2018-2019年美国“大联盟”(Math League)思维探索活动第一阶段三年级试卷及答案
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。
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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。
历届美国数学邀请赛试题集
历届美国数学邀请赛试题集历届美国数学邀请赛试题集是向全球数学爱好者提供的难题挑战。
参加这些比赛需要高超的数学水平和一些独特的思维技巧。
以下是一些历届数学邀请赛试题集的例子，让我们一起领略一下这些难题带来的魅力：第一份试题集：1. 计算$\frac{1}{1 \times 2 \times 3}+\frac{1}{2 \times 3 \times4}+\frac{1}{3 \times 4 \times 5}+\cdots+\frac{1}{99 \times 100 \times 101}$的值。
2. 求以 $(1, 2), (3, 5), (4, 5)$为顶点的三角形的面积。
3. $a_1=1, a_2=2, a_3=3, a_4=4$，对于 $n>4$，$a_n$ 是所有使得$a_1+a_2+\cdots+a_{n-1}+k \equiv 0 \pmod n$ 的 $k$ 的最小正整数解。
求$a_{2019}$。
4. 在平面上给定 $2n$ 点，其中没有三点共线。
证明存在一种方法将这$2n$ 点分成两组，每组包含各 $n$ 个点，并且这两组点可以每组内同时作一条直线使得这两条直线平行且距离相等。
第二份试题集：1. 已知 $\triangle ABC$ 中 $\angle A=60^\circ$，$AD$ 为 $BC$ 上的高，$M$ 是 $AB$ 上的满足 $\angle ACD = \angle AMD$ 的点。
证明：$\angle BMC = 90^\circ$。
2. 对于正整数 $n$，设 $S(n)$ 表示所有可以表示成 $a^2+b^2+c^2$，其中 $a, b, c$ 都是不同的正整数且 $a<b<c<n$ 的数的和，求 $S(17)$ 的值。
3. 设 $p$ 是一个奇素数，$b$ 是一个介于 $2$ 到 $p-2$ 之间的整数。
证明：$b^{p-1} \equiv 1 \pmod {p^2}$ 的充分必要条件是 $b \equiv \pm 1\pmod p$。
2018年美国“数学大联盟杯赛”（中国赛区）初赛四年级试卷（1）
2018年美国“数学⼤联盟杯赛”（中国赛区）初赛四年级试卷（1）2017-2018年度美国“数学⼤联盟杯赛”(中国赛区)初赛(四年级)(初赛时间：2017年11⽉26⽇，考试时间90分钟，总分200分)学⽣诚信协议：考试期间，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. 本试卷是四年级试卷，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。
考试完毕，请务必将英⽂词汇⼿册带回家，上⾯有如何查询初赛成绩、及如何参加复赛的说明。
其他材料均不能带⾛，请留在原地。
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.Which of the following is the smallest?A) 2.018 B) 20.18 C) 0.218 D) 20182.What is the least common multiple of 20 and 18?A) 90 B) 180 C) 240 D) 3603.The sum of the degree-measures of the exterior angles of a triangle is?A) 180 B) 360 C) 540 D) 7204.In the figure on the right, please put the numbers 1 – 11 in the elevencircles so that the three numbers in every straight line add up to 18.What is the number in the middle circle? Note: There are 5 straightlines in total in this figure.A) 6 B) 7 C) 8 D) 95.I am a lovely cat. When I multiply the digits of a whole numberand the product I get is 8, I put that whole number on my list offavorite numbers. Of the whole numbers from 1000 to 9999,how many would I put on my list of favorite numbers?A) 10 B) 12 C) 16 D) 206.Two planes take off at the same time from the same point to race to apoint and back. Place A travels at 180 miles per hour on the way out and240 miles per hour on the return trip. Plane B covers the entire distance at an averagespeed of 210 miles per hour. Which plane wins the race, or is it a tie?A) plane A wins B) plane B winsC) a tie D) non-deterministic 7.52 × 88 = 44 ×?A) 102 B) 96 C) 104 D) 1248.What is the smallest whole number that leaves a remainder of 4, 5, 6 when divided byeach of 5, 6, 7?A) 29 B) 209 C) 210 D) 20099.In △ABC, m∠A + m∠C = m∠B. What is the degree measure of ∠B?A) 80 B) 90 C) 100 D) 18010.I bought a toy for $10, sold it for $20, rebought it for $30, and resold it for $40. My totalprofit on the 4 transactions was ?A) 10 B) 20 C) 30 D) 4011.What is the greatest number of integers I can choose from the first ten positive integers sothat any 3 of the chosen integers could be the lengths of the three sides of a triangle?A) 4 B) 5 C) 6 D) 712.How many whole numbers between 200 and 400 have all their digits increasing in valuewhen read from left to right?A) 30 B) 36 C) 42 D) 4813.What is the value of 1% of 10% of 100?A) 0.01 B) 0.1 C) 1 D) 1014.If three cats can eat three bowls of food in three minutes, how many minutes will it take100 cats to eat 100 bowls of food?A) 1 B) 3C) 100 D) None of the above15.There are three squares. The area of the smallest one is 2. The side-length of the secondsquare is twice the side-length of the smallest one. And the side-length of the third square is three-times the side-length of the smallest one. The total area of the three squares isA) 12 B) 28 C) 36 D) 7216.A man, who had been married for three years, spent25of his yearly income on his family,14on business, and110on personal travel. If he saved $45000 during those three years, what was his annual income?A) $45000 B) $50000C) $65000 D) None of the above17.Given four different integers, at most how many different sums can be formed bychoosing two, three, or four of them and finding each sum?A) 8 B) 9 C) 10 D) 1118. Max places 100 eggs in 10 baskets, with each basket receiving at least1 egg, but no2 baskets receiving the same number of eggs. What is the greatest number of eggs that may be placed in a basket?A) 45 B) 47 C) 55 D) 6519. 2 + 3 × 4 – 5 =A) 0 B) 6 C) 9 D) 15 20. What is the highest power of 2 that divides 2 × 4 × 6 × 8 × 10? A) 25 B) 27 C) 28 D) 215 21. Which of the following is a prime number?A) 2017B) 2018C) 2015D) 201622. What is the greatest possible number of acute angles in a figure consisting of a triangleand a line passing through two sides of the triangle?A) 5B) 6C) 7D) 823. Amy can solve 5 questions every 3 minutes. Kate can solve 3 questions every 5 minutes.How many more questions Amy can solve than Kate in one hour?A) 15B) 32C) 60D) 6424. Using 3 Ts and 2 Js, in how many different orders can the five letters be arranged? Forexample, TTTJJ and TTJJT are two such different orders.A) 2B) 10C) 20D) 6025. Coastal Coconuts can divide all their coconuts evenly among 8, 9, or10 customers, with 1 coconut left over each time. If Coastal Coconuts has more than 1 coconut, what is the least number of coconuts they could have?A) 561 B) 721C) 831 D) None of the above 26. 35 ÷ 32 =A) 3 B) 9 C) 27 D) 81 27. If the sum of three prime numbers is 30, what is the least prime number?A) 2B) 3C) 5D) 728. Juxtaposing two identical squares to form a rectangle, the perimeter of the rectangle is 12less than the sum of the perimeter of the two squares. What is the side-length of the original square?A) 3B) 6C) 9D) 1229. It takes Mike 2 hours to finish a task. It takes 4 hours for Tom to finish the same task.Mike and Tom worked together on this task for one hour before Mike had to leave. How long will it take Tom to finish the rest of the task?A) 1 B) 2 C) 3 D) 4 30. The number of triangles in the figure on the right isA) 9 B) 10 C) 11 D) 12 31. What is the thousands digit of the product 1234560 × 2345670 × 3456780?A) 8B) 6C) 5D) 032. The sum of nine consecutive positive integers is always divisible byA) 2B) 5C) 7D) 933. You can put as many as 96 books in 6 backpacks. How many backpacks are necessary forA) 7B) 8C) 9D) 1034. The number of nickels I have is twice the number of dimes I have, and together thesecoins are worth more than $1. The least number of dimes that I can have isA) 5B) 6C) 8D) 1035. The ages of four kids are four consecutive positive integers. The product of their ages is3024. How old is the oldest kid?A) 8B) 9C) 10D) 1136. In the Game of Life, you earn 3 points for flipping a coin to “heads”, and 5 points forflipping a coin to “tails”. In all, how many positive whole number scores are IMPOSSIBLE to get after flipping it one or more times?A) 4B) 5C) 7D) 1137. Four monkeys can eat four bags of peanuts in three minutes. How many monkeys will ittake to eat 100 bags of peanuts in one hour?A) 4 B) 5 C) 20 D) 100 38. The tens digit of the product of the first 100 positive integers isA) 2B) 4C) 8D) 039. Someone put three dimes into my pile of quarters. If I add up the value of these coins,including the dimes, the sum could beA) $6.25B) $7.75C) $8.0540. Brooke's empty tub fills in 20 minutes with the drain plugged, and her full tub drains in 10 minutes with the water off. How many minutes would it take the full tub to drain while the water is on?A) 12B) 15 C) 20 D) 30。
2018-2019年美国“大联盟”(Math League)思维探索活动第一阶段四年级试卷及答案
2018-2019年度美国“大联盟”(Math League)思维探索活动第一阶段(四年级)(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. 本试卷是四年级试卷，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.(123 + 456) + 678 = (123 + 678) + ?A) 123 B) 456 C) 579 D) 6782.Bea sharpened 1200 pencils. Half the pencils had erasers, andhalf of all the erasers were pink. How many pencils with pinkerasers did Bea sharpen?A) 200 B) 300C) 400 D) 6003.I have a prime number of pairs of socks. The total number of socks I have could not beA) 26 B) 38 C) 46 D) 544.The product of 500 000 and 200 000 has exactly ? zeros.A) 5 B) 6 C) 10 D) 115.Divide 100 by 10, then multiply the result by 10. The final answer isA)0 B)1 C) 10 D) 1006.At most how many complete 8-minute songs can I sing in 3 hours?A) 22 B) 23 C) 24 D) 1807. 2 × (44 + 44 + 44) = 88 + 88 + ?A) 0 B) 44 C) 66 D) 888. A rectangle has sides of even lengths and perimeter 12. Its area isA) 6 B) 8 C) 9 D) 169.16 × (17 + 1) –? × (15 + 1) = 0A) 15 B) 16 C) 17 D) 1810.The crowd clapped for 840 seconds, stopping at 8:15 P.M. Theystarted clapping at ? P.M.A) 7:59 B) 8:01C) 8:08 D) 8:1411.If each digit of my 5-digit ID code is different, the sum of its digitsis at mostA) 15 B) 25 C) 35 D) 4512.At the museum, adult tickets cost $4 each and child tickets cost $3 each. With $50, I canbuy ? more child tickets than adult tickets.A) 1 B) 4 C) 12 D) 1613.Each day last week I read for a whole number of hours. I read forthe same number of hours each day except Sunday. If I read for 12hours last week, I read for ? hours on Sunday.A) 7 B) 6 C) 5 D) 414.The product 2 × 3 × 4 × 5 × 6 has the same value as the product ? × 3 × 5.A) 12 B) 36 C) 48 D) 6315.The average test grade in my class is a whole number, and the sum of the test grades is2400. Of the following, which could be the total number of test grades?A) 18 B) 21 C) 27 D) 3216.If twice a whole number is 120 less than five times the same whole number, then half thewhole number isA) 10 B) 20 C) 40 D) 6017.The number of bees I have doubles each day. If I had 1024 bees last Friday, the first daythe number of bees was more than 100 was aA) Tuesday B) WednesdayC) Thursday D) Friday18.Each of 6 dogs ate 3 treats from each of 4 bags. If each bag started with 30 treats, the 4bags together ended with ? treats.A) 36 B) 48 C) 72 D) 9619.What is the greatest possible product of two different even whole numbers whose sum is100?A) 196 B) 625 C) 2496 D) 250020. Ed built 3 times as many houses as Bob, who built half as many houses as Ally. If the 3 of them built 96 houses in all, Ed and Ally built a combined total of ? houses.A) 16 B) 32 C) 48D) 8021. How many factors of 2 × 4 × 8 × 16 are multiples of 4?A) 3B) 4C) 8D) 922. When I divide a certain number by 3 or 5, I get a remainder of 2. The sum of the digits ofthe least number for which this is true isA) 1B) 3C) 7D) 823. My 144 fish are split between 2 tanks so that 1 tank has twice as many fish as the other. How many fish must I move from one tank to the other so that both tanks have the same number of fish?A) 24 B) 48 C) 60D) 7224. A 3-digit number is the product of at most ? whole numbers greater than 1.A) 2B) 3C) 9D) 1025. Abby earns $2 for every clam she finds and $3 for every oyster. If Abby finds 5 times asmany oysters as clams, which of the following could be her total earnings?A) $150B) $160C) $170D) $18026. (The average value of the 10 smallest even whole numbers greater than 0) – (the average value of the 10 smallest odd whole numbers) =A) 0B) 1C) 10D) 1127. Ana planted seeds in rows. If the total number of rowsequaled the number of seeds in each row, the number of seeds planted could have beenA) 194B) 216C) 250D) 28928. What is the greatest possible sum of five 2-digit whole numbers if all 10 digits of the fivenumbers are different?A) 270B) 315C) 360D) 48529. I thought I wrote every whole number between 1 and 500 in order from least to greatest, but actually I skipped 3 numbers in a row. If I left out a total of 8 digits, what is the sum of the numbers I skipped?A) 100B) 150C) 300D) 39030. Written backwards, 123 becomes 321. How many whole numbers between 100 and 200 have a larger value when written backwards?A) 70B) 80C) 90D) 9831. The average of four different numbers is 18. And the least of the four numbers is 3. What is the least possible value of the biggest of the four numbers?A) 21B) 23C) 24D) 6032. 3 tigers can eat 36 Big Macs in 6 minutes. How many Big Macs can 12 tigers eat in 3 minutes?A) 18B) 36C) 72D) 28833. In the followi ng sequence, 2, 0, 1, 8, 2, 0, 1, 8, … (repeating), what is its 2018th term?A) 2B) 0C) 1D) 834. 2018a b c is a multiple of 9. What is the least possible value of abc ? (Note: abc is a three-digit number, which means a is not 0.)A) 7B) 100C) 106D) 99735. What is the least common multiple of 84 and 112?A) 28B) 196C) 336D) 940836. In triangle ABC , ∠C = 90°, ∠A = 15°, AB = 20. What is the area of this triangle?A) 20B) 50C) 100D) 20037. ABCD is a rectangle and its perimeter is 22, as shown at the right. EFGH is a square. AH = 6. CF = ?A) 4 B) 5 C) 6D) 838. How many leap years are there between 2018 and 2081?A) 16B) 17C) 18D) 1939. My class was lined up on the gym floor in 8 rows, with 2 students in each row. If our coach rearranged us so that the number of rows was the same as the number of students in each row, how many rows were there after we were rearranged? A) 4B) 6C) 10D) 1640. Of the 100 numbers from 1 to 100, how many of them don’t contain 7 as its digit?A) 65 B) 75C) 80 D) none of the above。
美国“数学大联盟杯赛”介绍演示文稿PPT课件
a) /education/best-highschools/national-rankings/spp+100
2. 美国“数学大联盟杯赛”第一届竞赛于1977年 举行， 至今已经连续举办了35年。
3. 每年有来自全球的超过一百万名中小学生参加。
2021/3/12
4
Math League History
Mr. Steven R. Conrad & Mr. Daniel Flegler: 美国著 名的数学教育家
students.php c) Lectures
a) Brian Conrad, /2012/finals/professor/professor1.php
b) Ravi Vakil, /2012/finals/professor/professor3.php
2. Math League in China
a) Intro of Committee Members b) Comparison c) 2012-2013 Math League in China d) Online Learning System
2021/3/12
2
Math League in North America & The World
b) /jointly/consequence.php
b) Harvard Twins Sisters
a) /jointly/consequence.php#
c) US newspaper coverage
2018-2019年美国“大联盟”(Math League)思维探索活动第一阶段六年级试卷及答案
B) Monday
C) Tuesday
D) Friday
12. Rounding a decimal to the nearest whole number yields a number that is at most ? greater than the original decimal.
A) 0.05
B) 0.1
C) 0.5
D) 0.9
13. The perimeter of a rectangle with area 2019 and integral side-lengths is greatest when its length and width diffC) 670
A) 6
B) 9
C) 12
D) 18
3. The number 36 is the product of -6 and
A) -6
B) 6
C) -30
D) 42
4. 20 ×18 = 20 ×19 + 20 × ?
A) -1
B) 0
C) 1
D) 20
5. Angel arrived 3 of an hour early for her noon appointment. At what time did Angel 10
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AMC8(美国数学竞赛)历年真题、答案及中英文解析
AMC8（美国数学竞赛）历年真题、答案及中英文解析艾蕾特教育的AMC8 美国数学竞赛考试历年真题、答案及中英文解析：AMC8-2020年：真题 --- 答案---解析（英文解析+中文解析）AMC8 - 2019年：真题----答案----解析（英文解析+中文解析）AMC8 - 2018年：真题----答案----解析（英文解析+中文解析）AMC8 - 2017年：真题----答案----解析（英文解析+中文解析）AMC8 - 2016年：真题----答案----解析（英文解析+中文解析）AMC8 - 2015年：真题----答案----解析（英文解析+中文解析）AMC8 - 2014年：真题----答案----解析（英文解析+中文解析）AMC8 - 2013年：真题----答案----解析（英文解析+中文解析）AMC8 - 2012年：真题----答案----解析（英文解析+中文解析）析）AMC8 - 2010年：真题----答案----解析（英文解析+中文解析）AMC8 - 2009年：真题----答案----解析（英文解析+中文解析）AMC8 - 2008年：真题----答案----解析（英文解析+中文解析）AMC8 - 2007年：真题----答案----解析（英文解析+中文解析）AMC8 - 2006年：真题----答案----解析（英文解析+中文解析）AMC8 - 2005年：真题----答案----解析（英文解析+中文解析）AMC8 - 2004年：真题----答案----解析（英文解析+中文解析）AMC8 - 2003年：真题----答案----解析（英文解析+中文解析）AMC8 - 2002年：真题----答案----解析（英文解析+中文解析）AMC8 - 2001年：真题----答案----解析（英文解析+中文解析）AMC8 - 2000年：真题----答案----解析（英文解析+中文解析）析）AMC8 - 1998年：真题----答案----解析（英文解析+中文解析）AMC8 - 1997年：真题----答案----解析（英文解析+中文解析）AMC8 - 1996年：真题----答案----解析（英文解析+中文解析）AMC8 - 1995年：真题----答案----解析（英文解析+中文解析）AMC8 - 1994年：真题----答案----解析（英文解析+中文解析）AMC8 - 1993年：真题----答案----解析（英文解析+中文解析）AMC8 - 1992年：真题----答案----解析（英文解析+中文解析）AMC8 - 1991年：真题----答案----解析（英文解析+中文解析）AMC8 - 1990年：真题----答案----解析（英文解析+中文解析）AMC8 - 1989年：真题----答案----解析（英文解析+中文解析）AMC8 - 1988年：真题----答案----解析（英文解析+中文解析）析）AMC8 - 1986年：真题----答案----解析（英文解析+中文解析）AMC8 - 1985年：真题----答案----解析（英文解析+中文解析）◆AMC介绍◆AMC（American Mathematics Competitions) 由美国数学协会（MAA）组织的数学竞赛，分为 AMC8 、 AMC10、 AMC12 。
2018-2019年美国“大联盟”(Math League)思维探索活动第一阶段五年级试卷
act is twice as long as the 3rd, what fraction of the play is the 3rd act?
A) 1/9
B) 2/9
C) 3/9
D) 4/9
15. If I double my speed of 12 000 m/hr., my new speed will be
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won every 6th game we played. If my team qualified for the finals after our
96th game, how many wins did we need?
A) 12
B) 16
C) 18
D) 90
12. What is the greatest common factor of 1 ×3 ×5 ×7 ×9 and 2 ×4 ×6 ×8 ×10?
A) 200 m/min.
B) 400 m/min.
C) 600 m/min.
D) 2400 m/min.
16. Which of the following could be the perimeter of an equilateral triangle with integral side-lengths?
2018-2019年美国“大联盟”(Math League)思维探索活动第一阶段四年级试卷
2018-2019年度美国“大联盟”(Math League)思维探索活动第一阶段(四年级)(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. 本试卷是四年级试卷，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.(123 + 456) + 678 = (123 + 678) + ?A) 123 B) 456 C) 579 D) 6782.Bea sharpened 1200 pencils. Half the pencils had erasers, andhalf of all the erasers were pink. How many pencils with pinkerasers did Bea sharpen?A) 200 B) 300C) 400 D) 6003.I have a prime number of pairs of socks. The total number of socks I have could not beA) 26 B) 38 C) 46 D) 544.The product of 500 000 and 200 000 has exactly ? zeros.A) 5 B) 6 C) 10 D) 115.Divide 100 by 10, then multiply the result by 10. The final answer isA)0 B)1 C) 10 D) 1006.At most how many complete 8-minute songs can I sing in 3 hours?A) 22 B) 23 C) 24 D) 1807. 2 × (44 + 44 + 44) = 88 + 88 + ?A) 0 B) 44 C) 66 D) 888. A rectangle has sides of even lengths and perimeter 12. Its area isA) 6 B) 8 C) 9 D) 169.16 × (17 + 1) –? × (15 + 1) = 0A) 15 B) 16 C) 17 D) 1810.The crowd clapped for 840 seconds, stopping at 8:15 P.M. Theystarted clapping at ? P.M.A) 7:59 B) 8:01C) 8:08 D) 8:1411.If each digit of my 5-digit ID code is different, the sum of its digitsis at mostA) 15 B) 25 C) 35 D) 4512.At the museum, adult tickets cost $4 each and child tickets cost $3 each. With $50, I canbuy ? more child tickets than adult tickets.A) 1 B) 4 C) 12 D) 1613.Each day last week I read for a whole number of hours. I read forthe same number of hours each day except Sunday. If I read for 12hours last week, I read for ? hours on Sunday.A) 7 B) 6 C) 5 D) 414.The product 2 × 3 × 4 × 5 × 6 has the same value as the product ? × 3 × 5.A) 12 B) 36 C) 48 D) 6315.The average test grade in my class is a whole number, and the sum of the test grades is2400. Of the following, which could be the total number of test grades?A) 18 B) 21 C) 27 D) 3216.If twice a whole number is 120 less than five times the same whole number, then half thewhole number isA) 10 B) 20 C) 40 D) 6017.The number of bees I have doubles each day. If I had 1024 bees last Friday, the first daythe number of bees was more than 100 was aA) Tuesday B) WednesdayC) Thursday D) Friday18.Each of 6 dogs ate 3 treats from each of 4 bags. If each bag started with 30 treats, the 4bags together ended with ? treats.A) 36 B) 48 C) 72 D) 9619.What is the greatest possible product of two different even whole numbers whose sum is100?A) 196 B) 625 C) 2496 D) 250020. Ed built 3 times as many houses as Bob, who built half as many houses as Ally. If the 3 of them built 96 houses in all, Ed and Ally built a combined total of ? houses.A) 16 B) 32 C) 48D) 8021. How many factors of 2 × 4 × 8 × 16 are multiples of 4?A) 3B) 4C) 8D) 922. When I divide a certain number by 3 or 5, I get a remainder of 2. The sum of the digits ofthe least number for which this is true isA) 1B) 3C) 7D) 823. My 144 fish are split between 2 tanks so that 1 tank has twice as many fish as the other. How many fish must I move from one tank to the other so that both tanks have the same number of fish?A) 24 B) 48 C) 60D) 7224. A 3-digit number is the product of at most ? whole numbers greater than 1.A) 2B) 3C) 9D) 1025. Abby earns $2 for every clam she finds and $3 for every oyster. If Abby finds 5 times asmany oysters as clams, which of the following could be her total earnings?A) $150B) $160C) $170D) $18026. (The average value of the 10 smallest even whole numbers greater than 0) – (the average value of the 10 smallest odd whole numbers) =A) 0B) 1C) 10D) 1127. Ana planted seeds in rows. If the total number of rowsequaled the number of seeds in each row, the number of seeds planted could have beenA) 194B) 216C) 250D) 28928. What is the greatest possible sum of five 2-digit whole numbers if all 10 digits of the fivenumbers are different?A) 270B) 315C) 360D) 48529. I thought I wrote every whole number between 1 and 500 in order from least to greatest, but actually I skipped 3 numbers in a row. If I left out a total of 8 digits, what is the sum of the numbers I skipped?A) 100B) 150C) 300D) 39030. Written backwards, 123 becomes 321. How many whole numbers between 100 and 200 have a larger value when written backwards?A) 70B) 80C) 90D) 9831. The average of four different numbers is 18. And the least of the four numbers is 3. What is the least possible value of the biggest of the four numbers?A) 21B) 23C) 24D) 6032. 3 tigers can eat 36 Big Macs in 6 minutes. How many Big Macs can 12 tigers eat in 3 minutes?A) 18B) 36C) 72D) 28833. In the followi ng sequence, 2, 0, 1, 8, 2, 0, 1, 8, … (repeating), what is its 2018th term?A) 2B) 0C) 1D) 834. 2018a b c is a multiple of 9. What is the least possible value of abc ? (Note: abc is a three-digit number, which means a is not 0.)A) 7B) 100C) 106D) 99735. What is the least common multiple of 84 and 112?A) 28B) 196C) 336D) 940836. In triangle ABC , ∠C = 90°, ∠A = 15°, AB = 20. What is the area of this triangle?A) 20B) 50C) 100D) 20037. ABCD is a rectangle and its perimeter is 22, as shown at the right. EFGH is a square. AH = 6. CF = ?A) 4 B) 5 C) 6D) 838. How many leap years are there between 2018 and 2081?A) 16B) 17C) 18D) 1939. My class was lined up on the gym floor in 8 rows, with 2 students in each row. If our coach rearranged us so that the number of rows was the same as the number of students in each row, how many rows were there after we were rearranged? A) 4B) 6C) 10D) 1640. Of the 100 numbers from 1 to 100, how many of them don’t contain 7 as its digit?A) 65 B) 75C) 80 D) none of the above。
共同出版了15本书TableofContentMath
Math League in China
裘宗沪先生，美国“数学大联盟杯赛”(中国赛 区)组委会主任。自1985年起，组织我国中学生 参加国际数学奥林匹克竞赛(IMO)，至今，15次 获得总分第一，共荣获107枚金牌和31枚银牌、 铜牌。 曾任中国数学会理事、中国数学会普及
工作委员会主任、中国数学会奥林匹克委员会 副主席、中国青少年科技辅导协会常务理事。 1993年被国家数学竞赛世界联盟授予艾尔多斯 奖(Erdos Award)，以表彰他通过数学竞赛等活 动对中国数学课外教育的杰出贡献。 1994年被 国务院妇女儿童工作委员会评为全国对少年儿 童工作有突出贡献的工作者，同时授予“荣誉 奖章”。 著作有《趣味数学三百题》、《你能 解吗？》、套书《数学奥林匹克教程》等。
Math League in North America
"What science can there be more noble, more excellent, more useful for men, more admirably high and demonstrative, than this of mathematics?"
Math League History
1. 美国“数学大联盟杯赛” (Math League) 是美 国及北美地区影响力最大的中小学数学赛事， 也是一项具有全球影响力的中小学生数学赛事。
2. 美国“数学大联盟杯赛”第一届竞赛于1977年 举行， 至今已经连续举办了38年。
3. 每年有来自全球的超过一百万名中小学生参加。
- Benjamin Franklin
“有什么样的科学比数学更崇 高、更完美、对人类更有用 和更加令人敬仰呢？”
本杰明·富兰克林
美国数学大联盟杯赛五年级试卷
2018-2019年度美国“数学大联盟杯赛”(中国赛区)初赛(五年级)(初赛时间：2018年11月14日,考试时间90分钟,总分200分)学生诚信协议：考试期间,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,我确定以下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。
如果您同意遵守以上协议请在装订线内签名选择题：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. A 6 by 6 square has the same area as a 4 by ? rectangle.A) 3 B) 6 C) 8 D) 92.Every prime has exactly ? positive divisors.A) 1 B) 2 C) 3 D) 4 or more3.If I answered 34 out of 40 questions on my math test correctly, I answered ? % of thequestions correctly.A) 75 B) 80 C) 85 D) 904.120 ÷ 3 ÷ 4 × 12 =A) 1 B) 10 C) 12 D) 1205.10 × 20 × 30 × 40 = 24 ×?A) 1000 B) 10 000 C) 100 000 D) 1000 0006.One of my boxes contains 1 pencil and the others each contain 5 pencils. If there are 101pencils in my boxes, how many boxes do I have?A) 19 B) 20 C) 21 D) 227.An electrical company imports 2016 light bulbs. Unfortunately, 25%of those are damaged. How many light bulbs are not damaged?A) 25 B) 504 C) 1512 D) 20168.50 × (16 + 24) is the square ofA) -40 B) -4 C) 4 D) 809.Which of the following numbers has exactly 3 positive divisors?A) 49 B) 56 C) 69 D) 10010.Ten people stand in a line. Counting from the left, Jerry stands at the 5th position.Counting from the right, which position is he at?A) 4 B) 5 C) 6 D) 7 11.On a teamwork project, Jack contributed 2/7 of the total amount of work, Jill contributed1/4 of the work, Pat contributed 1/5 of the work, and Matt contributed the rest. Whocontributed the most toward this project?A) Jack B) Jill C) Pat D) Matt12.Which of the following numbers is a factor of 2016?A) 5 B) 11 C) 48 D) 9913.2 × 4 × 8 × 16 × 32 × 64 =A) 210B) 215C) 221D) 212014.On a game show, Al won four times as much as Bob, and Bob won four times as much asCy. If Al won $1536, how much did Al, Bob, and Cy win together?A) $96 B) $384 C) $1920 D) $201615.The sum of two composites cannot beA) odd B) even C) 11 D) 1716.If a and b are positive integers such that a/b = 5/7, then a + b isA) 12 B) 24 C) 36 D) not able to be determined17.What is the greatest odd factor of the number of hours in all the days of the year 2015?A) 3 B) 365 C) 1095 D) 328518.If the current month is February, what month will it be 1 199 999 months from now?A) January B) February C) March D) April19.Two angles are complementary. One of these angles is 36° less than the other. What is themeasure of the larger angle?A) 36°B) 54°C) 63°D) 72°20.(The square root of 16) + (the cube root of 64) + (the 4th root of 256) =A) 12 B) 24 C) 32 D) 6421.In ∆ABC, m∠A–m∠B = m∠B–m∠C. What is the degree measure of ∠B?A) 30 B) 60 C) 90 D) 12022.For every 3 math books I bought, I bought 2 biology books. I bought 55 books in all. Howmany of those are math books?A) 11 B) 22 C) 33 D) 4423.John wrote a number whose digits consists entirely of 1s. This number was a compositenumber. His number could contain exactly ? 1s.A) 17 B) 19 C) 29 D) 3224.Weird Town uses three types of currencies: Cons, Flegs, and Sels. If 3 Sels = 9 Cons and 2Cons = 4 Flegs, then 5 Sels = ? Flegs.A) 12 B) 24 C) 30 D) 36第1页,共4页25.If the length of a rectangular prism with volume V is doubled while the width and theheight are halved, the volume of the new prism will beA) 4V B) V/2 C) V D) 2V26.Rick and Roy each stands at different ends of a straight road that is 64 m long. They runtoward each other. Rick’s speed is 3 m/s and Roy’s speed is 5 m/s. They will meet in? seconds.A) 1 B) 2 C) 4 D) 827.If the area of a certain circle is 2016, its radius isA) sqrt(2016/π) B) sqrt(4032/π)C) 2016/πD) 1008/π28.In a toy shop, the cost of a Teddy Bear is 200% as much as that of a toy train. The cost ofa toy train is 6/5 the cost of a pack of the wooden blocks. The cost of a pack of woodenblocks is $50. What is the cost, in dollars, of the Teddy Bear?A) 60 B) 100 C) 120 D) 20029.In the sequence 2016, 225, 141, 66, 432, 99, 1458 …, each term after the first term is thesum of the cubes of the digits of the previous term. What is the 100th term of thissequence?A) 153 B) 351 C) 370 D) 37130.What is the sum of all the positive divisors of 210?A) 210 – 1 B) 211 – 1 C) 212 – 1 D) 213 – 131.It takes 4 hours for Mike and Lucy to finish a task. It takes Lucy and Jerry 5 hours tofinish the same task. And it takes 6 hours for Mike and Jerry to finish the same task. Lucy and Jerry first work on the task for 1 hour and 45 minutes. Then Mike takes over the task on his own. How many more hours does it take for Mike to finish the task?A) 3 B) 4 C) 5 D) 632.If you sell a cloth at its current price, you get $40 profit. The total profit you get selling 10clothes at 70% of its current price is equal to the total profit you get selling 20 clothes at $82 per cloth. What is the current price of a cloth?A) 80 B) 100 C) 120 D) 12533.There are 6 identical squares in the figure on the right. The side length of eachsquare is 1. Of all the triangles constructed by connecting three of the 18vertices in the figure, how many of them are triangles whose area is 2 andwhich has at least one vertical or horizontal side?A) 12 B) 16 C) 24 D) 2834.Pick up N numbers from 1 to 2015 inclusively, such that the sum of any three of the Nnumbers is divisible by 24. What is the maximum value of N?A) 83 B) 84 C) 168 D) 252 35.汤姆有一件花了64美金买来的衬衫,他打算以比原价高出25%的价格出售,他会卖出多少钱？A) $16 B) $32 C) $48 D) $8036.满足以下条件的最小整数是多少：“除以3余2,除以5余4,除以7余6。
美国大联盟数学竞赛铜奖 小升初
美国大联盟数学竞赛铜奖小升初美国“大联盟”(Math League)思维探索研学活动是美国及北美地区具有卓越影响力的中小学数学思维探索研学活动，也是一项具有全球影响力的中小学数学思维探索研学活动。
第一届探索活动于1977年举行，至今已经连续举办了43届。
美国“大联盟”(Math League)思维探索研学活动每年举办一次，分为第一阶段、第二阶段、和研学嘉年华三个阶段。
第一阶段活动试题灵活、生动，富有趣味性，同时也贴近生活。
让学生理解数学、欣赏数学，激励学生创新，更能激发学生学习数学的兴趣，培养学生主动探索的精神。
第二阶段活动的目的是为了让中国学生体验美国数学教学的优点和特点，即美国启发式、重视应用、轻松、有趣、与实际生活紧密相关、研究、探索式的教学方式，启发学生如何thinking mathematically，培养学生creative thinking, critical thinking, and problem solving skills，即探索如何用数学解决实际生活中的问题，培养学生创造性思维、批判性思维、和实际解决问题的能力。
第二阶段活动还考察学生的英语能力(listening, speaking, communication, writing, and comprehension)。
研学嘉年华的决赛试题由美国"大联盟"(Math League)思维探索研学活动和普林斯顿大学(Princeton University)、哥伦比亚大学(Columbia University)、威廉姆斯学院(Williams College)联合命题，具有很高的挑战性，是数学超常少年展现才华的绝佳平台。
美国"大联盟"(Math League)思维探索研学活动会在其美方官网公布每年优胜者的成绩并颁发荣誉证书。
优异的成绩更是美国乃至世界各地学生申请北美重点高中和重点大学的重要参考依据。
美国数学大联盟常用英文词汇
contribute 贡献 cookie 饼干 coordinate 坐标 coordinate axes 坐标轴 corner 角 cost 花费 count 数；计算；计数；计算 couple 夫妻 cousin 堂[表]兄弟；堂[表]姐妹 cover 盖；覆盖 crack: 裂缝 crate 箱子 create 创建；创造；造成；创作 cross 交叉；穿越；越过 cube 立方体；立方 cubic 立方体的；立方的 customer 顾客；客户 cycle 循环；周期；自行车；骑自 行车 cylindrical 圆柱的
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dance 跳舞 darken 变黑 dart 飞镖 December 十二月 decimal 十进制的；小数的；小数 define 定义 degree 度数 degree-measure (角的)角度 denominator 分母 dessert 甜点 determine 确定 diagonal 对角线的；对角线 diagram 图表 diameter 直径 diamond 菱形 dice 骰子 die 死亡 difference 差额；差异 different 不同的 digit 数字；阿拉伯数字 digit sum In mathematics, the digit sum of a given integer is the
sum of all its digits (e.g. the digit sum of 84001 is calculated as 8+4+0+0+1 = 13). dime (美元)十美分 dimension 尺寸 dine 就餐；聚餐 dinner 晚餐 display 展览 distance 距离 distinct 不同的 distribute 分发；分配 divide 除；划分；分开；等分 divisible 可除尽的 division 除法 divisor 除数；约数 divorce 离婚 dodecahedron 十二面体 dollar (符号为＄)美元 double 两倍；使加倍 dough (用于制面包和糕点的)生 面团 download 下载
2014-2015年度美国”数学大联盟杯赛“(中国赛区)初赛-(十、十一、十二年级).doc
2014-2015年度美国”数学大联盟杯赛“（中国赛区）初赛（十、十一、十二年级）一、选择题（每小题10分，答对加10分，答错不扣分，共100分，请将正确答案A 、B 、C 或者D 写在每题后面的圆括号内。
）正确答案填写示例如下：=-⨯⨯20522 ？ （A ）A)5 B)15 C)25 D)301. Meg loves her megaphone! The large circular end has a circumference that is the reciprocal of its diameter. What is the area of the circle? ( )A)π14 B) π12 C) 14 D) 122. How many solutions does the equation x x +=233 have? ( )A)0 B)1 C)2 D)43. If y x =-1, which of the following is always true for any value of x ? ( )A) ()()x y -=-2211B) ()()x x y y -=-222211 C) ()()x x y y --=-222211 D) ()()()()x x y y -+=-+22221111 4. Lee the crow ate a grams of feed that was 1% seed, b grams of feed that was 2% seed, and c grams of feed that was 3% seed. If combined, all the feed he ate was 1.5% seed. What is a in terms of b and c ?( )A)b c +3B)b c +3 C)b c +23 D)b c +32 5. If <x 0 and <.x 2001, then x -1 must be ( )A)less than -10B)between-0.1 and 0 C)between 0and 0.1 D) greater than 106. At 9:00 A.M., the ratio of red to black cars in a parking lot was 1 to 5. An hour later the number of red cars had increased by 2, the number of black cars had decreased by 5, and the ratio of red to black cars was 1 to 4. How many black cars were in the lot at 10:00 A.M.? ( )A)13 B)15 C)60 D)657. If x ≠1and x ≠-1, then ()()()x x x x x --++-32241111=( ) A)x -21 B) x +21 C) x -241 D) x -341 8. The Camps are driving at a constant rate. At noon they had driven 300 km.At 3:30 P.M. they had driven 50% further than they had driven by 1:30 P.M.What is their constant rate in km/hr? ( )A)150 B)120 C)100 D)909. The letters in DIGITS can be arranged in how many orders without adjacent I ’s? ( )A)240 B)355 C)600 D)71510. Al, Bea, and Cal each paint at constant rates, and together they are painting a house. Al and Bea togethercould do the job in 12 hours; Al and Cal could to it in 15, and Bea and Cal could do it in 20. How many hours will it take all three working together to paint the house?( )A)8.5 B)9 C)10 D)10.5二、填空题（每小题10分，答对加10分，答错不扣分，共200分）11. What is the sum of the degree-measures of the angles at the outer points ,,,A B C D and E of a five-pointed star, as shown? Answer: . 12. What is the ordered pair of positive integers (,k b ), with the least value of k , which satisfiesk b ⋅⋅=34234?Answer: .13. A face-down stack of 8 playing cards consisted of 4 Aces (A ’s) and 4 Kings (K ’s).After I revealed and then removed the top card, I moved the new top card to thebottom of the stack without revealing the card. I repeated this procedure until thestack without revealing the card. I repeated this procedure until the stack was leftwith only 1 card, which I then revealed. The cards revealed were AKAKAKAK ,in that order. If my original stack of 8 cards had simply been revealed one card at atime, from top to bottom (without ever moving cards to the bottom of the stack),in what order would they have been revealed?Answer: .14. For what value of a is one root of ()x a x a -+++=222120 twice the other root?Answer: .15. Each time I withdrew $32 from my magical bank account, the account ’sremaining balance doubled. No other account activity was permitted. My fifth$32 withdrawal caused my account ’s balance to become $0. With how manydollars did I open that account?Answer: .16. In how many ways can I select six of the first 20 positive integers, disregarding the order in which these sixintegers are selected, so that no two of the selected integers are consecutive integers?Answer: .17. If, for all real ,()()xx f x f x =-21, what is the numerical value of f (3)?Answer: .18. How many pairs of positive integers (without regard to order) have a least common multiple of 540?Answer: .19. If the square of the smaller of consecutive positive integers is x , what is the square of the larger of thesetwo integers, in terms of x ?Answer: .20. A pair of salt and pepper shakers comes in two types: identical and fraternal.Identical pairs are always the same color. Fraternal pairs are the same colorhalf the time. The probability that a pair of shakers is fraternal is p andthat a pair is identical is .q p =-1 If a pair of shakers is of the same color, AE DCBword 格式-可编辑-感谢下载支持 determine, in terms of the variable q alone, the probability that the pair is identical. Answer: .21. As shown, one angle of a triangle is divided into four smaller congruentangles. If the lengths of the sides of this triangle are 84, 98, and 112, as shown,how long is the segment marked x ?Answer: .22. How long is the longer diagonal of a rhombus whose perimeter is 60, if threeof its vertices lie on a circle whose diameter is 25, as shown?Answer: .23. The 14 cabins of the Titanic Mail Boat are numbered consecutively from1 through 14, as are the 14 room keys. In how many different ways canthe 14 room keys be placed in the 14 rooms, 1 per room, so that, for everyroom, the sum of that room ’s number and the number of the key placed inthat room is a multiple of 3?Answer: .24. For some constant b , if the minimum value of ()x x b f x x x b -+=++2222is 12, what is the maximum value of ()f x ? Answer: .25. If the lengths of two sides of a triangle are 60cos A and 25sin A , what is the greatest possible integer-length of the third side?Answer: .26. {}n a is a geometric sequence in which each term is a positive number. If a a =5627, what is the value oflog log log ?a a a +++3132310Answer: . 27. What is the greatest possible value of ()=sin cos ?f x x x ++3412Answer: .28. Let C be a cube. Triangle T is formed by connecting the midpoints of three edges of cube C . What is the greatest possible measure of an angle of triangle T ?Answer: .29. Let a and b be two real numbers. ()sin f x a x b x =++34 and (lg log )f =3105. What is the value of (lg lg )f 3?Answer: .30. Mike likes to gamble. He always bets all his chips whenever the number of chips he has is <=5. He always bets n （10-）chips whenever the number of chips he has is greater than 5 and less than 10. He continues betting until either he has no chips or he has more than 9 chips. For every round, if he bets n chips. The probability that he wins or loses in each round is 50%. If Mike begins with 4 chips, what is the probability that he loses all his chips?Answer: .1129884xword格式-可编辑-感谢下载支持。
2018-2019年美国“大联盟”(Math League)思维探索活动第一阶段五年级试卷及答案
2018-2019年度美国“大联盟”(Math League)思维探索活动第一阶段(五年级)(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. 本试卷是五年级试卷，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. 2018 + 2019 = 20 + 18 + 20 + 19 + ?A) 0B) 2000C) 3960D) 40002. If 1/3 of my hats are red, and 36 are not red, I have ? hats.A) 18B) 54C) 72D) 1083. The sum of the measures of the 2 smallest angles of a triangle could beA) 151 degrees B) 135 degrees C) 121 degreesD) 61 degrees4. 2018 = 20 × ? + 18 × 1A) 1B) 10C) 18D) 1005. Every English letter appears in my 2018-letter password at least once. The letter A appears at most ? times.A) 77B) 78C) 1992D) 19936. Which of the following is the product of 2 consecutive integers?A) 182B) 195C) 208D) 2217. The least integer with a prime number of different prime factors isA) 6B) 8C) 12D) 158. I have 5 coins consisting of pennies, nickels, and dimes. If I have at least 1 of each type of coin, the least possible value of my 5 coins isA) 5¢B) 15¢C) 16¢D) 18¢9. Exactly ? different 3-digit area codes can be made using only 2s and 3s, with at leastone 2 and one 3 in each area code.A) 4B) 6C) 9D) 1210. How many multiples of 10 are factors of 102?A) 1B) 2C) 3D) 411. My team had to win a certain number of games to make it to the finals, and we won every 6th game we played. If my team qualified for the finals after our 96th game, how many wins did we need?A) 12 B) 16 C) 18D) 9012. What is the greatest common factor of 1 × 3 × 5 × 7 × 9 and 2 × 4 × 6 × 8 × 10?A) 1B) 3C) 5D) 1513. The expression 2400 is the product of exactly ? sixteens.A) 25B) 50C) 100D) 20014. The 2nd act of a 3-act play is 1/3 the length of the entire play. If the 1st act is twice as long as the 3rd, what fraction of the play is the 3rd act?A) 1/9 B) 2/9 C) 3/9 D) 4/9 15. If I double my speed of 12 000 m/hr., my new speed will beA) 200 m/min. B) 400 m/min. C) 600 m/min.D) 2400 m/min.16. Which of the following could be the perimeter of an equilateral triangle with integralside-lengths?A) 2017B) 2018C) 2019D) 202017. The greatest of 10 consecutive positive integers is a prime number. What is the least possible sum of these integers?A) 65B) 77C) 127D) 12918. One-fourth of Ed’s balloons popped, with 2 balloons popping every3 minutes for an hour. How many balloons did not pop?A) 40 B) 80 C) 120 D) 160 19. What is the greatest common factor of 68 and 86?A) 22B) 44C) 66D) 8820. The expression 1002018 can be written as the product of exactly ? prime numbers.A) 5 × 2018B) 4 × 2018C) 2 × 2018D) 201821. How many integers have a square root greater than 15 and less than 16?A) 0B) 1C) 29D) 3022.=A) 0B) 54C) 90D) 14423. Each day for a month, Sully wakes up 5 minutes earlier than he did the day before. IfSully woke up at 6:50 a.m. on a Monday, on what day did he wake up at 6:20 a.m.?A) Sunday B) Monday C) TuesdayD) Wednesday24. The product of all factors of 21 equals 21 × ? .A) 1B) 2C) 3D) 2125. (1234 + 0 + 1234 + 1 + 1234 + 2 + 1234 + 3 + 1234 + 4) ÷ 5 =A) 1234B) 1234 + 1C) 1234 + 2D) 1234 + 326. On a number line, two different integers are each the same distance from my favorite integer and have a sum of 144. What is my favorite integer?A) 31B) 36C) 48D) 7227. Last year I spent $180 for 80 pairs of shades. This year I spent $180for 5 fewer pairs of the same shades. How much did the price per pair increase since I bought them last year?A) 15¢B) 72¢C) 96¢D) 120¢28. I drove at a constant speed of 60 km/hr. without stopping. At exactly 5:00 p.m. I had traveled 318 km. At what time did I start driving?A) 10:42 a.m. B) 11:42 a.m. C) 12:42 p.m.D) 1:42 p.m.29. I added 3 of the numbers 11111, 22222, 33333, 44444, 55555, 66666, 77777, 88888, and 99999. My sum was one of these 9 numbers. When my sum was divided by 11, the remainder could not have beenA) 5B) 6C) 7D) 830. I wrote the 101 integers from 1 to 101 in order on paper. If I wrote 101 digits per line, what was the sum of the last 4 digits on the first line?A) 11B) 17C) 19D) 2131. The product of all the factors of an integer greater than 1 equals the cube of that integer.What is the least integer for which this is true?A) 24B) 18C) 12D) 832. On our last history test, at least one student scored each of the grades A, B, C, D, and F. If 8 got an A, 15 got a C or higher, 10 got a B or lower, and only one student got a D, how many students got an F?A) 1B) 2 C) 3 D) 5 33. (22 × 24 × 26 × … × 298 × 2100) ÷ (21 × 23 × 25 × … × 297 × 299) =A) 2B) 249C) 250D) 210034. Starting at 1:00 p.m., a ball was rolled in each of two lanes. A ballwas rolled once every 15 seconds in one lane and once every 18 seconds in the other. By 1:44 p.m., how many times had balls been rolled at the same time in both lanes?A) 29B) 30C) 40D) 4435. I counted backwards out loud from 2018 by ones. When I said my 50th multiple of 8, howmany numbers had I counted?A) 252B) 395C) 400D) 161836. If 30% of a number is 42, the 50% of the number is?A) 60B) 70C) 84D) 10837. The perimeter of a square is 12. The area of the square is?A) 3B) 6C) 9D) 1238. A year is called a lucky year if the sum of its digits is 18. For example, 1917 is alucky year since 1 + 9 + 1 + 7 = 18. How many lucky years are there between 2018 and 2108? A) 2B) 3C) 4D) 539. I ran 1 km in 6.0 minutes. Rabbit ran 50% faster and finished inA) 3.0 minutes B) 4.0 minutes C) 4.5 minutesD) 9.0 minutes40. If the pattern of the first 5 letters of ABCDEABCDEABCDE … continues to the right, the 2018th letter in the pattern would beA) A B) BC) CD) D or E。
小学数学：2022-2023年度美国大联盟(MathLeague)思维探索三年级试题(含参考答案)
2022-2023年度美国Math League思维探索第一阶段活动(三年级)(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。
1.2–0+2–1+2–2+2–3=A)2B)3C)4D)52.Sandy the Snail ran6cm in an hour!At that rate,how many cm could she runin4hours?A)10B)12C)24D)303.2022+2023=4044+?A)0B)1C)2D)44.The names of how many months begin with a vowel?A)0B)1C)2D)35.Tyson eats only chicken2days a week,only sausages3days a week,and onlyhamburgers the rest of the week.Tyson eats hamburgers?days a week.A)1B)2C)4D)56.When?is divided by6,the remainder is4.A)20B)22C)24D)267.I began with20hats9years ago.For the first5years after that,each year I doubled the number of hats Ihad.For the next4years,each year I gave away half of my hats.How many hats do I have now?A)10B)20C)30D)408.(60+40+20)–(50+30+10)=A)10B)20C)30D)609.Turbo Turtles are toys that can run one-half of1km in2minutes.The average hourly speed of a TurboTurtle isA)15km/hr.B)20km/hr.C)30km/hr.D)60km/hr.10.The sum of the hundreds and ones digits of the product of12and34isA)9B)10C)11D)1211.How many different possible perimeters could a rectangle that has area100cm2and side-lengths that areall whole numbers have?A)4B)5C)6D)712.20+31+42=23+34+?A)36B)39C)45D)4813.Five days ago my birthday was exactly three weeks away.Today my birthday is?days away.A)16B)18C)25D)2614.Sophia Squirrel started with44acorns and found4more,then divided all of heracorns equally among her4friends.How many acorns did each friend get?A)4B)7C)11D)1215.If my motorcycle travels135km on9L of gas,how far can it travel on2L at that same rate?A)20km B)25km C)30km D)35km16.I have an odd number of nickels and an odd number of quarters.What could be the total value of all mynickels and quarters?A)$0.75B)$1.20C)$2.55D)$5.2517.Two train cars are placed end-to-end,and one car is twice as long as the other.If the total length is48m,the longer car has lengthA)16m B)20m C)28m D)32m18.200+200+200+200=40×?A)5B)10C)16D)2019.Franny Farmer has an empty bucket that can hold up to30oranges.Each morninghe puts5oranges in it,and each evening he eats3oranges.In how many dayswill he need a2nd bucket to hold his oranges?A)13B)14C)15D)1620.10×100×1000×10000=100000×1000000÷?A)10B)100C)1000D)1000021.Victor has100same-sized Rubik’s Cubes,and boxes that hold8Cubes each.How many boxes must hehave to hold all the Cubes?A)12B)13C)14D)1522.Stevie counted even numbers backwards,and the22nd number he counted was22.At what number didhe start counting?A)64B)66C)68D)7023.The product of two whole numbers,each of which is less than1000,can have at most?digits.A)5B)6C)7D)824.Rhonda the Racehorse runs70km in1hour.Sammy the Sloth runs1km in4hours.At these rates,Rhonda runs?km farther than Sammy in8hours.A)552B)556C)558D)55925.Jonah found the difference between a two-digit number and the number formed by reversing the digits inthe two-digit number.This difference could beA)72B)74C)76D)7826.A square and a rectangle have equal perimeters.If the length of a side of the square is6cm and thelength of one side of the rectangle is2cm,the length of the longer side of the rectangle is?cm.A)10B)12C)18D)2027.The time on Noelle’s digital clock was5:55PM.How many minutes later didNoelle next see all identical digits?A)71B)316C)376D)43628.Between which numbers listed do the greatest number of primes fall?A)20and30B)30and40C)40and50D)50and6029.If a line segment divides a certain polygon into a square and a triangle with the square and the trianglesharing a common side,the greatest number of sides this polygon could have isA)5B)6C)7D)830.When talking about a number,Marcia said,“It is an odd number,”and Jan said,“Both of the digits aredifferent.”If only one of them is telling the truth,what could the number be?A)20B)21C)22D)2331.It cost me$30to buy2balls and2bats.At the same unit prices,it cost mysister$65for5balls and4bats.The cost of1bat isA)$5B)$10C)$15D)$2032.What is the sum of all the positive divisors of18?A)20B)28C)30D)3933.The number of orangutans living in my tree increases by a constant numbereach year.If there are40orangutans now and there were24two years ago,how many orangutans will be in my tree in six years?A)64B)72C)88D)13634.The ratio of game stands to food stands at the beach boardwalk is currently3:1.There are80food stands.How many game stands must be converted to food stands to make the ratio1:1?A)40B)80C)120D)16035.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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2018-2019年度美国“大联盟”(Math League)思维探索活动第一阶段(三年级)(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. 本试卷是三年级试卷，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。
1.20 + 20 + 20 = 18 + 18 + ?A) 14 B) 18 C) 22 D) 242.It takes four Beaver Brothers to carry a sharpened stick. How many Beaver Brothers doesit take to carry eight sharpened sticks?A) 12 B) 18C) 24 D) 323.On a number line, what is the difference between the two numbers that are 20 away from30?A) 10 B) 20 C) 40 D) 504.Seven weeks = ? daysA) 14 B) 49 C) 77 D) 3435.Which of the following numbers is a multiple of both 3 and 7?A)10 B)37 C) 42 D) 736.Which of the following has the greatest remainder?A) 3 ÷ 2 B) 11 ÷ 3 C) 13 ÷ 5 D) 15 ÷ 77.20 – 18 + 16 – 14 + 12 – 10 + 8 – 6 + 4 – 2 =A) 2 × 5 B) 3 × 5 C) 4 × 5 D) 5 × 58.Which number is 10 times as large as 2018?A)2028 B)2128 C) 3018 D) 20 1809.100 × 100 + ? = 1000 × 1000A) 900 B) 9000 C) 99 900 D) 990 00010.Yogi the Yelling Bear can yell for 15 seconds, but then needs a 10-secondbreak. For at most how many seconds can he yell in two minutes?A) 60 B) 75C) 90 D) 11011.Which of the following expressions has the lowest value?A) 1 ÷ (2 ÷ 3) B) (1 ÷ 2) ÷ 3C) (3 ÷ 2) ÷ 1 D) 3 ÷ (2 ÷ 1)12.The difference between two odd numbers cannot beA) 0 B) 6 C) 9 D) 1213.I have a pizza that can be cut into either 12 equal small slices or8 equal large slices. If I’ve already cut 4 large slices from thepizza, how many small slices can I cut from what remains?A) 2 B) 4C) 6 D) 814.The sum of 4 consecutive odd numbers is 80. What is the greatest of the 4 odd numbers?A) 21 B) 23 C) 25 D) 2715.I started with $20.18, and then spent 2 quarters, 0 dimes, 1 nickel, and 8 pennies. Howmuch money do I have left?A) $19.55 B) $19.60 C) $19.65 D) $19.7516.If Cyrus can buy 5 apples for 80 cents, then at that same rate he can buy 9 apples forA) $0.64 B) $1.44 C) $1.60 D) $1.8017.It is 1:00 p.m. now. I spent 20 minutes eating and finished eating 200 minutes ago. Atwhat time did I begin eating?A) 9:20 a.m. B) 9:40 a.m.C) 10:20 a.m. D) 10:40 a.m.18.Eighteen $10 bills and eighteen $100 bills equalsA) $1818 B) $1880 C) $1918 D) $198019.My square has an area of 4 cm2. The sides of your square are three times as long as thesides of my square. Your square has areaA) 9 cm2B) 16 cm2C) 24 cm2D) 36 cm220. There was no snow on the ground on December 31. Each evening after that, 4 cm of snow fell, and no other snow fell that day. If 2 cm of snow melted each morning, ? cm were on the ground the afternoon of January 4th.A) 4B) 6C) 8D) 1221. The product of 2 ÷ (1 ÷ 4) and 4 ÷ (1 ÷ 2) isA) 1B) 4C) 8D) 6422. A carton of milk and a muffin cost $2.60, and the milk costs 40 cents more than the muffin.How much does the milk cost?A) $0.75B) $1.10C) $1.50D) $2.2023. There are 30 students in my class. Eighteen of them can juggle, ten of them can ride a unicycle, and six of them can do neither. How many can do both?A) 2 B) 4 C) 6D) 824. (24 + 18 + 12 + 6) ÷ (16 + 12 + 8 + 4) =A) 1.5B) 2C) 4D) 625. A 4 m by 5 m rectangular floor is tiled with square tiles that have sides that are 1 m long. How many of the tiles are NOT surrounded by other tiles on all four sides?A) 12B) 14C) 16D) 1826. Of the following, which has the least value?A) 5500 ones B) 7 thousands C) 60 hundredsD) 500 tens27. Colton ran 1 km on Monday last week, and on each day after that he rantwice as many km as he had the day before. Cayden runs the same whole number of km every day. For the 7 days from Monday last week to Sunda y, if Cayden ran more total km than Colton did, Cayden’s daily run must be at least ? km.A) 2B) 7C) 19D) 3328. How many 2-digit whole numbers contain neither the digit 1 nor the digit 2?A) 56B) 64C) 70D) 8029. Every time Emma, Gaby, and Rosie play a game, the winner gets a quarter. After 3 gameswith no ties, some players may have no quarters, some 1 or 2 quarters, or one may have 3 quarters. In how many ways can the 3 quarters be distributed among the three players?A) 8B) 9C) 10D) 1230. If a teacher gives each of his students 4 crayons from his box of crayons, 16 crayons remain. If instead he gives each student 6 crayons from this box, 2 crayons remain. How many students are there?A) 7B) 8C) 12D) 1431. Which of the following results in an odd number?A) even × oddB) odd × even C) odd + evenD) odd + odd32. Zach multiplied the number on his shirt by itself. What could be his result?A) 24B) 25C) 26D) 2733. Find the missing term in the following sequence: 4, 7, 10, 13, _____, 19A) 3B) 16C) 19D) 2334. Ella wears different colored sneakers each day of the week, but her sneakers fall apart after four days of wear! At least how many pairs of sneakers does she need for the month of December?A) 7B) 8C) 10D) 1235. The largest whole-number multiple of 7 less than 100 isA) 91B) 93C) 97D) 9836. How many two-digit positive integers can be written using only even numbers?A) 16B) 20C) 25D) 5037. Simona has only dimes and quarters, and has exactly one dollar. If she has at least one dime and one quarter, how many coins must she have?A) 3B) 5C) 7D) 938. How many numbers between 100 and 200 are evenly divisible by 4 and 6?A) 4B) 6C) 8D) 1039. In Olive’s kingdom, castles have 8 beds and homes have 2 beds. If there are 48 beds in total and 3 castles, how many homes are there? A) 8 B) 12C) 16D) 2440. How many additional 2 × 2 non-overlapping squares can fit into a 6 × 6 square than a4 × 4 square? A) 4 B) 5C) 6D) 8。