2017-2018年度四年级美国数学大联盟杯赛(中国赛区)初赛(含答案)

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。

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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 for144 books?A) 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) $8.05D) $9.5040. Brooke's empty tub fills in 20 minutes with the drain plugged, andher full tub drains in 10 minutes with the water off. How manyminutes would it take the full tub to drain while the water is on?A) 12B) 15 C) 20 D) 30。

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。

2017年美国“数学大联盟杯赛”初赛四年级试卷2016-2017年度美国“数学大联盟杯赛”(中国赛区)初赛(四年级)(初赛时间：2016年11月20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.Which of the following is the greatest?A) 2.017 B) 20.17 C) 201.7 D) 20172.The sum of the degree-measures of the interior angles of a triangle isA) 180 B) 360 C) 540 D) 7203.100 + 200 + 300 + 400 + 500 = 300 ×?A) 3 B) 4 C) 5 D) 64.100 ÷ 4 = 200 ÷?A) 2 B) 4 C) 8 D) 165.In tonight’s talent show, Jack sang 3 songs. The number of songs that Jill sang is 8 lessthan 4 times the number of songs Jack sang. How many songs did Jill sing?A) 3 B) 4 C) 6 D) 76.Doubling a certain number is the same as adding that number and 36. What is thatnumber?A) 18 B) 36 C) 54 D) 727.The side-lengths of three square farms are 1 km, 2 km, and 3 km respectively. The sum ofthe areas of these three farms is ? km2.A) 6 B) 12 C) 13 D) 148.What is the greatest common factor of 2017 and 20 × 17?A) 1 B) 2 C) 3 D) 59.If a computer can download 2% of the files in 2 seconds, how many seconds does it taketo download all the files?A) 100 B) 200 C) 300 D) 40010.In yes terday’s giant-pie eating, all pies were the same size. Al ate 3/4 of a giant pie, Barbate 4/5 of a giant pie, Cy ate 5/6 of a giant pie, and Di ate 6/7 of a giant pie. Who ate the largest portion?A) Al B) Barb C) Cy D) Di 11.The product of two consecutive positive integers is alwaysA) odd B) evenC) prime D) composite12.In a 5-term sequence, the first term is 2. The value of each term after the first is twice thatof its previous term. What is the product of the 5 terms?A) 24B) 210C) 215D) 24513.Ace, Bo, and Cat performed in a talent show. Bo’s total score was twice that of Ace, andCat’s total score was three times that of Bo. If the sum of all three total scores was 900, what was Cat’s total score?A) 100 B) 200C) 300 D) 60014.The length of each side of triangle T is an integer. If twosides of T have lengths of 2016and 2017, what is the least possible value for the length of the third side?A) 1 B) 2 C) 4032 D) 403315.If the sum of three consecutive whole numbers is 2016, what is the sum of the next threeconsecutive whole numbers?A) 2032 B) 2025 C) 2020 D) 201716.If the sum of a prime and a composite is 2017, what is the least possible value for theproduct of the two numbers?A) 3000 B) 4030 C) 6042 D) 912017.What is the smallest whole number that leaves a remainder of 2 when divided by each of 3,4, 5, and 6?A) 58 B) 60 C) 62 D) 6418.What is the highest power of 2 that divides 1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 × 9?A) 25B) 26C) 27D) 2819.The product of the digits of 23 is 6. How many different whole numbers between 100 and999 have a product of 6?A) 12 B) 9 C) 6 D) 320.What is the value of 1% of 10% of 100%?A) 0.001 B) 0.01 C) 0.1 D) 121.In a box that contains only balls that are red, yellow, or green, 10% of the balls are red, 1/5of the balls are yellow, and 49 balls are green. How many balls are in the box?A) 70 B) 80 C) 90 D) 10022.Of the following, which has the greatest number of positive whole number divisors?A) 24 B) 26 C) 51 D) 2017第1页，共4页第2页，共4页23.If you subtract the sum of the digits of a whole numbergreater than 9 from the numberitself, the result must be divisible byA) 5 B) 6 C) 9 D) 1224.I bought a painting for $40, sold it for $50, rebought it for $60, and resold it for $70. Mytotal profit on the 4 transactions wasA) $10 B) $20 C) $30 D) $4025.What is the minimum number of whole number divisors of the product of two differentcomposite numbers?A) 5 B) 6 C) 8 D) 926.For each whole number from 1000 to 9999, inclusive, I write the product of its digits.How many of the products I write are even?A) 625 B) 3125 C) 5775 D) 837527.Lisa baked some cookies and cakes. Baking one cookie requires 4 cups of sugar and 3cups of flour, and baking one cake requires 7 cups of sugar and 5 cups of flour. At the end she used 83 cups of sugar and 61 cups of flour. How many cookies did she bake?A) 11 B) 12 C) 13 D) 1428.Working by oneself, Al can build a bridge in 3 years, Barb can build a bridge in 4 years,and Cy can build a bridge in 5 years. Working together, how long, in years, does it take them to build the bridge?A) 12B)6047C)6053D) 129.Jack is a gifted athlete who has trained hardfor the Olympic marathon. In the lasthundred yards he finds the inner strength toincrease his pace and overtakes the runner inthe second place.But then, with the finishing line just feetaway, he is overt aken by two other runners…What medal will Jack receive?A) Gold B) SilverC) Bronze D) None30.If we juxtapose three congruent squares, we get a rectangle with perimeter 64. What is thearea of one of the squares?A) 36 B) 49 C) 64 D) 8131.In a four-digit perfect square, the digits in the hundreds and thousands places are equal,and the digits in the tens and ones places are equal. What is this number?A) 6644 B) 7744 C) 8844 D) 9944 32.For how many of the integers from 100 to 999 inclusive is the product of its digits equal to9?A) 6 B) 7 C) 8 D) 933.What is the smallest positive integer x for which (x + 8) is divisible by 5 and (x + 17) isdivisible by 7?A) 30 B) 31 C) 32 D) 3334.Tom’s new tower was completed. The total value ofthe project, the sum of the cost of the construction andthe cost of the land, was one million dollars. The cost of the construction was $900,000 more than the cost of theland. So what did T om pay for the land?A) $25,000 B) $50,000C) $75,000 D) $90,00035.五个连续正整数的和总是可以被下面哪个数整除？A) 2 B) 3 C) 5 D) 736.从1开始，鲍勃一共喊了2017个数，从第一个数之后的每个数都比前一个数大4。

2016-2017年度美国“数学大联盟杯赛”(中国赛区)初赛(四年级)(初赛时间：2016年11月20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.Which of the following is the greatest?A) 2.017 B) 20.17 C) 201.7 D) 20172.The sum of the degree-measures of the interior angles of a triangle isA) 180 B) 360 C) 540 D) 7203.100 + 200 + 300 + 400 + 500 = 300 ×?A) 3 B) 4 C) 5 D) 64.100 ÷ 4 = 200 ÷?A) 2 B) 4 C) 8 D) 165.In tonight’s talent show, Jack sang 3 songs. The number of songs that Jill sang is 8 lessthan 4 times the number of songs Jack sang. How many songs did Jill sing?A) 3 B) 4 C) 6 D) 76.Doubling a certain number is the same as adding that number and 36. What is thatnumber?A) 18 B) 36 C) 54 D) 727.The side-lengths of three square farms are 1 km, 2 km, and 3 km respectively. The sum ofthe areas of these three farms is ? km2.A) 6 B) 12 C) 13 D) 148.What is the greatest common factor of 2017 and 20 × 17?A) 1 B) 2 C) 3 D) 59.If a computer can download 2% of the files in 2 seconds, how many seconds does it taketo download all the files?A) 100 B) 200 C) 300 D) 40010.In yesterday’s giant-pie eating, all pies were the same size. Al ate 3/4 of a giant pie, Barbate 4/5 of a giant pie, Cy ate 5/6 of a giant pie, and Di ate 6/7 of a giant pie. Who ate the largest portion?A) Al B) Barb C) Cy D) Di 11.The product of two consecutive positive integers is alwaysA) odd B) evenC) prime D) composite12.In a 5-term sequence, the first term is 2. The value of each term after the first is twice thatof its previous term. What is the product of the 5 terms?A) 24B) 210C) 215D) 24513.Ace, Bo, and Cat performed in a talent show. Bo’s total score was twice that of Ace, andCat’s total score was three times that of Bo. If the sum of all three total scores was 900, what was Cat’s total score?A) 100 B) 200C) 300 D) 60014.The length of each side of triangle T is an integer. If two sides of T have lengths of 2016and 2017, what is the least possible value for the length of the third side?A) 1 B) 2 C) 4032 D) 403315.If the sum of three consecutive whole numbers is 2016, what is the sum of the next threeconsecutive whole numbers?A) 2032 B) 2025 C) 2020 D) 201716.If the sum of a prime and a composite is 2017, what is the least possible value for theproduct of the two numbers?A) 3000 B) 4030 C) 6042 D) 912017.What is the smallest whole number that leaves a remainder of 2 when divided by each of 3,4, 5, and 6?A) 58 B) 60 C) 62 D) 6418.What is the highest power of 2 that divides 1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 × 9?A) 25B) 26C) 27D) 2819.The product of the digits of 23 is 6. How many different whole numbers between 100 and999 have a product of 6?A) 12 B) 9 C) 6 D) 320.What is the value of 1% of 10% of 100%?A) 0.001 B) 0.01 C) 0.1 D) 121.In a box that contains only balls that are red, yellow, or green, 10% of the balls are red, 1/5of the balls are yellow, and 49 balls are green. How many balls are in the box?A) 70 B) 80 C) 90 D) 10022.Of the following, which has the greatest number of positive whole number divisors?A) 24 B) 26 C) 51 D) 2017第1页，共4页第2页，共4页23.If you subtract the sum of the digits of a whole number greater than 9 from the numberitself, the result must be divisible byA) 5 B) 6 C) 9 D) 1224.I bought a painting for $40, sold it for $50, rebought it for $60, and resold it for $70. Mytotal profit on the 4 transactions wasA) $10 B) $20 C) $30 D) $4025.What is the minimum number of whole number divisors of the product of two differentcomposite numbers?A) 5 B) 6 C) 8 D) 926.For each whole number from 1000 to 9999, inclusive, I write the product of its digits.How many of the products I write are even?A) 625 B) 3125 C) 5775 D) 837527.Lisa baked some cookies and cakes. Baking one cookie requires 4 cups of sugar and 3cups of flour, and baking one cake requires 7 cups of sugar and 5 cups of flour. At the end she used 83 cups of sugar and 61 cups of flour. How many cookies did she bake?A) 11 B) 12 C) 13 D) 1428.Working by oneself, Al can build a bridge in 3 years, Barb can build a bridge in 4 years,and Cy can build a bridge in 5 years. Working together, how long, in years, does it take them to build the bridge?A) 12B)6047C)6053D) 129.Jack is a gifted athlete who has trained hardfor the Olympic marathon. In the lasthundred yards he finds the inner strength toincrease his pace and overtakes the runner inthe second place.But then, with the finishing line just feetaway, he is overt aken by two other runners…What medal will Jack receive?A) Gold B) SilverC) Bronze D) None30.If we juxtapose three congruent squares, we get a rectangle with perimeter 64. What is thearea of one of the squares?A) 36 B) 49 C) 64 D) 8131.In a four-digit perfect square, the digits in the hundreds and thousands places are equal,and the digits in the tens and ones places are equal. What is this number?A) 6644 B) 7744 C) 8844 D) 9944 32.For how many of the integers from 100 to 999 inclusive is the product of its digits equal to9?A) 6 B) 7 C) 8 D) 933.What is the smallest positive integer x for which (x + 8) is divisible by 5 and (x + 17) isdivisible by 7?A) 30 B) 31 C) 32 D) 3334.Tom’s new tower was completed. The total value ofthe project, the sum of the cost of the construction andthe cost of the land, was one million dollars. The cost of theconstruction was $900,000 more than the cost of theland. So what did Tom pay for the land?A) $25,000 B) $50,000C) $75,000 D) $90,00035.五个连续正整数的和总是可以被下面哪个数整除？A) 2 B) 3 C) 5 D) 736.从1开始，鲍勃一共喊了2017个数，从第一个数之后的每个数都比前一个数大4。

2013-2014年度美国“数学大联盟杯赛”(中国赛区)初赛答案(三、四年级)一、选择题1. A.Since 0 is a factor, 2 × 0 × 1 × 4 = 0.A) 0B) 7C) 8D) 20142. B.If 2 years ago I was 3 years old, I am now 3 + 2 = 5.A) 4B) 5C) 6D) 233. D.8 + (60 ÷ 4) = 8 + (15) = 23.A) 15B) 17C) 22D) 234. B.(1 + 7) + (2 + 6) + (3 + 5) = 3 × 8 = 24 = 4 + 20.A) 4B) 20C) 24D) 285. C.The prime numbers less than 10 are 2, 3, 5, and 7.A) 2B) 3C) 4D) 56. B.Caleb the dog dreams he has 12 dozen bones. Since 12 dozen = 12 × 12 = 144, there are 144 ÷ 2 = 72 pairs. Caleb will have to dig 72 holes.A) 24B) 72C) 144D) 2887. C.From 9:45 PM to 10:45 PM is 60 mins. From 10:45 PM to 11 PM is 15 mins. From 11 PM to 11:10 PM is 10 mins. That’s (60 + 15 + 10) mins.A) 65B) 75C) 85D) 958. D.From January 1st to January 31st, there are 16 odd-numbered dates. From February 1st to February 21st, there are 11 odd-numbered dates. That’s 27 × $2 = $54.A) $48B) $50C) $52D) $549. C.9 × 9 + 9 × 8 + 9 × 7 + 9 × 6 = 9 × (9 + 8 + 7 + 6).A) 20B) 24C) 30D) 3610.D.Manny weighs three times as much as Murray. Manny also weighs 8000 kg more than Murray, so 8000 kg is twice Murray’s weight. Thus Murray weighs 4000 kg and Manny weighs 12 000 kg.D) 12 00011.B.I have twice as many shirts as hats, and four times as many hats as scarves. If I have 24 shirts, I have 24÷ 2 = 12 hats and 12 ÷ 4 = 3 scarves.A) 2B) 3C) 6D) 1212.C.My coins have a total value of $6.20. If I have 1 of each coin, I have (1 + 5 + 10 + 25)¢ = 41¢. Subtract 41¢ from $6.20 repeatedly until there is 5¢ left. After 15 subtractions, there is 5¢ left. I have 15 + 5 or20 pennies.A) 10B) 15C) 20D) 2513.D.The diagrams demonstrate choices A, B, and C.A) 14 kmB) 10 kmC) 8 kmD) 1 km14.C.(2014 −1014) + (3014 − 2014) = 1000 + 1000 = 2000.A) 0B) 1000C) 2000D) 201415.A.10 + (9 ×8) − (7 × 6) = 10 + 72 − 42 = 40.A) 40B) 110The prime factorization of 72 is 2 × 2 × 2 × 3 × 3. The largest prime is 3.A) 3B) 7C) 36D) 7217.D.6 × 4 = 24 = 96 ÷ 4.A) 6B) 12C) 24D) 9618.C.If 6 cans contain 96 teaspoons of sugar, 1 can contains 96 ÷ 6 = 16 teaspoons of sugar. Thus 15 cans contain 16 × 15 = 240 teaspoons of sugar.A) 192B) 208C) 240D) 28819.C.The largest possible such sum is 98 + 99 = 197.A) 21B) 99C) 197D) 19820.B.Ann sent Wilson hearts with odd numbers with odd tens digits. The number on each heart he received must be two digits with both digits odd. There are 5 possible tens digits and 5 possible ones digits.That’s a total of 5 × 5 = 25 hearts.A) 23B) 25C) 30D) 4521.B.Since Rich ate his favorite sandwich 8 days ago, today is the 9th day of the month. Since the shortest month has 28 days, it is at least 28 − 9 = 19 days until the last day of the month. He must wait 1 more day.A) 1922.D.The factors of 49 are 1, 7, and 49. Since 49 has 3 factors, it has a prime number of factors.A) 6B) 12C) 36D) 4923.D.Dividing a certain two-digit number by 10 leaves a remainder of 9, so it is 19, 29, 39, 49, 59, 69, 79, 89, or 99. The only number listed with remainder 8 when divided by 9 is 89, so the number is 89 and 8 + 9 = 17.A) 7B) 9C) 13D) 1724.A.The whole numbers less than 1000 that can be written as such a product are 0 × 1 × 2, 1 × 2 × 3, 2 × 3 ×4, 3 × 4 × 5, 4 × 5 × 6, 5 × 6 × 7, 6 × 7 × 8, 7 × 8 × 9, 8 × 9 × 10, and 9 × 10 ×11. In all, that’s 10.A) 10B) 11C) 15D) 2125.B.The only such numbers are 5432, 5431, 5430, 5421, 5420, 5410, 5321, 5320, 5310, and 5210. In all, there are 10 such numbers.A) 3B) 10C) 69D) 12026.C.2014 × 400 = 805 600; the hundreds digit is 6.A) 0B) 5C) 6D) 827.B.Greta was 110 cm tall 2 years ago, when she was 10 cm taller than her brother. Her brother was 100 cmB) 130C) 140D) 15028.B.The number 789 678 567 456 is added to the number 987 876 765 654. Since we carry a 1 when adding the left-most digits, the sum has 12 + 1 digits.A) 12B) 13C) 24D) 2529.D.We must find which number among the choices is two more than a multiple of 5. Divide each choice by5 (or recogniz e that any number that ends in “2”or “7” is 2 more than a multiple of 5).A) 4351B) 5215C) 5616D) 646230.C.Of every 11 people, there are 2 adults and 9 children. Since 99 ÷ 11 = 9, there are 9 groups of 11 people.Of these, 9 × 2 = 18 are adults.A) 9B) 11C) 18D) 22二、填空题31.5.32.22.33.4.34.1.35.617.36.21.37.499.38.765.39.69.40.10.。

2017-2018年度美国“数学大联盟杯赛”(中国赛区)初赛(三年级)(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。

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1. 5 + 6 + 7 + 1825 + 175 =A) 2015 B) 2016 C) 2017 D) 20182.The sum of 2018 and ? is an even number.A) 222 B) 223 C) 225 D) 2273.John and Jill have $92 in total. John has three times as much money as Jill. How muchmoney does John have?A) $60 B) $63 C) $66 D) $694.Tom is a basketball lover! On his book, he wrote the phrase “ILOVENBA” 100 times.What is the 500th letter he wrote?A) L B) B C) V D) N5.An 8 by 25 rectangle has the same area as a rectangle with dimensionsA) 4 by 50 B) 6 by 25 C) 10 by 22 D) 12 by 156.What is the positive difference between the sum of the first 100 positive integers and thesum of the next 50 positive integers?A) 1000 B) 1225 C) 2025 D) 50507.You have a ten-foot pole that needs to be cut into ten equal pieces. If it takes ten secondsto make each cut, how many seconds will the job take?A) 110 B) 100 C) 95 D) 908.Amy rounded 2018 to the nearest tens. Ben rounded 2018 to the nearest hundreds. Thesum of their two numbers isA) 4000 B) 4016 C) 4020 D) 4040 9.Which of the following pairs of numbers has the greatest least common multiple?A) 5,6 B) 6,8 C) 8,12 D) 10,2010.For every 2 pencils Dan bought, he also bought 5 pens. If he bought 10 pencils, how manypens did he buy?A) 25 B) 50 C) 10 D) 1311.Twenty days after Thursday isA) Monday B) Tuesday C) Wednesday D) Thursday12.Of the following, ? angle has the least degree-measure.A) an obtuse B) an acute C) a right D) a straight13.Every student in my class shouted out a whole number in turn. The number the firststudent shouted out was 1. Then each student after the first shouted out a number that is 3 more than the number the previous student did. Which number below is a possible number shouted out by one of the students?A) 101 B) 102 C) 103 D) 10414.A boy bought a baseball and a bat, paying $1.25 for both items. If the ball cost 25 centsmore than the bat, how much did the ball cost?A) $1.00 B) $0.75 C) $0.55 D) $0.5015.2 hours + ? minutes + 40 seconds = 7600 secondsA) 5 B) 6 C) 10 D) 3016.In the figure on the right, please put digits 1-7 in the sevencircles so that the three digits in every straight line add upto 12. What is the digit in the middle circle?A) 3 B) 4 C) 5 D) 617.If 5 adults ate 20 apples each and 3 children ate 12 apples in total, what is the averagenumber of apples that each person ate?A) 12 B) 14 C) 15 D) 1618.What is the perimeter of the figure on the right? Note: Allinterior angles in the figure are right angles or 270°.A) 100 B) 110C) 120 D) 16019.Thirty people are waiting in line to buy pizza. There are 10 peoplein front of Andy. Susan is the last person in the line. How manypeople are between Andy and Susan?A) 18 B) 19C) 20 D) 2120.Thirty-nine hours after 9:00 AM isA) 1:00 AM B) 12:00 PM C) 8:00 PM D) 12:00 AM21.200 + 400 + 600 + 800 = (1 + 2 + 3 + 4) ×?A) 2 B) 20 C) 200 D) 200022.11…11 (the number consisting of 2016 1’s) is not a mult iple ofA) 11 B) 111 C) 1111 D) 1111123.The average of two thousands and two millions isA) 10000 B) 1000000 C) 1001000 D) 111100024.A triangle has the same area as a square. If the length of a base of the triangle is the sameas the side-length of the square, and the height of the triangle to the base is 4, what is thearea of the square?A) 1/2 B) 2 C) 4 D) 825.When V olta found a field in the shape of an isosceles triangle, she was soexcited that she ran a lap around all three sides. Two sides of the field havelengths of 505 m each, and the third side has a whole-number length.What is the greatest possible distance that V olta might have run in one lap?A) 2016 B) 2017 C) 2018 D) 201926.25 ×66 = 75 ×?A) 22 B) 44 C) 16 D) 3327.The number that has an odd number of whole number divisors isA) 15 B) 16 C) 17 D) 1828.In a sequence of 8 numbers, the average of the 8 terms is 15. If the average of the firstthree terms is 16 and the average of the next two terms is 15, what is the average of thelast three terms?A) 12 B) 13 C) 14 D) 1529.All years between 2000 and 2050 that are divisible by 4 are leap years.No other years between 2000 and 2050 are leap years. How many daysare there all together in the 17 years from 2010 to 2026?A) 6029 B) 6030 C) 5018 D) 501930.The sum of the hundreds digit and the tens digit of 2357 isA) 5 B) 8 C) 10 D) 1231.Which of the expressions below has the greatest value of (quotient × remainder)?A) 27 ÷ 4 B) 47 ÷ 6C) 57 ÷ 8 D) 87 ÷ 1232.I have some dimes and nickels, and together these coins are worth $3. If I replace everynickel with a quarter, I will have $5. How many dimes do I have?A) 10 B) 15 C) 20 D) 2533.I am a lovely cat. When I multiply the digits of a whole numberand the product I get is 9, I put that whole number on my list offavorite numbers. Of the whole numbers from 1000 to 9999, howmany would I put on my list of favorite numbers?A) 5 B) 10 C) 15 D) 2034.The sum of the tens digit and the units digit of the sum 1 + 12 + 123 + 12345+ … + 123456789 isA) 4 B) 5 C) 6 D) 735.The product of all prime numbers between 1 and 10 isA) 210 B) 105C) 1890 D) none of the above36.What is the average of 12, 14, 16, and 18?A) 13 B) 14 C) 15 D) 1637.When Jon shouts out a whole number, Al shouts out the product ofits digits, Barb shouts out the product of the digits of the number Alshouted out, and Cy shouts out the product of the digits of thenumber Barb shouted out. When Cy shouts out 18, what numbermight Jon have shouted out?A) 789 B) 799 C) 899 D) 99938.Each big box contains 3 medium boxes, each medium box contains2 small boxes, and each small box contains 5 apples. How many bigboxes are necessary for 1200 apples?A) 30 B) 40 C) 50 D) 6039.Eighteen years from now, my age will be 4 more than twice my currentage. My age now isA) 12 B) 14 C) 16 D) 1840.Each time Wanda waved her wand, 4 more stars appeared on herdress (which started with no stars). After several waves, Wandamultiplied the total number of stars then on her dress by thenumber of times she had waved her wand. This product cannot beA) 144 B) 256 C) 364 D) 676。

2017年第十五届“走美杯”小数数学竞赛初赛试卷（四年级B卷）一、填空题（共5小题，每小题8分，满分40分）1．（8分）计算：四十二亿九千四百九十六万七千二百九十七除以六百七十万零四百一十七等于（用数字作答）．2．（8分）将一个周角平均分成6000份，其中的一份作为角的度量单位，则可以得到一种新的度量角的单位：密位．显然，360°=6000密位，那么45°=密位，1050密位= °．3．（8分）两个标准骰子一起投掷1次，点数之和恰好为10的可能性（概率）为（用分数表示）．4．（8分）大于0的自然数，如果满足所有因数之和等于它自身的2倍，则这样的数称为完美数或完全数．比如，6的所有因数为1，2，3，6，1+2+3+6=12，6是最小的完美数．是否有无限多个完美数的问题至今仍然是困扰人类的难题之一．研究完美数可以从计算自然数的所有因数之和开始，78的所有因数之和为．5．（8分）“24点游戏”是很多人熟悉的数学游戏，游戏过程如下：任意从52张扑克牌（不包括大小王）中抽取4张，用这4张扑克牌上的数字（A=l，J=11，Q=12，K=13）通过加减乘除四则运算得出24，先找到算法者获胜．游戏规定4张牌扑克都要用到，而且每张牌只能用1次，比如2，3，4，Q，则可以由算法（2×Q）×（4﹣3）得到 24．如果在一次游戏中恰好抽到了以下两组排，请分别写出你的算法：（1）5，5，9，9，你的算法是（2）4，5，8，K，你的算法是．二、填空题（共5小题，每小题10分，满分50分）6．（10分）用5个边长为单位长度的小正方形（单位正方形）可以构成如图所示的5﹣联方（在中国又称为伤脑筋十二块）．在西方国家，人们用形象的拉丁字母来标记每一个5﹣联方．其中，既不是中心对称图形也不是轴对称图形的5﹣联方为：既是中心对称图形又是轴对称图形的5﹣联方为．7．（10分）将图中的圆圈染色，要求有连线的两个相邻的圆圈染不同的颜色，则最少需要种颜色．8．（10分）在中国古代的历法中，甲、乙、丙、丁、戊、己、庚、辛、壬、癸被称为“十天干”，子、丑、寅、卯、辰、已、午、未、申、酉、戌、亥叫作“十二地支，；十天干和十二地支进行循环组合：甲子、乙丑、丙寅．一直到癸亥，共得到60个组合，称为六十甲子．如此周而复始用来纪年的方法，称为甲子纪年法在甲子纪年中，以“丑”结尾的年份除了“乙丑”外，还有．9．（10分）在印度河畔的圣庙前，一块黄铜板上立着3根金针，针上穿着很多金盘．据说梵天创世时，在最左边的针上穿了由大到小的64片金盘，他要求人们按照“每次只能移动一片，而且小的金盘必须永远在大的金盘上面”的规则，将所有的64 片金盘移动到最右边的金盘上面．他预言，当所有64片金盘都从左边的针移动到右边的时候，宇宙就会湮（yan）灭．现在最左边金针（A）上只有6片金盘，如图（1）所示，要按照规则，移动成图（2）的状态，至少需要移动步．10．（10分）用3颗红色的珠子，2颗蓝色的珠子，1颗绿色的珠子串成圆形手链，一共可以串成种不同的手链．三、填空题（共5小题，每小题12分，满分60分）11．（12分）索玛立方体组块是丹麦物理学家皮特•海音（Piet Hein）发明的7个小立方体组块（如图所示，注意5号与6号组块，这是两个不同的组块）．因为利用这7个组块可以恰好组成一个立方体，所以称为索玛立方体组块．一个索玛立方体组块如果能够被某个平面分割成形状完全相同的两部分，则称这个组块是可平面平分的．那么，这些组块中有而且只有1种分割方法的可平面平分组块为，不可平面平分组块为（填0表示没有）．12．（12分）在平面上，用边长为1的单位正方形构成正方形网格，顶点都落在单位正方形的顶点（又称为格点）上的简单多边形叫做格点多边形．最简单的格点多边形是格点三角形，而除去三个顶点之外，内部或边上不含格点的格点三角形称为本原格点三角形，如图所示的格点三角形MBN．每一个格点多边形都能够很容易地划分为若干个本原格点三角形．那么，图中的格点四边形的面积为，可以划分为个本原格点三角形．13．（12分）如果一个长方形能够被分割为若干个边长不等的小正方形，则这个长方形称为完美长方形．已知下面的长方形是一个完美长方形，分割方法如图所示，已知其中最小的三个正方形的边长分别为1，2，7，那么，图中没有标示边长的小正方形的边长按照从小到大的顺序分别为．14．（12分）如果两个不同自然数的积被5除余1，那么我们称这两个自然数互为“模5的倒数”．比如，3×7=21，被5除余1，则3和7互为“模5的倒数”．即3与7都是有“模5的倒数”的数．那么8，9，10，11，12中有“模5的倒数”的数为，最小的“模5的倒数”分别为．15．（12分）将自然数1到16排成4×4的方阵，每行每列以及对角线上数的和相等，这样的方阵称为4阶幻方．幻方起源于中国，在世界上很多地方也都有发现．下面的4阶幻方是在印度耆那神庙中发现的，请将其补充完整：2017年第十五届“走美杯”小数数学竞赛初赛试卷（四年级B卷）参考答案与试题解析一、填空题（共5小题，每小题8分，满分40分）1．（8分）计算：四十二亿九千四百九十六万七千二百九十七除以六百七十万零四百一十七等于641 （用数字作答）．【分析】首先要把数四十二亿九千四百九十六万七千二百九十七和六百七十万零四百一十七写出来，然后计算即可．【解答】解：四十二亿九千四百九十六万七千二百九十七写作：4294967297六百七十万零四百一十七写作：67004174294967297÷6700417=641【点评】本题考查的数的读写，正确写出数，进行计算即可．2．（8分）将一个周角平均分成6000份，其中的一份作为角的度量单位，则可以得到一种新的度量角的单位：密位．显然，360°=6000密位，那么45°= 750 密位，1050密位= 63 °．【分析】根据题意可知1°=密位，1密位=°，据此解答即可．【解答】解：1°=密位，1密位=°，45°=45×=750密位，1050密位=1050×=63°【点评】本题考查的是单位换算，根据题意算出1°=密位，1密位=°，是解答本题的关键．3．（8分）两个标准骰子一起投掷1次，点数之和恰好为10的可能性（概率）为（用分数表示）．【分析】每个骰子的点数分别是1、2、3、4、5、6，所以投掷两个骰子的点数之和可能有：6×6=36种情况，其中相加等于10的有（4，6）、（6，4）、（5，5）这3种情况，据此解答即可．【解答】解：投掷两个骰子的点数之和可能有：6×6=36种情况，其中相加等于10的有（4，6）、（6，4）、（5，5）这3种情况．则点数之和恰好为10的可能性（概率）为：3÷36=【点评】本题考查的是概率问题，正确得出投掷两个骰子的点数之和可能情况一共有多少种是关键．4．（8分）大于0的自然数，如果满足所有因数之和等于它自身的2倍，则这样的数称为完美数或完全数．比如，6的所有因数为1，2，3，6，1+2+3+6=12，6是最小的完美数．是否有无限多个完美数的问题至今仍然是困扰人类的难题之一．研究完美数可以从计算自然数的所有因数之和开始，78的所有因数之和为168 ．【分析】要想求一个数的所有因数的和，首先要把这个数分解质因数，然后利用求一个数的所有的因数之和的公式解答即可．【解答】解：78=2×3×13所以78的所有的因数之和是：（1+2）×（1+3）×（1+13）=168【点评】本题考查的是如何求一个数的所有因数的和．把一个自然数M分解质因数，M=a b×c d×e f××…×m n，则自然数M的所有因数的和是（1+a1+a2+…+a b）×（1+c1+c2+…+c d）×（）…×（1+m1+m2+…+m n），据此解答即可．5．（8分）“24点游戏”是很多人熟悉的数学游戏，游戏过程如下：任意从52张扑克牌（不包括大小王）中抽取4张，用这4张扑克牌上的数字（A=l，J=11，Q=12，K=13）通过加减乘除四则运算得出24，先找到算法者获胜．游戏规定4张牌扑克都要用到，而且每张牌只能用1次，比如2，3，4，Q，则可以由算法（2×Q）×（4﹣3）得到 24．如果在一次游戏中恰好抽到了以下两组排，请分别写出你的算法：（1）5，5，9，9，你的算法是5×5﹣9÷9=24（2）4，5，8，K，你的算法是4×8+5﹣K=24 ．【分析】本题考查“24点游戏”，细心解答即可．【解答】解：（1）因为24=25﹣1，所以5×5﹣9÷9=24（2）4×8+5﹣K=24【点评】本题难度较低，细心解答即可．二、填空题（共5小题，每小题10分，满分50分）6．（10分）用5个边长为单位长度的小正方形（单位正方形）可以构成如图所示的5﹣联方（在中国又称为伤脑筋十二块）．在西方国家，人们用形象的拉丁字母来标记每一个5﹣联方．其中，既不是中心对称图形也不是轴对称图形的5﹣联方为F、L、N、P、Y ：既是中心对称图形又是轴对称图形的5﹣联方为I、X ．【分析】按题意，可以根据图形的对称性不难看出来，只有F、L、N、P、Y既不是中心对称图形也不是轴对称的图形，I、X既是中心对称图形又是轴对称图形．【解答】解：根据分析，可以根据图形的对称性不难看出来，只有F、L、N、P、Y既不是中心对称图形也不是轴对称的图形，I、X既是中心对称图形又是轴对称图形．故答案是：FLNPY，IX【点评】本题考查了图形的变换和对称性，突破点是：利用图形的对称性，不难看出符合题意的图形．7．（10分）将图中的圆圈染色，要求有连线的两个相邻的圆圈染不同的颜色，则最少需要 4 种颜色．【分析】要保证使用的颜色最少，则两个相邻的圆圈的颜色要尽可能多的相同，尝试2种颜色和3种颜色都不行，需要4种颜色，据此解答即可．【解答】解：尝试2种颜色和3种颜色都不行，需要4种颜色，如下图：【点评】本题考查染色问题．8．（10分）在中国古代的历法中，甲、乙、丙、丁、戊、己、庚、辛、壬、癸被称为“十天干”，子、丑、寅、卯、辰、已、午、未、申、酉、戌、亥叫作“十二地支，；十天干和十二地支进行循环组合：甲子、乙丑、丙寅．一直到癸亥，共得到60个组合，称为六十甲子．如此周而复始用来纪年的方法，称为甲子纪年法在甲子纪年中，以“丑”结尾的年份除了“乙丑”外，还有丁丑，己丑，辛丑，癸丑．【分析】首先分析题中的丑经过12年出现一次，共60年出现5次．枚举法即可．【解答】解：依题意可知：第一个是乙丑，丑出现时经过12+2=14年．24+2=26年，36+2=38年，48+2=50年．经过14，26，38，50年对应的天干是丁，己，辛，癸．故答案为：丁丑，己丑，辛丑，癸丑【点评】本题考查对周期问题的理解和掌握，关键是找到对应的数字．问题解决．9．（10分）在印度河畔的圣庙前，一块黄铜板上立着3根金针，针上穿着很多金盘．据说梵天创世时，在最左边的针上穿了由大到小的64片金盘，他要求人们按照“每次只能移动一片，而且小的金盘必须永远在大的金盘上面”的规则，将所有的64 片金盘移动到最右边的金盘上面．他预言，当所有64片金盘都从左边的针移动到右边的时候，宇宙就会湮（yan）灭．现在最左边金针（A）上只有6片金盘，如图（1）所示，要按照规则，移动成图（2）的状态，至少需要移动24 步．【分析】这是一个汉诺塔的变形问题，根据汉诺塔的推理结果，把n个盘从一个柱子上全部转移到另一个柱子上需要的步数是2n﹣1，据此解答即可．【解答】解：设6片金盘从小到大的编号依次是①、②、③、④、⑤、⑥，由图可知，图（2）中A上是③和④号金盘，C上是①、②、⑤、⑥金盘．第一次：把①、②、③、④4个金盘全部转移到图（2）B上，需要24﹣1=15（步）第二次：把⑤、⑥2个金盘全部转移到图（2）C上，需要22﹣1=3（步）第三次：把图（2）B上的①、②2个金盘全部转移到图（2）C上，需要22﹣1=3（步）第四次：把图（2）B上的③、④2个金盘全部转移到图（2）A上，需要22﹣1=3（步）综上所述：需要的步数是：15+3×3=24（步）【点评】本题考查的汉诺塔问题，重点是要理解有关汉诺塔的公式：把n个盘从一个柱子上全部转移到另一个柱子上需要的步数是2n﹣110．（10分）用3颗红色的珠子，2颗蓝色的珠子，1颗绿色的珠子串成圆形手链，一共可以串成 5 种不同的手链．【分析】因为是圆形手链，所以旋转和翻转相同的只能算一种，因为红色的珠子有3颗，所以可以让3颗红色的珠子相邻，也可以让2个红色的珠子相邻，也可以让红色的珠子不相邻这三种情况考虑，据此解答即可．【解答】解：①3颗红色的珠子相邻，则只有2种；②只有2颗红色的珠子相邻，有2种；③3颗红色的珠子都不相邻，有1种；2+2+1=5（种）答：一共可以串成5种不同的手链．【点评】本题考查的排列组合问题．三、填空题（共5小题，每小题12分，满分60分）11．（12分）索玛立方体组块是丹麦物理学家皮特•海音（Piet Hein）发明的7个小立方体组块（如图所示，注意5号与6号组块，这是两个不同的组块）．因为利用这7个组块可以恰好组成一个立方体，所以称为索玛立方体组块．一个索玛立方体组块如果能够被某个平面分割成形状完全相同的两部分，则称这个组块是可平面平分的．那么，这些组块中有而且只有1种分割方法的可平面平分组块为5、6 ，不可平面平分组块为7号（填0表示没有）．【分析】对1～7号组块进行逐一分析，看每一个组块有几种方法分割成两个完全相同的部分．【解答】解：1号有如下两种分割方法：2号有如下两种分割方法：3号有如下两种分割方法：4号有如下两种分割方法：5号只有如下一种分割方法：6号只有如下一种分割方法：7号不能分割成完全相同的两部分．故答案为：5、6；7号．【点评】对各个组块进行分析，易错点是7号不能分割成两个完全相同的部分．12．（12分）在平面上，用边长为1的单位正方形构成正方形网格，顶点都落在单位正方形的顶点（又称为格点）上的简单多边形叫做格点多边形．最简单的格点多边形是格点三角形，而除去三个顶点之外，内部或边上不含格点的格点三角形称为本原格点三角形，如图所示的格点三角形MBN．每一个格点多边形都能够很容易地划分为若干个本原格点三角形．那么，图中的格点四边形的面积为7.5 ，可以划分为15 个本原格点三角形．【分析】根据皮克公式：设格点多边形的面积是S，该多边形各边上的格点个数为a个，内部格点个数为b个，则S=a+b﹣1，即可求出图中的格点四边形的面积．【解答】解：皮克公式：S=a+b﹣1图中的格点四边形中，各边上的格点数a=5，内部的格点数b=6，所以格点四边形的面积是：×5+6﹣1=7.5根据题意，本原格点三角形内部没有格点，那么S=×3+0﹣1=0.5，所以7.5÷0.5=15（个），故答案为7.5，15．【点评】本题考查皮克公式的灵活运用．13．（12分）如果一个长方形能够被分割为若干个边长不等的小正方形，则这个长方形称为完美长方形．已知下面的长方形是一个完美长方形，分割方法如图所示，已知其中最小的三个正方形的边长分别为1，2，7，那么，图中没有标示边长的小正方形的边长按照从小到大的顺序分别为9、11、13、21、22、24、36、37、44 ．【分析】本题考察平面图形的计算．【解答】解：剩下的小正方形的编号分别是从①到⑨，如下图：正方形①的边长是：2+7=9正方形②的边长是：9+2=11正方形③的边长是：11+2=13正方形④的边长是：9+11+1=21正方形⑤的边长是：21+1=22正方形⑥的边长是：22+1=23正方形⑦的边长是：23+13=36正方形⑧的边长是：9+21+7=37正方形⑨的边长是：37+7=44．故填：9、11、13、21、22、24、36、37、44．【点评】本题较为繁琐，可操作性低，难度也低．14．（12分）如果两个不同自然数的积被5除余1，那么我们称这两个自然数互为“模5的倒数”．比如，3×7=21，被5除余1，则3和7互为“模5的倒数”．即3与7都是有“模5的倒数”的数．那么8，9，10，11，12中有“模5的倒数”的数为8和12 ，最小的“模5的倒数”分别为2和3或1和6 ．【分析】因为5的倍数的末尾是0或5，所以被5除余1的数的末尾是1或6，据此解答即可．【解答】解：因为5的倍数的末尾是0或5，所以被5除余1的数的末尾是1或6在8，9，10，11，12这四个数中，只有8×12=96符合要求．因为1×6=6，2×3=6，所以最小的“模5的倒数”分别是2和3或1和6．【点评】本题关键要理解因为5的倍数的末尾是0或5，所以被5除余1的数的末尾是1或6，据此解答即可．15．（12分）将自然数1到16排成4×4的方阵，每行每列以及对角线上数的和相等，这样的方阵称为4阶幻方．幻方起源于中国，在世界上很多地方也都有发现．下面的4阶幻方是在印度耆那神庙中发现的，请将其补充完整：【分析】首先算出1+2+3+4+…+16的和，从而求出每行、每列以及对角线上4个数的和，然后再根据幻方的“模块特性”求出空缺的数，据此解答即可．【解答】解：（1+2+3+4+…+16）÷4=34幻方的“模块特性”取出任意一个2×2的小正方形，4个数之和也是34，则有：【点评】本题考查的是幻方以及幻方的一些性质．。

四年级(初赛)试题 解答姓名_____________ 学校_____________ 得分____________一、填空题I （每小题6分，共60分）1. 计算：20.13⨯12+16.4⨯7.2+0.36=__________；答案：360解：原式=36⨯(6.71+3.28+0.01)=36⨯10=360.2. 一个等差数列，首项是1，公差为3，共20项，这20项的和为__________；答案：590解：第20项为1+(20-1)⨯3=58，原式=(1+58)⨯20÷2=590.3. 已知数列：2、0、1、3、4、7、1、⋯，规律是从第5个数开始，每个数都是它前面两个数之和的个位数字，那么这个数列第100个是__________； 答案：3解：从第3个数开始，12个数为一个周期，由于(100-2)=12⨯8+2，所以第100个数相当于周期中的第2个，即第100个为3.4. 请在下面的横式中填上两对小括号，使得等式成立；9 ⨯ 8 ⨯ 7 ÷ 6 - 5 ⨯ 4 - 3 ⨯ 2 - 1 =2013 答案：（9⨯8⨯7÷(6-5)⨯4-3⨯(2-1)=2013） 解：从9⨯8⨯7=504，504⨯4=2016着手．5. 幼儿园小班的小朋友做捡豆子游戏，有3个小朋友分别捡7、8、9粒，其他小朋友每人捡6粒，刚好捡完．如果有2个小朋友分别捡2、3粒，其他小朋友每人捡7个，也刚好捡完．那么共有__________粒豆子； 答案：45解：本题相当于每人捡6个多6粒，每人捡7粒少9粒，共有(6+9)÷(7-6)=15人，豆子有15⨯6+6=96粒． 6. 我们约定一种标记方法：n p 表示各位数码都是p 的n 位数，例如34=3333，5263=55666，如果2221212341453489a b c d +=，那么a b c d +++=__________；答案：8解：a =4，b =1，c =2，d =1．7. 像2013这样，由四个连续数字组成，且前两位数字是偶数，后两个数字是奇数，那么这样的四位数共有__________个； 答案：26 解：由0、1、2、3组成的四位数中符合条件的有2个，由1、2、3、4，2、3、4、5，…，6、7、8、9组成的四位数中符合条件的各有4个，所以共有2+6⨯4=26个．2 23 34 41 1 1 1 4 53 34 4 8 9 +8.把一个边长为7的大正方形纸片如图折叠，其中一层的纸片是面积为1的小正方形，那么中正方形的边长为__________；答案：5解：在中间再补上一个面积为1的小正方形纸片，这样都是两层，所以中正方形的面积为(7⨯7+1)÷2=25，边长为5.另解：把边长7分成差1的两段，这两段长为3和4，根据勾股定理，中正方形边长为5.9.右边乘法竖式中的乘积为__________；答案：10776解：12⨯898=10776.10.四、五两个年级学生各排成一队，经过一座长50米的桥时，四年级队伍完全过桥需要30秒，五年级队伍完全过桥需要40秒，两个队伍之间没有间隔前后合成一队，完全经过这座桥需要60秒，那么四年级队伍长__________米；答案：100解：两个班级共用30+40=70秒，而合成一队过桥需要60秒，相差70-60=10秒，相差一个桥长，所以过桥速度为50÷10=5，四年级队伍长30⨯5-50=100米．二、填空题II（每小题68分，共40分）11.如图，如果每个小三角形的面积为1，那么图中可以数出的三角形面积之和为__________；答案：54解：面积为1的有12个，面积为4的有6个，面积为9的有2个，所有三角形的面积为1⨯9+4⨯6+9⨯2=54.12.今年甲年龄是乙年龄的3倍，3年后，甲年龄为乙年龄的2倍多6岁，那么甲今年__________岁；答案：36解：如果甲年龄增加3，乙年龄增加9，那么甲年龄仍是乙年龄的3倍，也是乙年龄的2倍多6+(9-3)=12岁，即3年后乙年龄为12÷(3-2)=12岁，今年甲年龄为(12-3)⨯3=27岁．另解：设乙今年x岁，则332(3)6x x+=++，解得x=9，甲今年年龄为9⨯3=27岁．13.在4⨯4的方格中分别填入1、2、3、4，使得每行、每列、每条对角线上的四个数都互不相同，那么a+b+c的最大值为__________；答案：11解：三个数之和最大为4+4+4=12，试填时发现右上角也是4，这样左下角到右上角的四个数中有2个4，不符合．那么其中有一个是3，可以如右图填写．14.一次考试，有甲、乙两个考点，其中甲考点报考数量是乙考点报考数量的3倍，实际上甲考场比乙考场只多80个座位，必须将20名考生转移到乙考场，才能使两考场恰好坐满，那么共有__________名考生参加考试；8⨯答案：240解：甲考点比乙考点多报考80+20⨯2=120人，实际差3-1=2份，故共有120÷2⨯(1+3)=240人参加. 15. 如图，把2、0、1、3在正方体的角上写两次，然后每次把棱两个端点上的数同时加上或减去相同的数，称作一次操作，经过许多次后变成右图，那么右图中的A 处的数应为__________； 答案：4 解：如图，把对角上的四个顶点处染成黑色，每次黑色上数字之和与白色上数字之和的差不变，相差(2+0+1+3)-(2+0+1+3)=0，故A =(1+3+5+7)-4-6-2=4．四年级（初赛）试题答案。

2016-2017年度美国“数学大联盟杯赛”（中国赛区）初赛（五年级）1.Which of the has the greatest value?A) 2017 B) 2017C) 20 × 17 D) 20 + 172.Which of the leaves a remainder of 2 when divided by 4?A) 2014 B) 2015 C) 2016 D) 20173.Which of the is a product of two consecutive primes?A) 30 B) 72 C) 77 D) 1874.A Bizz-Number is a integer that either contains the 3 or is a multiple of 3. What is the of the 10th Bizz-Number?A) 24 B) 27 C) 30 D) 315.The of an isosceles triangle with side-lengths 1 and 1008 isA) 1010 B) 1012 C) 2017 D) 20186.How integers less than 2017 are divisible by 16 but not by 4?A) 0 B) 126 C) 378 D) 5047.Jon has a number of pens. If he distributed them evenly among 4 students,he have 3 left. If he distributed them evenly among 5 students, he have 4 left. The minimum number of pens that Jon have isA) 14 B) 17 C) 19 D) 248.Which of the numbers is not divisible by 8?A) 123168 B) 234236 C) 345424 D) 4566249.Which of the is both a square and a cube?A) 36 × 58B) 36 × 59C) 36 × 512D) 39 × 51210.The of two prime numbers cannot beA) odd B) even C) prime D) composite11.At the end of day, the amount of water in a cup is twice what it was atthe beginning of the day. If the cup is at the end of 2017th day, then it was1/4 at the end of the ? day.A) 504th B) 505th C) 2015th D) 2016th12.The grades on an exam are 5, 4, 3, 2, or 1. In a class of 200 students, 1/10of got 5’s, 1/5 of got 4’s, 25% of got 3’s, and 15% of got 2’s. How many students got 1’s?A) 40 B) 60 C) 80 D) 10013.22000 × 52017 = 102000 × ?A) 517B) 51000C) 52000D) 5201714.1% of 1/10 of 10000 is ? percent than 10A) 0 B) 9 C) 90 D) 90015.What is the of the of Circle C to the of Square S if the of adiameter of C and a of S are equal?A) π:1 B) π:2 C) π:3 D) π:416.Which of the is not a prime?A) 2003 B) 2011 C) 2017 D) 201917.If the sum of prime numbers is 30, what is the possible value of any of the primes?A) 19 B) 23 C) 27 D) 2918.For $3 I spend on books, I spend $4 on and $5 on toys. If I spent $20 on food, how much, in dollars, did I spend in total?A) 60 B) 90 C) 120 D) 15019.How positive odd factors does 25 × 35 × 55 have?A) 25 B) 36 C) 125 D) 21620.The of scalene triangles with perimeter 15 and side-lengths isA) 3 B) 5 C) 6 D) 721.Which of the when rounding to the nearest thousands, hundreds, and tens, 3000, 3500, and 3460, respectively?A) 3210 B) 3333 C) 3456 D) 351722.Which of the below has exactly 5 positive divisors?A) 16 B) 49 C) 64 D) 10023.Each after the 1st in the sequence 1, 5, 9, … is 4 than the previousterm. The greatest in sequence that is < 1000 and that leaves a of1 when divided by 6 isA) 991 B) 995 C) 997 D) 99924.For integer from 100 to 999 I the of the integer’s digits. Howmany of the products I are prime?A) 4 B) 8 C) 12 D) 1625.If a machine paints at a of 1 m2/sec, its is alsoA) 600 cm2/min B) 6000 cm2/minC) 60000 cm2/min D) 600000 cm2/min26.The of Square A is 1. The of Square B is times ofSquare A. The of Square C is times of Square B. The of Square C is ? times of Square A.A) 3 B) 6 C) 36 D) 8127.If the 17 minutes ago was 19:43, what will be the 17 minutes from now?A) 20:00 B) 20:17 C) 20:34 D) 20:1528.Pick any greater than 100 and subtract the sum of its from theinteger. The largest that must the result isA) 1 B) 3 C) 9 D) 2729.The number of needed in a room so there are always atleast five in the room born in the same month isA) 48 B) 49 C) 60 D) 6130.If M, A, T, and H are digits such that MATH + HTAM = 12221, is the value of M + A + T + H?A) 8 B) 20 C) 22 D) 2431.If 10 forks, 20 knives, and 30 $360, and 30 forks, 20 knives, and10 $240, what is the of 5 forks, 5 knives, and 5 spoons?A) 15 B) 75 C) 150 D) 22532.Write, in reduced form, the value ofA) 0.5 B) 1 C) 1.5 D) 233.Al, Barb, Cal, Di, Ed, Fred, and participated in a chess tournament. Eachplayer play each of his six opponents exactly once. So far, Al has 1match. Barb has 2 matches. Cal has 3 matches. Di has 4matches. Ed has 5 matches, and has 6 matches. How manymatches has at this point?A) 1 B) 3 C) 5 D) 734.What is the number of different integers I can choose from the 100positive integers so that no of these integers could be the of the sides of the same triangle?A) 8 B) 9 C) 10 D) 1135.What is the value of change that you can have in US (pennies, nickels, dimes, and quarters) without being able to someone exact change for a one-dollar bill?A) $0.90 B) $0.99 C) $1.19 D) $1.2936.小罗星期一工作了2个小时。

2016-2017年度美国“数学大联盟杯赛”（中国赛区）初赛（五年级）1.Which of the has the greatest value?A)2017B)2017C)20×17D)20+172.Which of the leaves a remainder of2when divided by4?A)2014B)2015C)2016D)20173.Which of the is a pr oduct of two consecutive primes?A)30B)72C)77D)1874.A Bizz-Number is a integer that either contains the3or is a multiple of3.What is the of the10th Bizz-Number?A)24B)27C)30D)315.The of an isosceles triangle with side-lengths1and1008isA)1010B)1012C)2017D)20186.How integers less than2017are divisible by16bu t not by4?A)0B)126C)378D)5047.Jon has a n u mbe r of pens.If he distributed them evenly among4students, he have3left.If he distributed them evenly among5students,he have 4left.The minimum n u mbe r of pens that Jon have isA)14B)17C)19D)248.Which of the numbers is not divisible by8?A)123168B)234236C)345424D)4566249.Which of the is both a square and a cube?A)36×58B)36×59C)36×512D)39×51210.The of two prime numbers cannot beA)odd B)even C)prime D)composite11.At the end of day,the amount of water in a cup is twice what it was at the beginning of the day.If the cup is at the end of2017th day,then it was1/4at the end of the?day.A)504th B)505th C)2015th D)2016th12.The grades on an exam are5,4,3,2,or1.In a class of200students,1/10of got5’s,1/5of got4’s,25%ofgot3’s,and15%of got2’s.How many students got1’s?A)40B)60C)80D)10013.22000×52017=102000×?A)517B)51000C)52000D)5201714.1%of1/10of10000is?percent than10A)0B)9C)90D)90015.What is the of the of Circle C t o the of Square S if the ofa diameter of C and a of S are equal?A)π:1B)π:2C)π:3D)π:416.Which of the is not a prime?A)2003B)2011C)2017D)201917.If the su m of prime numbers is30,what is the possible value of any of the primes?A)19B)23C)27D)2918.For$3I s pe n d on books,I s pe n d$4on and$5on toys.If I spent$20 on food,how much,in dollars,did I s pen d in total?A)60B)90C)120D)15019.How positive odd factors do e s25×35×55have?A)25B)36C)125D)21620.The of scalene triangles with perimeter15and side-lengths isA)3B)5C)6D)721.Which of the when rounding t o the nearest thousands,hundreds,and tens,3000,3500,and3460,respectively?A)3210B)3333C)3456D)351722.Which of the below has exactly5positive divisors?A)16B)49C)64D)10023.Each after the1st in the sequence1,5,9,…is4than the previous term.The gr eatest in sequence that is<1000and that leavesa of1when divided by6isA)991B)995C)997D)99924.For integer from100t o999I the of the integer’s digits.How many of the products I are prime?A)4B)8C)12D)1625.If a machine paints at a of1m2/sec,its is alsoA)600cm2/min B)6000cm2/minof Square C is timesC)60000cm2/min D)600000cm2/min26.The of Square A is1.The of Square B is times of Square A.The of Square B.The of Square C is?times of Square A.A)3B)6C)36D)8127.If the17minutes ago was19:43,what will be the17minutes from now?A)20:00B)20:17C)20:34D)20:1528.Pick any greater than100and subtract the su m of its from the integer.The largest that must the result isA)1B)3C)9D)2729.The n u mbe r of needed in a room so there are always at least five in the room born in the s ame month isA)48B)49C)60D)6130.If M,A,T,and H are digits such that MA TH+HT AM=12221,is the value of M+A+T+H?A)8B)20C)22D)2431.If10forks,20knives,and30$360,and30forks,20knives,and10$240,what is the of5forks,5knives,and5spoons?A)15B)75C)150D)22532.Write,in r educed form,the value ofA)0.5B)1C)1.5D)233.Al,Barb,Cal,Di,Ed,Fred,and participated in a chess tournament.Each player play each of his six o ppo n en t s exactly once.So far,Al has1 match.Barb has2matches.Cal has3matches.Di has4 matches.Ed hasmatches has5matches,andat this point?has6matches.How many A)1B)3C)5D)7of these integers could be the34. What is the n u mber of different integers I can choose from the100positive integers so that noof the sides ofthe s a me triangle? A) 8 B) 9 C) 10 D) 1135. What is thevalue of change that you can have in US(pennies,nickels, dimes, and quarters) without being able t o someone exact change for aone-dollar bill? A) $0.90 B) $0.99 C) $1.19 D) $1.2936. 小罗星期一工作了 2 个小时。

2017-2018年度美国“数学大联盟杯赛”(中国赛区)初赛(十、十一、十二年级)(初赛时间：2017年11月26日，考试时间90分钟，总分300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。

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填空题(每小题10分，答对加10分，答错不扣分，共300分。

)1.Each pirate wants his own treasure chest, but there is 1 more pirate than thereare treasure chests. If the pirates would agree to pair up so each pirateshares a treasure chest with another pirate, then 1 treasure chest wouldnot be assigned to any pirate. How many treasure chests are there?Answer: ________________.2.If m and nare positive integers that satisfy 10=, what is the greatest possiblevalue of m + n?Answer: ________________.3.There are an infinite number of points with positive coordinates(x,y) the sum of whose coordinates is the square of an integer.Among all such points (x,y), which one satisfies y = 2x and hasx as small as possible?Answer: ________________.4.As shown, a small square is inscribed in one of the triangles formed whenboth diagonals of a larger square are drawn. If the area of the larger squareis 144, what is the area of the smaller square?Answer: ________________.5.Trisection points on opposite sides of a rectangle are joined, as shown. Ifthe area of the shaded region is 2018, what is the area of the rectangle?Answer: ________________.6. A unit fraction is a fraction whose numerator is 1 and whosedenominator is a positive integer. What is the largest rationalnumber that can be written as the sum of 3 different unitfractions?Answer: ________________.7.What is the greatest possible perimeter of a rectangle whose length and width are differentprime numbers, each less than 120?Answer: ________________.8.Mom, Dad, and I each write a positive integer. My number is leastand Dad's is greatest. The average of all 3 numbers is 20. Theaverage of the 2 smallest numbers is 8. If Dad's number is d andif my number is m, what is the greatest possible value of d–m?Answer: ________________.9.If 8 different integers are chosen at random from the first 15 positive integers, what is theprobability that an additional number chosen at random from the remaining 7 positiveintegers is smaller than every one of the 8 originally chosen positive integers?Answer: ________________.10.What sequence of 5 positive integers has these three properties:1) All but one of the numbers is a multiple of 5.2) Every number after the first is 1 more than the sum of all the preceding numbers.3) The first number is as small as possible.Answer: ________________.11.Three beavers (one not shown) take turns biting a tree until it falls. Thesecond beaver is twice as likely as the first to make the tree fall. Thethird is twice as likely as the second to make the tree fall. What isthe probability that a bite taken by the third beaver causes thetree to fall?Answer: ________________.12.What is the ratio, larger to smaller, of a rectangle's dimensions if halfof the rectangle is similar to the original rectangle?Answer: ________________.第1页，共4页第2页，共4页A rectangle is partitioned into 9 different squares, as shown at the right. The area of the smallest square, shown fully darkened, is 1. Two other squares have areas of 196 and 324, as shown. What is the area of the shaded square? Answer: ________________.When the square of an eight-digit integer is subtracted from the square of a differenteight-digit integer, the difference will sometimes have eight identical even digits. What are both possible values of the repeated digit in such a situation? Answer: ________________.If the perimeter of an isosceles triangle with integral sides is 2017, how many different lengthsare possible for the legs? Answer: ________________.What are all ordered triples of positive primes (p ,q ,r ) which satisfy p q + 1 = r ? Answer: ________________.The reflection of (6,3) across the line x = 4 is (2,3). If m ≠ 4, what is the reflection of (m ,n )across the line x = 4? Answer: ________________.The vertices of a triangle are (8,7), (0,1), and (8,1). What are thecoordinates of all points inside this triangle that have integralcoordinates and lie on the bisector of the smallest angle of the triangle? Answer: ________________.In a regular 10-sided polygon, two pairs of different vertices (four different verticesaltogether) are chosen at random, so that all points chosen are distinct from each other. What is the probability that the line segments determined by each pair of points do not intersect? Answer: ________________.A line segment is drawn from the upper right vertex of aparallelogram, as shown, dividing the opposite side into segments with lengths in a 2:1 ratio. If the area of the parallelogram is 90, what is the area of the shaded region?Answer: ________________.21. If 0 < a ≤ b ≤ 1, what is the maximum value of ab 2 – a 2b ? Answer: ________________.22. What are all ordered pairs of integers (x ,y ) that satisfy 5x 3 + 2xy – 23 = 0? Answer: ________________.23. If two altitudes of a triangle have lengths 10 and 15, what is the smallest integer that couldbe the length of the third altitude?Answer: ________________.24. If h is the number of heads obtained when 4 fair coins are each tossed once, what is theexpected (average) value of h 2? Answer: ________________.25. What is the largest integer N for which 7x + 11y = N has no solution in non-negativeintegers (x ,y )? Answer: ________________.26. There are only two six-digit integers n greater than 100 000 for which n 2 has n as its finalsix digits (or, equivalently, for which n 2 – n is divisible by 106). One of the integers is 890 625. What is the other?Answer: ________________.27. A hexagon is inscribed in a circle as shown. If lengths of three sidesof the hexagon are each 1 and the lengths of the other three sides are each 2, what is the area of this hexagon? Write your answer in its exact format or round to the nearest tenth. Answer: ________________.28. If x is a number chosen uniformly at random between 0 and 1, what is the probability thatthe greatest integer ≤ 21log x ⎛⎫⎪⎝⎭ is odd?Answer: ________________.29. In the interval -1 < x < 1, sin θ is one root of x 4 – 4x 3 + 2x 2 – 4x + 1 = 0. In that sameinterval, for what ordered pair of integers (a ,b ) is cos 2θ one root of x 2 + ax + b = 0? Answer: ________________.30. Let P (x ) = 2x 10 + 3x 9 + 4x + 9. If z is a non-real solution of z 3 = 1, what is the numericalvalue of 23111P P P z z z ⎛⎫⎛⎫⎛⎫++ ⎪ ⎪ ⎪⎝⎭⎝⎭⎝⎭?Answer: ________________.第3页，共4页第4页，共4页。

2017-2018年度美国“数学大联盟杯赛”(中国赛区)初赛(六年级)(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。

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1. Pick any integer greater than 1. Double it twice, then triple the result. The final outcomeis ? of your starting integer.A) 700%B) 1100%C) 1200%D) 1300%2. Barry listened to the radio for 3 hours and 36 minutes. Rounded to thenearest 10 minutes, for how many minutes was Barry listening? A) 210 B) 220 C) 330 D) 340 3. Divide 99 by 22 to get a quotient and remainder. Divide that remainder by that quotient, and the new remainder is A) 4 B) 3 C) 2 D) 14. A man had five pieces of chain, each made up of three links, figure below. He wanted to join the five pieces together to make a big chain of fifteen links and went to a blacksmith to see how much it would cost. “Well,” said the blacksmith, “I will charge you 50 cents for cutting a link and $1.00 for welding a link. Any bending that is required is free.” Given those prices, what is the smallest amount of money for which the job could be done? Note: In a chain, each link is connected to one or two other links. A) $4.50 B) $5.00 C) $5.50 D) $6.00 5. A bee sat on the head of a horse rider whose horse was trotting eastbound at a steady five miles per hour. Some distance ahead on the same path, another horse and rider were approaching westbound, also at five miles per hour. When the two horses were 20 miles apart, the bee left the first horse rider and flew toward the second horse at a rate of ten miles per hour. Upon reaching the second horse, the bee immediately turned around and flew back at the same rate to the first horse. If the bee kept up this performance until the two riders met, how far (in miles) did he travel from the moment he left the first horse rider?A) 10B) 20C) 30D) None of the above6. Which of the following is the sum of the prime factors of 2018?A) 11 B) 219 C) 1011 D) 20197. If the length of the longest side of a triangle is 18, which of the following could not be the length of its second-longest side? A) 9 B) 10 C) 12 D) 17 8. My final score in a competition is the average of my scores on 5 rounds. To get a final score of 88 after getting 84, 80, and 92 on the first 3 rounds, what must be my average score for the last 2 rounds? A) 88 B) 90 C) 92 D) 96 9. Mr. Rice had breakfast one day at a restaurant with Mr. Wheat. When it came time to pay the bill, it was found that Mr. Rice had as many one-dollar bills as Mr. Wheat had quarters. (Mr. Rice had one-dollar bills only, and Mr. Wheat had quarters only.) Rather than each man paying separately, Mr. Rice paid his share of the bill, $6, to Mr. Wheat. At that point, Mr. Wheat had four times as much money as Mr. Rice. How much money did Mr. Rice have at the beginning? A) $6 B) $8 C) $9 D) $12 10. Professor Peach teaches chemistry to clever kids. The ratio of freshmen to other students in his class is 3:8. The total number of students in Professor Peach’s class could be A) 42 B) 45 C) 56 D) 77 11. 440 ÷ 220 = A) 22B) 24C) 220D) 26012. Mr. Bogsworth once left a will which read:To Bob, twice as much as to Betty. To Brian, twice as much as to Bob. To Bill, twice as much as to Brian.If his estate was valued at $45000, how much money did Betty, one of his four heirs, receive? A) $1000 B) $2000 C) $3000 D) $6000 13. I paid $5 and got 5 quarters, 5 dimes, and 5 nickels in change. I spent A) $3.00 B) $3.25 C) $3.45 D) $3.75 14. One side of Todd ’s truck is a perfect rectangle with an area of 12 m 2. If its length is 3 times its width, then its perimeter is A) 8 m B) 12 m C) 16 m D) 20 m15. If a bird in the hand is worth two in the bush, and a bird in the bush is worth four in the sky, then 4 birds in the hand are worth ? birds in the sky.A) 1B) 4C) 16D) 3216. On each of the four shelves of my bookcase is a different prime number of books. Therecould be a total of ? books on my shelves.A) 15B) 21C) 22D) 24第1页，共4页 第2页，共4页Seven years ago I realized that my age would be tripled twelve years from then. How old am I now? A) 11 B) 13 C) 16 D) 18How many fractions with a numerator of 1 and a whole-number denominator are greater than 0.01 and less than 1? A) 98 B) 99 C) 100 D) 101If I write the letters R-E-P-E-A-T repeatedly, stopping when I have written exactly 100 letters, how many times do I write the letter E? A) 16 B) 18 C) 32 D) 34On my map, 1 cm represents 100 km. If a park shown on the map is a rectangle that is2.5 cm by 4 cm, the area of the actual park is ? km 2. A) 100 B) 1000 C) 10 000 D) 100 000Gloomy Gus’s Tu esday rain cloud shows up every Tuesday at 8:30 A.M.and every 50 minutes after that. Its last appearance on Tuesday is at ? P.M.A) 11:00 B) 11:10 C) 11:30 D) 11:50If my lucky number divided by its reciprocal is 100, then the square ofmy lucky number isA) 100 B) 10 C) 1D) 1100A boy and his sister were walking down the street one afternoon when they met a kind old man. When the old man asked them about the size of their family, the boy quicklyanswered. “I have as many brothers as I have sisters,” he proudly stated. Not to be left out, the girl added, “I have three times as many brothers as I have sisters.” Can you tell how many boys and girls in total there were in their family? A) 5 B) 6 C) 7 D) 8A farmer was asked how many pigs he had. “Well,” he said, “if I had just as m any more again, plus half as many more, plus another 1.5 times more, I would have three dozen.” How many pigs did he have?A) 6 B) 9 C) 12 D) 15 15% of 80 is 40% ofA) 30B) 55C) 105D) 210It took me 90 minutes to cycle 45 km to the beach. Later I got a ride from the beach tothe park at twice my cycling speed. If the ride to the park took 15 minutes, what distance did I travel from the beach to the park?A) 15 kmB) 30 kmC) 45 kmD) 135 kmTed, Rick, and Sam painted a wall together. Ted painted 80% more ofthe wall than Sam painted. Sam painted 40% less than Rick. Ted painted ? of the amount that Rick painted. A) 102% B) 108% C) 120% D) 140% The greatest integer power of 20 that is a divisor of 5050 isA) 2020B) 2025C) 2050D) 2012529. The ones digit of the sum of all even integers from 2 to 1492 is A) 2B) 4 C) 8 D) 030. The median of 12, 13, 14, 15, 16, and 17isA) 18 B) 223840 C) 514 D) 94031. The average of all positive even integers from 2 to 2018 is A) 1000 B) 1009 C) 1010 D) 1014 32. Pirate Percy has 300 coins in his chest. Of the Spanish coins,20% are gold. If 100 of the coins are gold but not Spanish and 70 of the coins are neither gold nor Spanish, how many Spanish gold coins are in Percy’s chest?A) 20 B) 26 C) 30 D) 3433. When I divided the population of my city by the number of streets in the city, I got a remainder of 18. If the exact quotient on my calculator was 123.06, how many streets are there in my city? A) 60 B) 120 C) 186 D) 30034. What is the greatest number of 3-by-7 rectangles that can be placed inside an 80-by-90 rectangle with no overlapping?A) 312B) 330C) 334D) 34235. How many four-digit whole numbers have four different even digitsand a ones digit greater than its thousands digit?A) 36B) 54C) 60D) 9036. Both arcs AB and AD are quarter circles of radius 5, figure on the right.Arc BCD is a semi-circle of radius 5. What is the area of the region ABCD ? A) 25 B) 10 + 5π C) 50 D) 50 + 5π 37. In the figure on the right, the side-length of the smaller square is 4. The four arcs are four semi-circles. Each side of square ABCD is tangent to one of the semi-circles. The area of ABCD is A) 32 B) 36 C) 48 D) 6438. A million is a large number, a “1” followed by 6 zeros. A googol is a large number, a “1” followed by one hundred zeros. Agoogo lplex is a large number, a “1” followed by a googol of zeros. A googolplexian is a large number, a “1” followed by a googolplex of zeros. A googolplexian isA) 1010 B) 1001010 C) 100101010 D) none of the above 39. If the total number of positive integral divisors of n is 12, what is the greatest possibletotal number of positive integral divisors of n 2? A) 23 B) 24 C) 33 D) 4540. Of all the isosceles triangles whose perimeter is 20 and whose side-lengths are integers, what is the length of the base of the triangle with the largest area? A) 2 B) 5 C) 6 D) 8第3页，共4页第4页，共4页。