分配给xx+yy与xx+yy−1方式需有十种则bb′至少4种质因子．bb′⩾2×3×5×7=210106×105=210×53，105×104=210×5270×69=210×23，141×140=210×9485×84=210×34，126×125=210×7591×90=210×39，120×119=210×6836×35=210×6，175×174=210×14521×20=210×2．故210+1=211【标注】15 、【答案】315;【解析】题中所述矩形可按边是否与正十二边形的边平行分为两类．（1）平行：如图所示（对称情况需×3）AA1AA5AA7AA11中有AA32AA52=30个矩形AA2AA4AA8AA10中也有AA32AA52=30个矩形重复了AA1AA2AA3AA4中的AA32AA32=9个此类共有(30+30−9)×3=153个（2）不平行：如图所示（对称情况需×3）ArrayAA1AA4AA7AA10中有AA42AA42=36个矩形AA5AA6AA11AA12中有AA62个矩形其中AA42个在AA1AA4AA7AA10中，还多AA62−AA42=9个此类共有(36+9+9)×3=162个总计153+162=315个【标注】第11页，共11页。
AMC10美国数学竞赛真题2006B卷
What is ?SolutionProblem 2For real numbers and , define . What is?SolutionProblem 3A football game was played between two teams, the Cougars and the Panthers. The two teams scored a total of 34 points, and the Cougars won by a margin of 14 points. How many points did the Panthers score?SolutionProblem 4Circles of diameter 1 inch and 3 inches have the same center. The smaller circle is painted red, and the portion outside the smaller circle and inside the larger circle is painted blue. What is the ratio of the blue-painted area to the red-painted area?SolutionProblem 5A rectangle and a rectangle are contained within a square without overlapping at any point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square?SolutionA region is bounded by semicircular arcs constructed on the side of a square whose sides measure , as shown. What is the perimeter of this region?SolutionProblem 7Which of the following is equivalent to when ?SolutionProblem 8A square of area 40 is inscribed in a semicircle as shown. What is the area of the semicircle?SolutionFrancesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water to make lemonade. There are 25 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar. Water contains no calories. How many calories are in 200 grams of her lemonade?SolutionProblem 10In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15. What is the greatest possible perimeter of the triangle?SolutionProblem 11What is the tens digit in the sumSolutionProblem 12The lines and intersect at the point . What is ?SolutionProblem 13Joe and JoAnn each bought 12 ounces of coffee in a 16 ounce cup. Joe drank 2 ounces of his coffee and then added 2 ounces of cream. JoAnn added 2 ounces of cream, stirred the coffee well, and then drank 2 ounces. What is the resulting ratio of the amount of cream in Joe's coffee to that in JoAnn's coffee?SolutionProblem 14Let and be the roots of the equation . Suppose that and are the roots of the equation . What is ?SolutionProblem 15Rhombus is similar to rhombus . The area of rhombus is and . What is the area of rhombus ?SolutionProblem 16Leap Day, February 29, 2004, occurred on a Sunday. On what day of the week will Leap Day, February 29, 2020, occur?SolutionProblem 17Bob and Alice each have a bag that contains one ball of each of the colors blue, green, orange, red, and violet. Alice randomly selects one ball from her bag and puts it into Bob's bag. Bob then randomly selects one ball from his bag and puts it into Alice's bag. What is the probability that after this process the contents of the two bags are the same?SolutionProblem 18Let be a sequence for which, , and for each positive integer .What is ?SolutionProblem 19A circle of radius is centered at . Square has side length . Sidesand are extended past to meet the circle at and , respectively. What is the area of the shaded region in the figure, which is bounded by , , and the minor arc connecting and ?SolutionProblem 20In rectangle , we have , , , for some integer . What is the area of rectangle ?SolutionProblem 21For a particular peculiar pair of dice, the probabilities of rolling , , , , , and , on each die are in the ratio . What is the probability of rolling a total of on the two dice?SolutionProblem 22Elmo makes sandwiches for a fundraiser. For each sandwich he uses globs of peanut butter at per glob and blobs of jam at per blob. The cost of the peanut butter and jam to make all the sandwiches is . Assume that , , and are positive integers with . What is the cost of the jam Elmo uses to make the sandwiches?SolutionProblem 23A triangle is partitioned into three triangles and a quadrilateral by drawing two lines from vertices to their opposite sides. The areas of the three triangles are 3, 7, and 7 as shown. What is the area of the shaded quadrilateral?SolutionProblem 24Circles with centers and have radii and , respectively, and are externally tangent. Points and on the circle with center and points and on the circle with center are such that and are common external tangents to the circles. What is the area of the concave hexagon ?SolutionProblem 25Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and the last two digits just happen to be my age." Which of the following is not the age of one of Mr. Jones's children?。
美国初中数学习题
Unit3–Square Roots and Surface Area(textbook Chpt. 1)
Identifying Perfect Squares and their Square Root Value
- whole values, decimal values, and fractional values
a)b)
Additional Review/Practice can be found in the textbook on the following pages…
Study Guide – pg.44
Extra Practice -pg.21; pg.45 - 47
48 cm
17 cm62 cm
8 cm
Calculate the Surface Area of Rectangular Prisms, Triangular
Prisms, and Cylinders
Calculate the surface area of each solid.Show process!
Using the information above, estimate the square root values.
a) b) c)
Using Square Roots in Real World Applications
Solve for the value asked for.
a)A = 265.69 cm2b)c)
a) 17b) 2.7c) d) 28.1
Estimate Square Roots of Non-Perfect Squares
美国数学大联盟杯赛一二年级竞赛例题
美国“数学大联盟杯赛”一、二年级竞赛例题1. 2 + 0 + 1 + 0 = 2 + 1 + ?A) 3B) 2C) 1D) 02. I was 5 old 4 ago. How old will I be 3 from now?A) 8B) 9C) 12D) 133. There are 3 dinosaur eggs in a nest. How many eggs is that?A) 15B) 36C) 42D) 724. The of 7 and 123 isA) 116B) 130C) 741D) 8615. Of the following, is not a number?A) 3 + 2B) 3 − 2C) 3 × 2D) 3 ÷ 26. Which is two less than one more than ten thousand?A) 10 002B) 10 001C) 9999D) 99987. Of the following, product is an odd number?A) 212 × 12B) 12 × 121C) 2 × 1212D) 1 × 21218. If is Tuesday, 20 days from now will be aA)B)C)D) Saturday9. Arnold the Dog is holding a square board has a perimeter of12 m. How long is one of the board?A) 2 mB) 3 mC) 4 mD) 5 m10. 10 = ? nickelsA) 20B) 12C) 5D) 211. How whole numbers are greater than 10 and than 30?A) 18B) 19C) 20D) 2112. 40 + 80 + 120 = 4 × ?A) 60B) 50C) 40D) 3013. I 1 dollar, 2 quarters, 3 dimes, and 4 nickels. This money is worth the amount as ? pennies.A)150B)175C)184D)20014. 37 after 6 o’clock is ? o’clock.A) 1B) 4C) 7D) 1215. An elf’s increases by 10 cm each year. Agnome’s increases by that amount each year. If the elf and the gnome were the 2 years ago, how taller than the gnome will the elf be 2 years from today?A) 10 cmB) 15 cmC) 20 cmD) 25 cm16. The of 2, 0, 1, and 1 isA) 0B) 1C) 2D) 417. The sum of all numbers between 1 and 9 isA) 8B) 12C) 20D) 3518. Didi the dog danced for exactly one week. For how hours did Didi dance?A) 24B) 84C) 128D) 16819. How 25¢ gumballs can I buy for $5?A) 10B) 20C) 40D) 8020. I 60 marbles. If I put as many of these marbles as I can 7 equal piles, how many marbles are over?A) 1B) 4C) 621. Which is 1000 tens minus 100 tens?A) 9000B) 9900C) 9990D) 999922. I 4567 to the nearest 10, then I 4567. I end up withA) 3B) 33C) 333D) 543323. If the of a rectangle is 20 and two of its sideseach a of 4, what is the of each of the other two sides?A) 16B) 12C) 8D) 624. Otto did a handstand, and 1 out of 5 pennies he had in his pockets fell out. If a of 19 pennies fell out, Otto must started with _?_ pennies in his pockets.A) 24B) 76C) 95D) 11425. I to take a 75-minute nap starting at 5:15. The used to end my nap should be set forA) 6:00B) 6:15C) 6:3026. The of 4 odd numbers is alwaysA) evenB) oddC) primeD) than 2027. The Bears' picnic has too ants! If the number of ants isthe whole number greater than 0 is divisible by both 12 and 42, are _?_ ants.A) 54B) 84C) 124D) 14228. 15 + 15 + 15 + 15 +15 + 15 = 5 × _?_A) 18B) 15C) 6D) 529. A zoo has 3 as many female as animals. If there are a total of 24 in the zoo, how many of these are female?A) 21B) 18C) 16D) 630. Each of my triangle is twice as as the diameter of my circle. If the radius of my is 4, what is the perimeter of my triangle?A) 6B) 24C) 32D) 48。
小学数学：2022-2023年度美国大联盟(MathLeague)思维探索四年级试题(含参考答案)
2022-2023年度美国Math League思维探索第一阶段活动(四年级)(答题时间：75分钟，总分：175分)学生诚信协议：答题期间，我确定没有就所涉及的问题或结论，与任何人、用任何方式交流或讨论，我确定我所填写的答案均为我个人独立完成的成果，否则愿接受本次成绩无效的处罚。
选择题：每小题5分，答对加5分，答错不扣分，共175分。
1.If each cone has3scoops of ice cream,24cones have?scoops in total.A)8B)27C)48D)722.Eighteen pairs of shoes is6×?shoes.A)3B)6C)12D)363.How many divisors of2022are multiples of2?A)1B)2C)3D)44.If10days after my last birthday was a Tuesday,on what day of the week was my last birthday?A)Thursday B)Friday C)Saturday D)Sunday5.The sum of the digits of a3-digit whole number is11.What is the greatest possible digit of this wholenumber?A)3B)4C)8D)96.If I missed the basket on every3rd throw of48throws,I missed the baskettimes.A)3B)16C)32D)457.If the sum of the lengths of three sides of a square is18,then the perimeter of the square isA)6B)24C)36D)728.The total number of hours in20days is?more than20hours.A)400hours B)456hours C)460hours D)476hours9.The greatest integer on a list of consecutive integers is30more than the smallest on the list.How manyintegers are on the list?A)15B)16C)30D)3110.Each year,Ann saves3times as much money as she spends.If Ann saved$150last year,how muchmore money did she save last year than she spent last year?A)$50B)$75C)$100D)$45011.What is the greatest possible value of one of two primes whose product is less than20?A)7B)11C)15D)1912.Sue swam a total of18laps in4consecutive days.On each day she swam onemore lap than on the day before.How many laps did Sue swim on the third day?A)4B)5C)17D)1913.The smallest multiple of6that is2more than a multiple of7isA)24B)28C)30D)3614.When I switch the order of the digits of a2-digit integer,the new integer is9less than the originalinteger.The new integer could beA)12B)13C)14D)1515.Kay counted to72by2s.May counted to72by3s.How many more numbers did Kay count than Maycounted?A)1B)12C)24D)3616.How many factors of1×2×3×4×5are factors of1×2×3×4?A)3B)5C)6D)817.If the area and perimeter of a square with integer side-lengths are numerically equal,the product of thearea and perimeter must beA)4B)16C)32D)25618.A total of480kids wait on8equal lines to ride a rollercoaster.The first3kidsfrom each line get on each ride.If no new kids get on any line,how many ridesmust run until each kid has ridden the rollercoaster?A)20B)24C)60D)18019.The sum of the digits of the greatest3-digit integer with one digit a multiple of2,another a multiple of3,and a3rd digit a multiple of5isA)17B)18C)21D)2220.I have an equal number of nickels and quarters.The total value of these coins,in cents,is?times thetotal number of these coins.A)5B)15C)20D)3021.The number of days it rained in my town last year is the same number of days in an odd number ofwhole weeks.If it rained more than100days last year,the fewest days it could have rained isA)103B)105C)107D)10922.Birds from a flock of200birds began landing on a10m-long wire.If the wire isdivided into50cm-long sections and at most8birds can occupy each section,how many birds from the flock will not have room to land on the wire?A)0B)20C)40D)8023.I wrote consecutive multiples of6,starting with6and ending with72.Which of the following is not afactor of any multiple I wrote?A)21B)22C)27D)2824.My cat sleeps from midnight to6AM each day,and then he sleeps for20minutesof every3hours the rest of each day.What is the average number of minutes mycat sleeps per hour each day?A)10B)20C)30D)3625.The sum of the100smallest even whole numbers greater than0is the product of what two numbers?A)100&25B)100&50C)100&100D)100&10126.Sally went shelling.Every third shell Sally found was dirty.Every fourth shell she found was cracked.How many of the84shells Sally found were dirty but not cracked?A)7B)14C)16D)2127.How many whole numbers greater than99and less than1000consist of exactly2different digits?A)243B)270C)297D)30028.It cost me$30to buy2balls and2bats.At the same unit prices,it cost mysister$65for5balls and4bats.The cost of1bat isA)$5B)$10C)$15D)$2029.The product of a2-digit number and a4-digit number may have?digits.A)6B)7C)8D)930.Cy and her5friends,all working at the same rate,can make a snowman in30minutes.If Cy and all her friends work at this same rate,the fewest numberof friends needs to help Cy make a snowman in20minutes or less isA)6B)7C)8D)931.The four digits of2022can be arranged to create?different4-digit whole numbers.A)3B)4C)6D)1232.The number of orangutans living in my tree increases by a constant number eachyear.If there are40orangutans now and there were24two years ago,how manyorangutans will be in my tree in six years?A)64B)72C)88D)13633.The greatest number of obtuse angles that a rhombus can have isA)1B)2C)3D)434.How many positive integers less than60have both4and6as factors?A)2B)3C)4D)535.The product of three whole numbers is36.What could not be the sum of the three?A)10B)11C)12D)13。
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美国数学题21、Jump Like a fleaA flea can jump 350 times its body length. If humans could jump like fleas, how far could you jump?SolutionAnswers will vary, depending on the height of the person. The average height of an 11–14-year-old male is 60 inches. The average height of a female of the same age is 62 inches, according to the National Institute of Health. A student who is 5 feet tall could jump about 1750 feet, or 583 yards. For varying answers, multiply the height by 350 and approximate that distance with a known landmark.From Menu of Problems, April 20012、Product of Primes2006 is the product of three primes, 2 × 17 × 59. Find the first year that occurs after 2006, which is the product of three consecutive primes.Solution2431, which is the product of 11 × 13 × 17.From Menu of Problems, April 2006Adding DigitsIf you add the digits in a number, how many numbers between 0 and 1000 will have a sum of 15?Solution73. Only two ways are possible to add two digits to get a sum of 15: 6 + 9 and 7 + 8. Each combination results in six numbers: 69, 96, 609, 690, 906, and 960. Eight ways exist to combine three distinct digits for a sum of 15: 1 + 5 + 9, 1 + 6 + 8, 2 + 4 + 9, 2 + 5 + 8, 2 + 6 + 7, 3 + 4 + 8, 3 + 5 + 7, and 4 + 5 + 6. Again, each combination results in six numbers. Four other ways to get 15 are 1 + 7 + 7, 3 + 3 + 9, 3 + 6 + 6, and 4 + 4 + 7. Each combination results in three numbers. Finally, 555 fits the bill. All told, we have 12 + 48 + 12 + 1 = 73.From April's Menu of Problems, April 19993、TwinsThe product of the ages of twins and their younger brother is 36. How old are the children?Solution1, 6, and 6. The possible ages of the children are 1, 1, 36; 2, 2, 9; 3, 3, 4; and 1, 6, 6. The only choice that has the twins as the oldest children is 1, 6, 6.From May's Menu of Problems, May 19994、5, 7, and 11 Minute GlassesHow can you measure 13 minutes exactly with a 5-minute, 7-minute, and 11-minute hourglass?SolutionStart the 5- and 11-minute timers at the same time. When the 5-minute runs out, start timing for the 13 minutes. When the 6 minutes remaining on the 11-minute timer are finished, start the 7-minute timer.From Menu of Problems, April 20025、Make the Statement TrueAdd one straight line segment to make the mathematical sentence 5 + 5 + 5 = 550 a true statement.SolutionChange one of the plus signs to a number 4 by connecting the left and top endpoints. There might be other possibilities as well.From April's Menu of Problems, April 20076、Smallest Odd NumberWhat is the smallest odd number you can obtain from the product of four different prime numbers?Solution1155. Remember that 1 is not a prime number. Two is the first prime number, but if it is one of the prime numbers used, the product will be even. Therefore, to obtain the smallest value, you would use 3, 5, 7, and 11, the product of which is 1155. From April's Menu of Problems, April 20077、Multiples of 6Find six consecutive multiples of 6 whose sum is the least common multiple (LCM) of 13 and 18.Solution24, 30, 36, 42, 48, and 54. The LCM of 13 and 18 is 13 × 18 = 234, since the numbers are relatively prime. Since we are looking for 6 numbers, we can divide 234 by 6 to get 39. Therefore, we know that 39 is greater than 3 of the numbers and less than 3 of the numbers; 24, 30, and 36 are the multiples of 6 less than 39, and 42, 48, and 54 are the multiples of 6 greater than 39. The sum of 24, 30, 36, 42, 48, and 54 is 234. From March's Menu of Problems, March 20078、Drawing RectanglesHow many different rectangles can be drawn using the hour marks on a clock's face as vertices?Solution15.The rectangles that can be drawn are shown below.From February's Menu of Problems, February 2004 MTMS.9、Paper MoneyIn how many different ways can you receive $20 from your bank if you ask for paper money only?SolutionTen, excluding the use of $2 bills.Students can solve this problem by making an organized list.Using $2 bills creates another thirty-one choices.From November's Menu of Problems, November 1999 MTMS.Contributed by Cynthia Barb, Kent State University--Stark Campus, Canton, Ohio; and Anne Larson Quinn, Edinboro University of Pennsylvania, Edinboro, Pennsylvania. 10、Half of a Half of a HalfWhat would you get if you divided a half of a half of a half by a half of a fourth of a half?Solution2. A half of a half of a half is 1/2×1/2×1/2 = 1/8, and a half of a fourth of a half is1/2×1/4×1/2 = 1/16; 1/8 divided by 1/16 is 2.From November's Menu of Problems, November 200611、Prime NumbersFind the sum of the least and greatest two-digit prime numbers whose digits are also prime.Solution96.Consider prime numbers such that each digit is 2, 3, 5, or 7. The smallest two-digit number that can be formed with these digits is 22, but it is not prime. Therefore, 23 is the least value.The greatest two-digit numbers that can be formed with these digits are 77 and 75, but they are not prime. The number with the greatest value that meets the conditions is 73; therefore, 23 + 73 = 96.From February's Menu of Problems, February 2005 MTMS.Courtesy of David Rock at the University of Mississippi.12、Multiple of 12Find the greatest ten-digit positive multiple of 12 using each digit once and only once.Solution9,876,543,120. For a number to be a multiple of 12, it must be divisible by both 3 and 4. The sum of the ten digits 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 + 0 = 45, which is divisible by 3. To be divisible by 4, the last two digits must be divisible by 4. The largest number obtained using all ten digits once and only once is 9,876,543,210, but 10, the last two digits, is not divisible by 4. The next greatest value is 9,876,543,201, but 01 is not divisible by 4. The next greatest value is 9,876,543,120; since 20 is divisible by 4, the number is therefore both divisible by 3 and 4 and 12.From March's Menu of Problems, March 200713、Positive PrimesWhat is the last digit (ones) of the product of the positive prime numbers less than 50?Solution0. Prime numbers less than 50 end in 1, 3, 7, or 9, with the exception of 2 and 5. Since 2 and 5 will have to be factors and 2×5 equals 10, the product must has a ones digit of 0.From May's Menu of Problems, May 200714、Hands of a ClockDetermine the smaller angle between the hands of a clock at 8:20, 12:20, and 1:30. Solution12:20, 110°; 8:20, 130°; 1:30, 135°.The hour hand moves 360°/12 = 30° each hour, or 10° each 20 minutes and 5° each 10 minutes. The minute hand moves 30° every 5 minutes.At 12:20, the angle between the hour and minute hands is 4(30°) - 10° = 110°, or3(30°) + 20°.At 8:20, the hour hand is 10° past the 8. The angle between the 4 and the 8 at 20 minutes past 8 is 4(30°) - 120°, so the angle between the hour and minute hands at 8:20 is 130°.At 1:30, the angle between hands is 6(30°) - 45° = 135°, or 4(30°) + 15°.Students should share other ways that they use to calculate these angles.From January's Menu of Problems, January 2003 MTMS.15、4-digit Even NumbersHow many four-digit numbers consist of only even digits?Solution500. The even digits are 0, 2, 4, 6, and 8. The first digit cannot be a 0, because the number would not be a four-digit number, so you may choose only 2, 4, 6, or 8 for the first digit. Five digits can be chosen for the remaining columns. Therefore, there are 4 × 5 × 5 × 5, or 500, ways to choose the digits. (Note that if the second digit were also 0, it would not be a three-digit number.)From October's Menu of Problems, October 200616、Squares on a BoardHow many squares are on a traditional 8 × 8 checkerboard? (By the way, the answer is not 64.)Solution204 squares. You have one 8 × 8 square, four 7 × 7 squares, nine 6 × 6 squares, sixteen 5 × 5 squares, twenty-five 4 × 4 squares, thirty-six 3 × 3 squares, forty-nine 2 × 2 squares, and sixty-four 1 × 1 squares, or 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 = 204. By constructing a table, you can see a pattern. Refer to print version to view table. From Menu of Problems, September 200617、RemainderWhat is the remainder when the product (1492)(1776)(1812)(1999) is divided by 5? Solution1. When considering the remainder when dividing by 5, we need to be concerned with only the units digit of each number. The product of the units digits is 2 × 6 × 2 × 9, or 216, which leaves a remainder of 1 when divided by 5.From Menu of Problems, September 199918、Prime FactorsFind the difference between the least and greatest prime factors of 33,660. Solution15. The prime factorization of 33,660 is 2 × 2 × 3 × 3 × 5 × 11 × 17. The difference between 17 and 2 is 15.From January's Menu of Problems, January 200719、How Many MinutesHow many minutes represent 10 percent of one full week?Solution1,008 minutes; 7 days × 24 hours × 60 minutes = 10,080 minutes in a week; 10 percent is 1,008 minutes.From April's Menu of Problems, April 200720、Candy CornEach Halloween, 20 million pounds of candy corn are sold. If each piece weighs about 1 gram, how many pieces are bought?SolutionRoughly 9 billion. Each pound of candy corn is 453 grams, so 20 million x 453 grams ≅9 billion pieces of candy corn.From October's Menu of Problems, October 2002 MTMS.。