2017年美国“数学大联盟杯赛”初赛四年级试卷
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 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。
美国2017年度四年级数学测试题赏析
美国2017年度四年级数学测试题赏析殷莉莉(编译)【摘要】美国国家教育进展评估(NAEP)是美国衡量中小学生各学科学业成就的一个评价测量体系,规模最大,持续时间最长,1969年首次在美国全国范围内实施。
NAEP的结果被教师、校长、政策制定者和研究人员用于评估美国教育的进展,改进美国的教育。
数学学科主要评估四年级和八年级学生的知识、技能,以及在解决问题时运用知识的能力。
包括五个方面的内容,即数的特征和运算,度量,几何,数据分析、统计和可能性,代数。
根据问题的复杂程度,可将题目分为低、适度和高三个层次。
【期刊名称】《小学教学：数学版》【年(卷),期】2019(000)002【总页数】5页(P60-64)【关键词】美国教育;数学学科;四年级;试题赏析;中小学生;NAEP;政策制定者;测量体系【作者】殷莉莉(编译)【作者单位】[1]教育部北京师范大学基础教育课程中心数学工作室【正文语种】中文【中图分类】G571.2美国国家教育进展评估（NAEP）是美国衡量中小学生各学科学业成就的一个评价测量体系，规模最大，持续时间最长，1969年首次在美国全国范围内实施。
NAEP 的结果被教师、校长、政策制定者和研究人员用于评估美国教育的进展，改进美国的教育。
数学学科主要评估四年级和八年级学生的知识、技能，以及在解决问题时运用知识的能力。
包括五个方面的内容，即数的特征和运算，度量，几何，数据分析、统计和可能性，代数。
根据问题的复杂程度，可将题目分为低、适度和高三个层次。
低复杂度的问题通常规定了学生要做什么，执行常规的数学程序；适度复杂的问题往往需要多步骤解决，涉及更多的灵活性思维；高复杂度的问题则对学生的思维提出了更高的要求，需要在新的情境中进行抽象的推理、分析。
2017年度，NAEP 首次采用数字化测试，将测试题呈现在平板电脑上，部分问题还为学生提供了屏幕计算器或虚拟的便签本。
以下给出题目赏析。
第1题（领域：数的特征和运算；复杂程度：低；难度水平：简单）：如图1，每个正方形由100 个大小相同的小方格组成。
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。
2017年第十五届”走美杯“小数数学竞赛上海赛区初赛试卷(四年级)后附答案解析
2017年第十五届“走美杯”小数数学竞赛上海赛区初赛试卷（四年级）一、填空题（共5小题，每小题8分，满分40分）1．（8分）24点游戏，用适当的运算符号（包括括号）把3，3，9，9这四个数组成一个算式，是结果等于24．．2．（8分）每个月的周一、周二、周三、周四、周五、周六、周日都有4天或5天，某月，周三比其他日期恰好都多一天，这个月28日是星期．3．（8分）图中共有个长方形．4．（8分）一堆棋子有黑、白两色，黑棋子的个数是白棋子的2倍，现在从这堆棋子中每次取出黑子5个、白子3个，若干次后，白子恰好取完，而黑子还有11个，白棋子原有个．5．（8分）2017除以9余1，2017年的每一天都可以用一个八位数表示，比如2017年1月3日可以表示为20170108，这个数除以9余1，2017年全年365天都用八位数表示，其中能被9整除的八位数共有个．二、填空题（共5小题，每小题10分，满分50分）6．（10分）两个长方形如图摆放，M为AD的中点，三角形ACM是等腰直角三角形，阴影部分的面积是35，长方形AEFC的面积为．7．（10分）A、B两个纸片都被分成了4个区域，用黄、蓝、红三种颜色分别给它们涂色，要求相邻的区域涂色不能相同，A，B两个纸片中的涂法较多，有种不同的涂法．8．（10分）甲、乙两人骑车分别从A、B两地同时出发，相向而行，相遇时，甲比乙多行了3千米，已知甲骑车从A到B需2小时，乙骑车从B到A需3小时，A，B两地相距千米．9．（10分）将2013拆成3个互不相同的整数，使这三个数的和为2013，且其中任意两个数的和除以3都余1，这三个数中，最大的数最小是．10．（10分）有一种五位数，从左向右第三位数字开始，每一个数字都是它前面两个数字的和，这样的五位数共有个．二、填空题11．（12分）圆上的50个点A1，A2，A3，…，A50将该圆分为50段等弧，以这50个点中的某些点为顶点，一共可以得到个不同的正多边形．12．（12分）将260个桃子分装到若干个相同的筐中，每个筐中最少放10个，最多放25个，放完后，每个框中的桃子数都不相同，有种放法，可能有个筐．13．（12分）一个宝库有16个藏宝室，成4×4状排列，但只有一个进口和一个出口，分别开在如图所示的藏宝室，每个藏宝室至多只能进去一次，相邻的两个藏宝室之间都有门想通，每个藏宝室中的宝贝价值已标在图中，大盗买通守护，夜间进入宝库，他能带走的宝物价值最多是．14．（12分）现有1×1×2的积木（A）、1×1×3的积木（B）、1×2×2的积木（C）（如图），分别有6块、11块、10块，从这些积木中选出若干个，拼成3×3×3的实心正方体，至多可以拼出个3×3×3的实心正方体，写出这几个正方体的拼法分别所用的A、B、C的个数（如1A+7B+1C）：15．（12分）请在下面的每个箭头里填上适当的数字（图中已经填出两个数字），使得每个数字都表示该箭头所指方向的箭头里含有不同数字的个数，其中双向箭头表示箭头所指的两个的箭头里不同数字的个数，图中第三行从左到右所填数字组成的四位数是．2017年第十五届”走美杯“小数数学竞赛上海赛区初赛试卷（四年级）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）24点游戏，用适当的运算符号（包括括号）把3，3，9，9这四个数组成一个算式，是结果等于24．3×9﹣9÷3=24 ．【分析】结合4个数字和24之间的关系进行试运算，可以联想24相关的加减乘除运算，据此解答．【解答】解：3+3+9+9=24，3×9﹣9÷3=24．故答案为：3+3+9+9=24，3×9﹣9÷3=24等．【点评】本题考查24点游戏，重点在于有一定的联想能力，可以想到4个数字和24之间的关系，属于简单题．2．（8分）每个月的周一、周二、周三、周四、周五、周六、周日都有4天或5天，某月，周三比其他日期恰好都多一天，这个月28日是星期二．【分析】首先分析这个月一个有多少天，周三比其他都多一天说明这个月是4个星期多一天共29天，继续分析即可求解．【解答】解：依题意可知：周三比其他都多一天说明这个月是4个星期多一天共29天，最后一天是星期三，那么28日就是星期二．故答案为：二【点评】本题考查对周期问题的理解和运用，关键问题是找到这个月的天数，问题解决．3．（8分）图中共有7 个长方形．【分析】此题采用分类的方法解答．（1）由1个图形构成的有4个；（2）由2个图形构成的有1个；（3）由3个图形构成的有1个；（4）由4个图形构成的有1个；【解答】解：（1）由1个图形构成的有4个；（2）由2个图形构成的有1个；（3）由3个图形构成的有1个；（4）由4个图形构成的有1个；答：图中共有 7个长方形．故答案为：7．【点评】本题考查了对平面图形的认识，在数长方形的个数时，要有规律地进行分类．4．（8分）一堆棋子有黑、白两色，黑棋子的个数是白棋子的2倍，现在从这堆棋子中每次取出黑子5个、白子3个，若干次后，白子恰好取完，而黑子还有11个，白棋子原有33 个．【分析】根据题意，若每次取白子3个，黑子6个，白子取完时，黑子也恰好取完，但每次取5个黑子，最后剩下11个黑子，说明取了11次，所以白子原有3×11=33（个）【解答】解：根据分析，若每次取白子3个，黑子6个，白子取完时，黑子也恰好取完，但每次取5个黑子，最后剩下11个黑子，说明取了11次，所以白子原有3×11=33（个）故答案是：33个．方法二：设白棋子原有x个，取了n次，可列方程：解得：故答案是：33个．【点评】本题考查了等量关系与方程，突破点是：根据题意逻辑推理，可以分析出白子的数量．5．（8分）2017除以9余1，2017年的每一天都可以用一个八位数表示，比如2017年1月3日可以表示为20170108，这个数除以9余1，2017年全年365天都用八位数表示，其中能被9整除的八位数共有59 个．【分析】按题意，根据被9整除的特征，可知数字之和能被9整除，而2017年的年份2017的数字之和为10，被9除余1，八位数能被9整除，则只要满足月份日期的四位数除以9余8即可．【解答】解：根据分析，根据被9整除的特征，可知数字之和能被9整除，而2017年的年份2017的数字之和为10，被9除余1，八位数能被9整除，则只要满足月份日期的四位数除以9余8即可．满足这个条件的四位数有：0107、0116、0125、0206、0215、0224、0306、0314、0523、0404、0413、0422、0503、0512、0521、0530、0602、0611、0620、0629、0701、0710、0719、0728、0809、0827、0908、0917、0926、1007、1016、1025、1106、1115、1124、1205、1214、1223；综上，满足条件的八位数个数有：59个．故答案是：59．【点评】本题考查数的整除特征，突破点是：根据数的整除特征，求得能被9整除的八位数的个数．二、填空题（共5小题，每小题10分，满分50分）6．（10分）两个长方形如图摆放，M为AD的中点，三角形ACM是等腰直角三角形，阴影部分的面积是35，长方形AEFC的面积为42 ．【分析】可以将阴影部分分割成5个与△ACM一样的等腰直角三角形，然后算得每个小等腰直角三角形的面积，再求长方形的面积．【解答】解：根据分析，将阴影部分分割成5个与△ACM一样的等腰直角三角形，如图所示：长方形AEFG的面积为：35÷5×6=42．故答案是：42．【点评】本题考查了三角形的面积，突破点是：将阴影部分分割成6个与△ACM 一样的等腰直角三角形，不难求得长方形AEFG的面积．7．（10分）A、B两个纸片都被分成了4个区域，用黄、蓝、红三种颜色分别给它们涂色，要求相邻的区域涂色不能相同，A，B两个纸片中 B 的涂法较多，有12 种不同的涂法．【分析】A的涂色区域只能是最上方区域和左下方区域图同色，其排列数为；图B的涂色区域中涂同色的区域有2类，一是最上方区域和左下方区域；二是最上方区域和右下角区域，涂色种类数为+．【解答】解：图A的涂色方法有=3×2×1=6（种）图B的涂色方法有+=6+6=12（种）故：B的涂法多，有12种不同涂法．【点评】此题的解题关键是能否想到合并能涂同色的区域，而且要把这种情况找全．8．（10分）甲、乙两人骑车分别从A、B两地同时出发，相向而行，相遇时，甲比乙多行了3千米，已知甲骑车从A到B需2小时，乙骑车从B到A需3小时，A，B两地相距15ɛ千米．【分析】根据甲乙行相同的路程，所需时间之比为2：3，就是告诉：甲乙2人是速度之比为3：2（时间之比与速度之比互为倒数）．甲乙2人是速度之比为3：2，也就是说在相同时间里，甲乙2人走的总路程中甲占3份，乙为2份，总路程是5份．即：在相同的时间里（他们相遇时），甲走了全程的3/5，乙走了全程的2/5．甲比乙多走了全程的1/5，就是那3千米．这样就可求出全程的长了．【解答】解：甲乙的时间比2：3，所以时间之比3：23÷（3/5﹣2/5 ）=15（千米）答：A，B两地相距15千米．【点评】本题有点绕，必须弄懂时间比与速度比的关系．才能明白在相同的时间里（他们相遇时），甲走了全程的3/5，乙走了全程的2/5．甲比乙多走了全程的1/5，就是那3千米．9．（10分）将2013拆成3个互不相同的整数，使这三个数的和为2013，且其中任意两个数的和除以3都余1，这三个数中，最大的数最小是674 ．【分析】根据题目条件“任意两个数的和除以3都余1”可知，三个数除以3的余数均为2，若要求最大的数最小，则三个数的差最小，结合题目情况可得答案．【解答】解：根据题目条件可知，三个数除以3的余数均为2，若要求最大的数最小，则三个数的差最小，2013÷3=671，即若三个数相等，则分别是671，671，671，而671÷3=223…2，即第二个数已经满足条件，因此只需将第一个数减去一个最小的整数给第三个数，从而使第一个数和第三个数除以3的余数均为2即可．易知需要减去的最小整数为3，因此这三个数分别为668，671，674，所以这三个数中，最大的数最小是674．故答案为：674．【点评】本题首先要理解最大的数最小时所对应的情况，然后在三数相等的情况稍作变化即可．10．（10分）有一种五位数，从左向右第三位数字开始，每一个数字都是它前面两个数字的和，这样的五位数共有8 个．【分析】按题意，可以利用每一个数字都是它前面两个数字的和，把这几个五位数分别列举出来，一共有8个．【解答】解：根据分析，从首位1开始算起，由1+0=1，故有10112；由1+1=2，有11235；由1+2=3，故有12358；由2+0=2，故有20224；由2+1=3，故有21347；由3+0=3，故有30336；由3+1=4，故有31459；由4+0=4，故有40448．综上，这样的五位数有：10112、11235、12358、20224、21347、30336、31459、40448共8个．故答案是：8．【点评】本题考查了数字问题，突破点是：列举符合题意的数，不难求得五位数的个数．填空题11．（12分）圆上的50个点A1，A2，A3，…，A50将该圆分为50段等弧，以这50个点中的某些点为顶点，一共可以得到18 个不同的正多边形．【分析】由于题目要求是正多边形，因此正多边形的边数必须是50的约数，根据50的约数情况进行分情况加和即可．【解答】解：50=2×5×5，因此大于3的50的约数有5、10、25、50．当多边形为五边形时，可以得到50÷5=10个；当多边形为正十边形时，可以得到50÷10=5个；当多边形为正二十五边形时，可以得到50÷25=2个；当多边形为正五十边形时，可以得到50÷50=1个．共10+5+2+1=18个．故答案为：18．【点评】本题的突破口在于能想到正多边形的边数必须为50的约数，难度中等．12．（12分）将260个桃子分装到若干个相同的筐中，每个筐中最少放10个，最多放25个，放完后，每个框中的桃子数都不相同，有 1 种放法，可能有15 个筐．【分析】首先可根据10﹣25的数据个数（16）按最小公差1计算最小的桃子个数，看是否在给定的桃子数量范围内，若不符合要求，则可减少筐的数量进一步讨论，据此解答．【解答】解：10+11+12+13+14+15+…+23+24+25=280＞260，则不可能有16个筐，若为14个筐，则桃子最多可能有12+13+14+15+…+23+24+25=259，则不可能有14个筐，因此只能有15个筐，由于280﹣260=20，因此没有筐里放20个．故答案为：1；15．【点评】本题的突破口在于能根据最少个数和最大个数推断出筐的数量，难度中等．13．（12分）一个宝库有16个藏宝室，成4×4状排列，但只有一个进口和一个出口，分别开在如图所示的藏宝室，每个藏宝室至多只能进去一次，相邻的两个藏宝室之间都有门想通，每个藏宝室中的宝贝价值已标在图中，大盗买通守护，夜间进入宝库，他能带走的宝物价值最多是133 ．【分析】本题首先能想到根据染色问题进行分析，可将房间黑白相间染色，根据进口和出口所染颜色相同可知大盗应该经过了奇数个房间，因此最多经过15个房间，且有一个白格无法走到，据此解答．【解答】解：借助染色解题，给3×3的方格黑白相同染色（如图），进口为黑格，若全部走完16个方格，出口应为白格，而图中出口为黑格，故至少有一个白格不能走到，标数最小的白为2，因此首先考虑2进行试走，发现若不走2，则无法到达12，因此舍去，接下来考虑3，进行试走，可行的路线为1﹣7﹣5﹣6﹣15﹣10﹣13﹣9﹣16﹣2﹣12﹣11﹣14﹣8﹣4．因此大盗最多能带走的宝物价值1+7+5+6+15+10+13+9+16+2+12+11+14+8+4=133故答案为：133．【点评】本题的突破口是能想到用染色方法确认大盗最多经过的房间数，确认后最小标数并不一定能走通，因此需要试走通过才可．14．（12分）现有1×1×2的积木（A）、1×1×3的积木（B）、1×2×2的积木（C）（如图），分别有6块、11块、10块，从这些积木中选出若干个，拼成3×3×3的实心正方体，至多可以拼出 3 个3×3×3的实心正方体，写出这几个正方体的拼法分别所用的A、B、C的个数（如1A+7B+1C）：2A+1B+5C、1A+3B+4C、1A+7B+1C或4A+1B+4C、1A+3B+4C、1A+7B+1C【分析】首先计算出1×1×2的积木（A）、1×1×3的积木（B）、1×2×2的积木（C）能提供的总块数为85，3×3×3的实心正方体需要的积木块数为27，85÷27=3…4，因此首先可以判断至多能拼出3个3×3×3的实心正方体，然后根据奇偶性判断A、B、C各自所用的块数，据此解答．【解答】解：6块、11块、10块A、B、C积木总共能提供的块数是2×6+3×11+4×10=85，一个3×3×3的实心正方体需要的块数为27，因此最多拼成3个，且剩下块数为85﹣27×3=4，可以为2个A积木或1个C积木．27=2A+3B+4C，考虑27为奇数，因此B必须为奇数，因此B只能为1，3，5，7，B的总块数为11，因此3个实心正方体所用B的数目可以为1，5，5或1，3，7．①所用B的数目可以为1，5，5：拼法1：1B拼法2：4A+5B+1C拼法3：2A+5B+2C则拼法1中已经没有积木A可用，不符合题意；①所用B的数目可以为1，3，7：拼法1：2A+1B+5C（或4A+1B+4C）拼法2：1A+3B+4C拼法3：1A+7B+1C两种方法均符合题意．因此这几个正方形的拼法可以是 2A+1B+5C、1A+3B+4C、1A+7B+1C或4A+1B+4C、1A+3B+4C、1A+7B+1C．故答案为：3；2A+1B+5C、1A+3B+4C、1A+7B+1C或4A+1B+4C、1A+3B+4C、1A+7B+1C．【点评】本题考查拼接方法，需要掌握这种题的答题技巧，难度较大．15．（12分）请在下面的每个箭头里填上适当的数字（图中已经填出两个数字），使得每个数字都表示该箭头所指方向的箭头里含有不同数字的个数，其中双向箭头表示箭头所指的两个的箭头里不同数字的个数，图中第三行从左到右所填数字组成的四位数是1212 ．【分析】首先可以推断有已知数据所在行或列，然后根据已推断数据进一步推断未知数据．【解答】解：首先判断第一列，i箭头向下，向下只有一个数据，因此i填1，第一列第四行是3，则上面三个是不同数据，e是双向箭头，且上下共有3个数据，因此e填3，则a填2；然后判断第四行，n箭头向右，向右只有一个数据，则n填1，m填2；接着看第四列，h箭头向下，向下只有两个数据，l向上，向上只有两个数据，因此l为1或2，h填2，接着看第二行，f箭头向右，向右有两个数据，则f为1或2，g箭头向左，有两个数据，且不同，则g填2，则f填1，接着看第三列，k箭头向上，则k 为1或2，则c只能填2，k填1，接着看第一行，b只能为1或2，若b为1，则d为2，p为1，j为1，从而m 为1，而上面已推出m为2，矛盾，则b只能为2，则d为1，p为2，j为2，综上可得，第三行从左到右所填数字组成的四位数是1212．故答案为：1212．【点评】本题考查数据的推理，该题突破口在于已知数据和快速找出易推断数据．。
2017年美国“数学大联盟杯赛”初赛四年级试卷
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。
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。
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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.(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。
美国数学试卷小学四年级(2篇)
一、选择题（每题2分，共20分）1. 下列哪个数是偶数？A. 17B. 20C. 23D. 272. 下列哪个数是质数？A. 6B. 11C. 15D. 183. 小明有3个苹果，妈妈又给了他5个，小明现在有多少个苹果？A. 5B. 8C. 10D. 134. 下列哪个图形是正方形？A. 矩形B. 三角形C. 圆形D. 平行四边形5. 小华的房间长8米，宽5米，她的房间面积是多少平方米？B. 40C. 24D. 336. 小红有8个红色气球，小蓝有6个蓝色气球，他们一共有多少个气球？A. 14B. 16C. 12D. 107. 小明骑自行车从家到学校用了15分钟，他平均每分钟骑多少米？A. 60B. 80C. 90D. 1008. 下列哪个数是2的倍数？A. 13B. 16C. 19D. 229. 小刚有15个糖果，他每天吃掉2个，他可以吃几天？A. 7B. 8C. 910. 下列哪个图形是长方形？A. 正方形B. 三角形C. 圆形D. 平行四边形二、填空题（每题2分，共20分）11. 7加上5等于______。
12. 9减去3等于______。
13. 12乘以2等于______。
14. 18除以3等于______。
15. 3加上4再加上5等于______。
16. 8乘以7等于______。
17. 15除以5等于______。
18. 6乘以3等于______。
19. 20减去8等于______。
20. 10加上5再加上3等于______。
三、解答题（每题5分，共25分）21. 小明有24个铅笔，他每天用掉3个，他可以用几天？22. 小华有12个苹果，她要平均分给她的3个朋友，每个朋友可以分到几个苹果？23. 小李骑自行车从家到公园用了20分钟，他平均每分钟骑多少米？24. 小明有6个红色气球和8个蓝色气球，他总共有多少个气球？25. 小刚有18个糖果，他每天吃掉3个，他可以吃几天？四、应用题（每题10分，共20分）26. 小华和小丽一共有15本书，小华有8本书，小丽有多少本书？解答：小丽有的书 = 总书数 - 小华的书数小丽有的书 = 15 - 8小丽有的书 = 7所以，小丽有7本书。
2017年第十五届“走美杯”小数数学竞赛初赛试卷(四年级B卷)后附答案解析
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．。
20162017年度美国“数学大联盟杯赛”(中国赛区)初赛(五年级).doc
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)名师制作优质教学资料
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年美国“数学大联盟杯赛”(中国赛区)初赛高中年级试卷及答案
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。
美国数学大联盟杯
美国数学大联盟杯∙美国“数学大联盟杯赛”分为初赛，复赛，决赛三个层次。
初赛，复赛均在国内进行，采用美国大联盟杯赛原题（与美国同时同题）加上中国组委会命题的挑战题。
初赛采用在线考试方式，优秀者进入复赛，复赛为纸质试题。
复赛的优胜者进入决赛并，决赛在美国斯坦福大学举行，同时在那里参加为期5天的数学夏令营。
三年级的数学竞赛试题采用美国“数学大联盟杯赛”试题(一级)，加上组委会命题的挑战题。
竞赛时间为60分钟。
全部采用选择题或填空题(共40题)，总分200分。
其中40%的题目翻译成中文，其余60%为英文原题。
学生可以携带正规出版社出版的纸质简明英汉数学字典。
(禁止携带任何电子版的英汉字典、电子词典或计算器等。
)∙四年级的数学竞赛试题采用美国“数学大联盟杯赛”试题(二级)，加上组委会命题的挑战题。
竞赛时间为60分钟。
全部采用选择题或填空题(共40题)，总分200分。
其中30%的题目翻译成中文，其余70%为英文原题。
学生可以携带正规出版社出版的纸质简明英汉数学字典。
(禁止携带任何电子版的英汉字典、电子词典或计算器等。
)∙五年级的数学竞赛试题采用美国“数学大联盟杯赛”试题(三级)，加上组委会命题的挑战题。
竞赛时间为90分钟。
全部采用选择题或填空题(共50题)，总分250分。
其中20%的题目翻译成中文，其余80%为英文原题。
学生可以携带正规出版社出版的纸质简明英汉数学字典。
(禁止携带任何电子版的英汉字典、电子词典或计算器等。
)∙六年级、七年级(初一)的竞赛采用美国“数学大联盟杯赛”试题(四级)，加上组委会命题的挑战题。
竞赛时间为90分钟。
全部采用选择题或填空题(共50题)，总分250分。
所有题目英文命题。
学生可以携带正规出版社出版的纸质简明英汉数学字典。
(禁止携带任何电子版的英汉字典、电子词典或计算器等。
)∙初二、初三的竞赛采用美国“数学大联盟杯赛”试题(五级)，加上组委会命题的挑战题。
竞赛时间为120分钟。
全部采用选择题或填空题(共60题)，总分300分。
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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。