美国普特南数学竞赛试题2015(题目)

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(最新整理)数学建模美赛试题

(最新整理)数学建模美赛试题

2015数学建模美赛试题编辑整理:尊敬的读者朋友们:这里是精品文档编辑中心,本文档内容是由我和我的同事精心编辑整理后发布的,发布之前我们对文中内容进行仔细校对,但是难免会有疏漏的地方,但是任然希望(2015数学建模美赛试题)的内容能够给您的工作和学习带来便利。

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地球资源的消耗速度快,越来越多的人关注人类社会的未来.自1960年以来,已经有许多专家研究可持续发展.然而大多数人的研究对象是整个世界,一个国家或一个地区。

几乎没有人选择48个最不发达国家(LDC)在联合国为研究对象列表。

然而,LDC国家集团共享许多相同的点。

他们的发展道路也有法律的内涵。

本文选择这些国家为研究对象针对发现常规的可持续发展道路。

本文组织如下.第二部分介绍研究的背景和本研究的意义。

第三节描述了我们对可持续发展的理解细节和显示我们的评估系统的建立过程和原理,那么我们估计每一个国家的LDC和获得可持续发展的能力和等级。

第四节提供了一个最糟糕的国家毛里塔尼亚计划指数在第三节。

第五节演示了在第四节的合理性和可用性计划。

最后在第六节总结本文的主要结论和讨论的力量和潜在的弱点。

地球上的资源是有限的。

三大能源石油、天然气和煤炭可再生。

如何避免人类的发展了资源枯竭和实现可持续发展目标是现在的一个热门话题.在过去的两个世纪,发达国家已经路上,先污染,再控制和达到高水平的可持续发展。

发展中国家希望发展和丰富。

然而,因为他们的技术力量和低水平的经济基础薄弱,浪费和低效率的发展在这些国家是正常的.所以本文主要关注如何帮助发展中国家特别是48在联合国最不发达国家实现可持续发展是列表可持续发展的理解是解决问题的关键.可持续发展的定义经历了一个长期发展的过程.在这里,布伦特兰可持续发展委员会的简短定义的"能力发展可持续- — - — - -以确保它既满足现代人的需求又不损害未来的能力代来满足自己的需求"[1]无疑是最被广泛接受的一个在各种内吗定义.这个定义方面发挥了重要作用在很多国家的政策制定的过程.然而,为了证明一个国家的现状是否可持续不可持续的,更具体的定义是必要的更具体的概念,我们认为,如果一个国家的发展是可持续的,它应该有一个基本的目前的发展水平,一个平衡的国家结构和一个光明的未来。

普特兰数学竞赛题

普特兰数学竞赛题
William Lowell Putnam Mathematical Competition 1 The 69th PMC 2008
A1 Let f : R2 → R be a function such that f (x, y )+f (y, z )+f (z, x) = 0 for all real numbers x, y , and z . Prove that there exists a function g : R → R such that f (x, y ) = g (x) − g (y ) for all real numbers x and y . Answer: The function g (x) = f (x, 0) works. Substituting (x, y, z ) = (0, 0, 0) into the given functional equation yields f (0, 0) = 0, whence substituting (x, y, z ) = (x, 0, 0) yields f (x, 0) + f (0, x) = 0. Finally, substituting (x, y, z ) = (x, y, 0) yields f (x, y ) = −f (y, 0) − f (0, x) = g (x) − g (y ). Remark: A similar argument shows that the possible functions g are precisely those of the form f (x, 0) + c for some c. A2 Alan and Barbara play a game in which they take turns filling entries of an initially empty 2008 × 2008 array. Alan plays first. At each turn, a player chooses a real number and places it in a vacant entry. The game ends when all the entries are filled. Alan wins if the determinant of the resulting matrix is nonzero; Barbara wins if it is zero. Which player has a winning strategy? Answer: Barbara wins using one of the following strategies. First solution: Pair each entry of the first row with the entry directly below it in the second row. If Alan ever writes a number in one of the first two rows, Barbara writes the same number in the other entry in the pair. If Alan writes a number anywhere other than the first two rows, Barbara does likewise. At the end, the resulting matrix will have two identical rows, so its determinant will be zero. Second solution: (by Manjul Bhargava) Whenever Alan writes a number x in an entry in some row, Barbara writes −x in some other entry in the same row. At the end, the resulting matrix will have all rows summing to zero, so it cannot have full rank. A3 Start with a finite sequence a1 , a2 , . . . , an of positive integers. If possible, choose two indices j < k such that aj does not divide ak , and replace aj and ak by gcd(aj , ak ) and lcm(aj , ak ), respectively. Prove that if this process is repeated, it must eventually stop and the final sequence does not depend on the choices made. (Note: gcd means greatest common divisor and lcm means least common multiple.) Answer: We first prove that the process stops. Note first that the product a1 · · · an remains constant, because aj ak = gcd(aj , ak )lcm(aj , ak ). Moreover, the last number in the sequence can never decrease, because it is always replaced by its least common multiple with another number. Since it is bounded above (by the product of all of the numbers), the last number must eventually reach its maximum value, after which it remains constant throughout. After this happens, the next-to-last number will never decrease, so it eventually becomes constant, and so on. After finitely many steps, all of the numbers will achieve their final values, so no more steps will be possible. This only happens when aj divides ak for all pairs j < k . We next check that there is only one possible final sequence. For p a prime and m a nonnegative integer, we claim that the number of integers in the list divisible by pm never changes. To see this, suppose we replace aj , ak by gcd(aj , ak ), lcm(aj , ak ). If neither of aj , ak is divisible by pm , then neither of gcd(aj , ak ), lcm(aj , ak ) is either. If exactly one aj , ak is divisible by pm , then lcm(aj , ak ) is divisible by pm but gcd(aj , ak ) is not. gcd(aj , ak ), lcm(aj , ak ) are as well. 1

2015年美国数学建模竞赛B题一等奖

2015年美国数学建模竞赛B题一等奖

Team# 39600
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Controlling Epidemic Situation of Ebola
Contents
1 Introduction .................................................................................................................................... 3 2 Model Overview ............................................................................................................................ 4 3 Common Assumptions ................................................................................................................... 5 4 Model ............................................................................................................................................. 6 4.1 Improved SIR Model .......................................................................................................... 6 4.1

2015美国数学竞赛10年级试题

2015美国数学竞赛10年级试题

American Mathematics Competitions
WRITE TO US!
Correspondence about the problems and solutions for this AMC 10 and orders for publications should be addressed to:
Students who score 120 or above or finish in the top 2.5% on this AMC 10 will be invited to take the 33rd annual American Invitational Mathematics Examination (AIME) on Thursday, March 19, 2015 or Wednesday, March 25, 2015. More details about the AIME and other information are on the back page of this test booklet.
MAA American Mathematics Competitions
are supported by Academy of Applied Science Akamai Foundation American Mathematical Society American Statistical Association Art of Problem Solving, Inc. Casualty Actuarial Society Conference Board of the Mathematical Sciences The D.E. Shaw Group Google IDEA MATH, LLC Jane Street Capital Math for America Inc. Mu Alpha Theta National Council of Teachers of Mathematics Simons Foundation Society for Industrial & Applied Mathematics

2015美国大学生数学建模竞赛D题

2015美国大学生数学建模竞赛D题

1.2 Our work




We tackle four main sub problems: Factors affecting the evaluation of sustainable development of a country are analyzed based on the theory of sustainable development. Develop a model for the sustainability of a country. This model should provide a measure to distinguish more sustainable countries and policies from less sustainable ones. Choose from forty-eight poorest countries LDC country, according to the model of a task1 has been established for the selected countries to create a more sustainable development plan in the next 20 years in the development process, so that the country toward a more sustainable future. Evaluate the effect our 20-year sustainability plan has on our country’s sustainability measure created in Task 1. And predicted under the evaluation system to implement our plan will happen the change over the next 20 years. According to the selected country, we should consider the environmental factors, Climate change, development aid, foreign investment, natural disasters, and the instability of the regime, etc. We determine which project or policy for the sustainable development measures of the state will have the greatest effect. Write a report to explain the established model, including sustainable development, sustainable development plans, according to the model and the national environmental situation, analysis the effect of the plan. For the ICM provides a sustainable development of intervention strategy about investment in LDC countries.

美国数学学会中学生数学竞赛真题和答案解析2015AMC8 Solutions

美国数学学会中学生数学竞赛真题和答案解析2015AMC8  Solutions

This Solutions Pamphlet gives at least one solution for each problem on this year’s exam and shows that all the problems can be solved using material normally associated with the mathematics curriculum for students in eighth grade or below. These solutions are by no means the only ones possible, nor are they necessarily superior to others the reader may devise.We hope that teachers will share these solutions with their students. However, the publication, reproduction, or communication of the problems or solutions of the AMC 8 during the period when students are eligible to participate seriously jeopardizes the integrity of the results. Dissemination at any time via copier, telephone, email, internet or media of any type is a violation of the competition rules.Correspondence about the problems and solutions should be addressed to:Prof. Norbert Kuenzi, AMC 8 Chair934 Nicolet AveOshkosh, WI 54901-1634Orders for prior year exam questions and solutions pamphlets should be addressed to:MAA American Mathematics CompetitionsAttn: PublicationsPO Box 471Annapolis Junction, MD 20701© 2015 Mathematical Association of AmericaWe thank the following donors for their generous support of the MAA American Mathematics Competitions, MOSP and the IMOPatron’s CircleAkamai FoundationSimons FoundationWinner’s CircleAmerican Mathematical SocietyThe D.E. Shaw GroupDropboxMathWorksTwo SigmaTudor Investment CorporationAchiever’s CircleArt of Problem SolvingJane Street CapitalMath for AmericaSustainer’s circleAcademy of Applied ScienceArmy Educational Outreach ProgramCollaborator’s CircleAmerican Statistical AssociationCasualty Actuarial SocietyConference Board of the Mathematical SciencesExpii, Inc.IDEA MATHMu Alpha ThetaNational Council of Teachers of MathematicsSociety for Industrial and Applied MathematicsStar League。

2000到2015年美国数学竞赛

2000到2015年美国数学竞赛

2015 美国数学竞赛(十年级)中图分类号:G424.79 文献标识码iA 文章编号:1005—6416(2015)07—0026—07 1.计算:(2。

一1+5 一0) X5的值为( ).(A)一125 (B)一120 (C)25(D) (E 12.箱子中放有三角形和正方形的瓷砖共25块,共有84条边.则箱子中的正方形瓷砖有( )块.(A)3 (B)5 (C)7 (D)9 (E)113.如图l,安琪儿用18根牙签拼三层楼梯.照这样计算,要想拼五层楼梯,她还需( )根牙签.(A)9 (B)l8 (C)2O(D)22 (E)24图14.巴勃罗、索菲亚和米娅在一次聚会上各分得一些糖果.巴勃罗的糖果数为索菲亚糖果数的3倍,索菲亚的糖果数为米娅糖果数的2倍.巴勃罗决定将自己的糖果分给索菲亚和米娅一部分,这样三个人的糖果数相等.则巴勃罗分给索菲亚的糖果数占自己原来糖果数的( ).(A) (B1(c1(D) (E)寻5.帕特里克先生是l5名学生的数学老师.一次测验后他发现,去掉佩顿的成绩,其余人的平均成绩为80分,加上佩顿的成绩后,全班的平均成绩为81分.则在这次考试中,佩顿的成绩为( )分.(A)81 (B)85 (C)91 (0)94 (E)95注意到,1D,1C =90。

+_1_ C = C.于是,D、,。

、、C四点共圆.则E= DC,2,CI2F= cD,故GEF= EDIl+ DILE= ADI 一DB + DCIz1= ÷厶( ADC+ BcD)一ADB,GFE :FCI2+ FlC= BCI2一+ ,.1= ÷( Dc+ BcD)一ADB.从而,∞F= G阳,得钮=G又GI3平分AGB,故上F.14.若一共有13个数,则可排列如下:a l,a2,a3,…,a 8,a9,a 2 ,a 3 ,a 4 ,…,a 9 ,a lo ,a 3 ,a 4 ,a 5 ,…,a lo ,a l1,04 ,a5 ,a6,…,0 11,a 12,U 5 ,a 6 ,a 7 ,…,a 12 ,a 13·由已知,其中每行数之和为正,从而,数表中所有数之和为正;另一方面,数表中每列数之和为负,从而,数表中所有数之和为负.矛盾.这表明,满足要求的数串至多有12项.考虑如下的一串数字:一4、一4、一4、15、一4、一4、一4、一4、15、一4、一4、一4,这串数满足题中要求且有12项,故知满足题中要求的,的最大值为12.(张宇鹏提供)2015年第7期276.已知两个正数的和是差的5倍.则较大数与较小数之比为( ).(A5(B3(c9(D)2(E)詈7.等差数列l3,16,19,…,70,73中,共有( )项.(A)20 (B)21 (C)24 (D)60 (E)618.两年前,皮特的年龄为其表弟克莱尔年龄的3倍;四年前,皮特的年龄为克莱尔年龄的4倍.( )年后,皮特和克莱尔的年龄比为2:1.(A)2 (B)4 (C)5 (D)6 (E)89.已知两个圆柱的体积相同,第二个圆柱半径比第一个圆柱半径多10%.下列叙述正确的为( ).(A)第二个圆柱比第一个圆柱低10%(B)第一个圆柱比第二个圆柱高10%(c)第二个圆柱比第一个圆柱低21%(D)第一个圆柱比第二个圆柱高21%(E)第二个圆柱的高是第一个圆柱高的80%10.对于字母排列abed,有( )种不同的重排,使得原来排列中相邻的两个字母重排后不能相邻(如ab、ba在重排后不能出现).(A)0 (B)1 (C)2 (D)3 (E)411.已知矩形的长与宽之比为4:3,对角线的长为d.若用|l} 表示矩形的面积,则k 的值为( ).(A)号(B)号(c) (D) (E)12.已知A(√7c,a)、B(√7c,b)为曲线Y + =2x Y +I_ L的两不同点.则Ia—bI的值为( ).(A)1 (B)詈(c)2(D) (E)1+13.克劳迪娅有5分和10分的硬币共12枚,用这些硬币中的一部分或者全部硬币恰可以组合成17种不同的币值.则克劳迪娅有( )枚1O分的硬币.(A)3 (B)4 (C)5 (D)6 (E)714.钟表圆盘的半径为2O厘米,它与另一个半径为10厘米的小圆盘在12点位置外切,小圆盘上有一个固定指针,开始时指针竖直指向上方.当小圆盘按顺时针方向沿大表盘外沿滚动,且始终保持相切,直到小圆盘上的指针再一次竖直指向上方时停止.此时,两圆盘外切的切点位于大表盘上的( )点位置.(A)2 (B)3 (C)4 (D)6 (E)815.考虑分数(x,y为两个互素的正整Y数)组成的集合.若分子和分母均增加1,则分数的值增加10%.那么,集合中这样的分数有( )个.(A)0 (B)1 (C)2 (D)3(E)无数多16.若Y+4=( 一2),+4=(Y一2),且x#y,则+ 的值为( ).(A)10 (B)15 (C)20 (D)25 (E)3017.一条经过坐标原点的直线,与直线= 1、Y=1+ 围成一个等边三角形.则这个等边三角形的周长为( ).(A)2./6 (B)2+2,3 (c)6(D)3+2√3 (E)6+18.已知十六进制数是由O一9十个数码和A至F六个字母构成的,其中,A,B,…,F分别代表1O,11,…,15.在前1000个正整数中,找出所有只用数码表示的十六进制数.求出这些数的个数n,此时,凡的各位数码之和为( ).(A)17 (B)18 (C)19 (D)20 (E)2119.在等腰Rt△ABC中,已知BC=AC,28 中等数学C=90。

2015年AMC12B竞赛真题及答案

2015年AMC12B竞赛真题及答案

2015 AMC 12B竞赛真题Problem 1What is the value of ?Problem 2Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishes the second task at 2:40 PM. When does she finish the third task?Problem 3Isaac has written down one integer two times and another integer three times. The sum of the five numbers is 100, and one of the numbers is 28. What is the other number?Problem 4David, Hikmet, Jack, Marta, Rand, and Todd were in a 12-person race with 6 other people. Rand finished 6 places ahead of Hikmet. Marta finished 1 place behind Jack. David finished 2 places behind Hikmet. Jack finished 2 places behind Todd. Todd finished 1 place behind Rand. Marta finished in 6th place. Who finished in 8th place?Problem 5The Tigers beat the Sharks 2 out of the 3 times they played. They then played more times, and the Sharks ended up winning at least 95% of all the games played. What is the minimum possible value for ?Problem 6Back in 1930, Tillie had to memorize her multiplication facts from to . The multiplication table she was given had rows and columns labeled with the factors, and the products formed the body of the table. To the nearest hundredth, what fraction of the numbers in the body of the table are odd?Problem 7A regular 15-gon has lines of symmetry, and the smallest positive angle for which it has rotational symmetry is degrees. What is ?Problem 8What is the value of ?Problem 9Larry and Julius are playing a game, taking turns throwing a ball at a bottle sitting on a ledge. Larry throws first. The winner is the first person to knock the bottle off the ledge. At each turn the probability that a player knocks the bottle off the ledgeis , independently of what has happened before. What is the probability that Larry wins the game?Problem 10How many noncongruent integer-sided triangles with positive area and perimeter less than 15 are neither equilateral, isosceles, nor right triangles?Problem 11The line forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?Problem 12Let , , and be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation ?Problem 13Quadrilateral is inscribed in a circle withand . What is ?Problem 14A circle of radius 2 is centered at . An equilateral triangle with side 4 has a vertex at . What is the difference between the area of the region that lies inside the circle but outside the triangle and the area of the region that lies inside the triangle but outside the circle?Problem 15At Rachelle's school an A counts 4 points, a B 3 points, a C 2 points, and a D 1 point. Her GPA on the four classes she is taking is computed as the total sum of points divided by 4. She is certain that she will get As in both Mathematics andScience, and at least a C in each of English and History. She thinks she has a chance of getting an A in English, and a chance of getting a B. In History, shehas a chance of getting an A, and a chance of getting a B, independently ofwhat she gets in English. What is the probability that Rachelle will get a GPA of at least 3.5?Problem 16A regular hexagon with sides of length 6 has an isosceles triangle attached to each side. Each of these triangles has two sides of length 8. The isosceles triangles are folded to make a pyramid with the hexagon as the base of the pyramid. What is the volume of the pyramid?Problem 17An unfair coin lands on heads with a probability of . When tossed times, theprobability of exactly two heads is the same as the probability of exactly three heads. What is the value of ?Problem 18For every composite positive integer , define to be the sum of the factors in the prime factorization of . For example, because the prime factorization of is , and . What is the range of the function , ?Problem 19In , and . Squares and are constructed outside of the triangle. The points , , , and lie on a circle. What is the perimeter of the triangle?Problem 20For every positive integer , let be the remainder obtained when is divided by 5. Define a functionrecursively as follows:What is ?Problem 21Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than 5 steps left). Suppose that Dash takes 19 fewer jumps than Cozy to reach the top of the staircase. Let denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of ?Problem 22Six chairs are evenly spaced around a circular table. One person is seated in each chair. Each person gets up and sits down in a chair that is not the same chair and is not adjacent to the chair he or she originally occupied, so that again one person is seated in each chair. In how many ways can this be done?Problem 23A rectangular box measures , where , , and are integers and. The volume and the surface area of the box are numerically equal. How many ordered triples are possible?Problem 24Four circles, no two of which are congruent, have centers at , , , and , and points and lie on all four circles. The radius of circle is times the radius of circle , and the radius of circle is times the radius of circle .Furthermore, and . Let be the midpoint of . What is ?Problem 25A bee starts flying from point . She flies inch due east to point . For, once the bee reaches point , she turns counterclockwise and then flies inches straight to point . When the bee reaches she isexactly inches away from , where , , and are positiveintegers and and are not divisible by the square of any prime. What is?2015 AMC 12B竞赛真题答案1.C2.b3.a4.b5.b6.a7.d8.d9.c 10.c 11.e 12.d 13.b 14.d 15.d 16.c 17.d 18.d 19.c 20.b 21.d 22.d 23.b 24.d 25.b。

15年竞赛小题分类

15年竞赛小题分类

集合、命题2.集合{2135}A x a x a =+≤≤+,{333}B x x =≤≤,()A A B ⊆ , 则a 的取值范围是___________18.对四位数(19,0,,9)abcd a b c d ≤≤≤≤,若,,a b b c c d ><>,则称abcd 为P 类数,若,,a b b c c d <><,则称abcd 为Q 类数,用()N P 与()N Q 分别表示P 类数与Q 类数的个数,则()()N P N Q -的值为34.已知集合230123112310{22222|{0,1},0,1,2,,2}k k a k a k a a k k k k i A a a a a a i k --+-+-+---=+⋅+⋅++⋅+⋅∈=-用k n 表示集合k A 中所有元素的和,则20151kk n==∑35.设集合{1,0,2}A =-,集合{|B x x A =-∈且2}x A -∉,则集合B 等于A.{1}B.{2}-C.{1,2}--D.{1,0}- 66设22{|,,}M a a x y x y Z ==-∈,则对任意的整数n ,形如4,41,42,43n n n n +++的数中,不是M 中的元素的数为A.4nB.41n +C.42n +D.43n + 71.记集合3124234{0,1,2,3,4,5,6},|,1,2,3,47777i a a a a T M a T i ⎧⎫==+++∈=⎨⎬⎩⎭,将M 中的元素按从大到小顺序排列,则第2015个数是74“实数,,,a b c d 依次成等差数列”是“a d b c +=+”成立的A.充分而不必要条件B.必要而不充分条件C.充要条件D.既不充分又不必邀条件91若集合{31,65}A x y =与集合{5,403}B xy =(其中,)x y R ∈仅有一个公共元素,则集合A B 中所有元素之积的值是1.设241i a ==∑(其中[]x 表示不超过x 的最大整数,集合*{|A x x N =∈且|}x a ,则集合A中的元素个数是A.4B.6C.8D.121.已知集合{|112}M x N x =∈≤≤,三元素集合{,,}A a b c =满足:A M ≠⊂且a b c ++为平方数,则这样的集合的个数是1.已知集合{|112}M x N x =∈≤≤,三元素集合{,,}A a b c =满足:A M ≠⊂且a b c ++为平方数,则这样的集合的个数是 12.对任意正整数n ,定义函数()n μ如下:(1)1μ=,且当12122ii n p p p ααα=≥ 时,12(1),1()0,t i n αααμ⎧-=====⎨⎩ 否则,其中121,,,,i t p p p ≥ 是不同的质数.若记12{,,,}k A x x x = 为全部不同正因数的集合,则1()ki i x μ==∑1.设,,A B C 是三个集合,则B 和C 都是A 的子集是()()A B A C B C = 成立的 A.充分而不必要条件 B.必要而不充分条件 C.充要条件 D.既不充分又不必邀条件三角函数、解三角形1.已知ABC ∆的外接圆半径为R ,且B b a C A R sin )2()sin (sin 222-=-(其中a 、b 分别是A ∠、B ∠的对边). 那么C ∠的大小为___________. 12.若实数θ满足cos tan θθ=,则41cos sin θθ+的值为 17.设ω为正实数,若存在,(2)a b a b ππ≤<≤,使得sin sin 2a b ωω+=,则实数ω的取值范围是21.若tan 3tan (0)2παββα=<<<,则αβ-的最大值是22.已知ABC ∆为等腰直角三角形,其中C ∠为直角,1,AC BC ==过点B 作平面ABC 的垂线DB ,使得1DB =,在,DA DC 上分别取点,E F ,则BEF ∆周长的最小值为33.设函数2sin ()((0,1))xf x x x π=∈,则()()(1)g x f x f x =+-的最小值为 37.函数cos sin ||(0)sin xy x x xπ=⋅<<的图象大致是A. B. C. D.49,[,],44x y a R ππ∈-∈且331sin 20,4sin 202x x a y y a +-=++=,则cos(2)x y +的值为54知顶角为20的等腰三角形的底边长为a ,腰长为b ,则332a b ab += 67已知三角形的三边长为连续自然数,且其最大角是最小角的两倍,则该三角形的周长为 81已知函数()sin()sin()(0)24f x x x ππωωω=++>的最小正周期为π,则()f x 在区间[0,]2π上的值域为86函数())4f x x π=-在4324x π=处的值是4.1sin10cos10-=2.已知ABC ∆的三边,,a b c 成等比数列,边,,a b c 所对的角分别为,,A B C 且sin sin sin sin cos 21A B B C B ++=,则角B 为 A.4π B.3π C.2π D.23π2.若1sin cos 5αα+=,则|sin cos |αα-=2.若1sin cos 5αα+=,则|sin cos |αα-=2.在ABC ∆中,三内角,,A B C 的对边分别是,,a b c ,若cos 2cos 3cos a b cA B C==,则A ∠的大小为 A.6π B.4π C.3π D.512π10.设sin cos 10,2cos 240x x x y y π+-=-++=,则sin(2)x y -= 6.设ABC ∆的周长为12,内切圆的半径为1,则A.ABC ∆必为直角三角形B.ABC ∆必为锐角三角形C.ABC ∆必为直角三角形或锐角三角形D.该三角形不能确定为何种三角形11.设复数44cos sin 77z i ππ=+,则23243||111z z z z z z++=+++(用数字作答)向量7.已知,a b两个互相垂直的单位向量,且1c a c b ==,则对任意的正实数t ,1||c ta b t++ 的最小值是____________.14.在矩形ABCD 中,2,1AB AD ==,边DC 上(包括点,)D C 的动点P 与CB 延长线上(包括点)B 的动点Q 满足||||DP BQ = ,则向量PA 与向量PQ 的数量积PA PQ ⋅的最小值为24.已知向量,,a b c 满足:*||:||:||2::3()a b c k k N =∈ ,且()2()b a c b -=- ,若α为,a c 的夹角,则cos α的值为48ABC ∆中,点O 是BC 的中点,过点O 的直线分别交直线,AB AC 于不同两点,M N ,若,AB mAM AC nAN ==,则m n +的值为56已知点P 在直角ABC ∆所在的平面内,90,BAC CAP ∠=∠ 为锐角,||2,2,1AP AP AC AP AB =⋅=⋅= ,当||AB AC AP ++取得最小值时,tan CAP ∠=76已知向量,a b的夹角为60 ,且||1,|2|a a b =-=则||b等于B.32 C. D.2 92设向量(cos ,sin ),(sin ,cos )a b αααα==-,向量127,,,x x x 中有3个为向量a ,其余为向量b ,向量127,,,y y y 中有2个为向量a ,其余为向量b ,则71i i i x y =⋅∑的可能取值中最小的是6.如图,,M N 分别为正六边形ABCDEF 的对角线,AC CE 的內分点,且AM CNAC CEλ==,若,,B M N三点共线,则λ的值是A.3B.13C.2D.1211.在矩形ABCD 中,3,4,AB AD P ==为矩形ABCD 所在平面上的任意一点,满足:2,PA PC ==则PB PD ⋅=441234,,,A A A A 是平面直角坐标系中两两不同的四个点,若1312,A A A A λ=1412(,)A A A A R μλμ=∈ 且112λμ+=,则称34,A A 调和分割12A A ,已知平面上的点,C D 调和分割AB ,则下列说法正确的是A.C 可能是线段AB 的中点B.D 可能是线段AB 的中点C.,C D 可能同时在线段AB 上D.,C D 不可能同时在线段AB 的延长线上函数、方程 8.若关于x 的方程24xkx x =+有四个不同的实数解,则k 的取值范围为____________.10.设()f x 是定义在整数集上的函数,满足条件:⑴(1)1,(2)0f f ==;⑵对任意的,x y 都有()()(1)(1)()f x y f x f y f x f y +=-+-,则(2015)f =___________.11.设,a b 为不相等的实数,若二次函数2()f x x ax b =++满足()()f a f b =,则(2)f 的值是19.已知函数())1(0)f x ax a =+>,则1(ln )(ln )f a f a+=23.已知函数3()3f x x x =+,对任意的[2,2],(8)(2)0x m f mx f ∈--+<恒成立,则正实数x 的取值范围是30.已知实数,x y 满足2ln(1)ln(1)4x x y x y =+-+--+,则2320152016x y +的值是 50函数()y f x =对定义域内的每一个值1x ,在其定义域内都存在唯一的实数2x ,使12()()1f x f x =成立,则称该函数为“依赖函数”,给出以下命题: ①21y x =是“依赖函数”;②函数sin ([,]22y x x ππ=∈-是“依赖函数”; ③函数2x y =是“依赖函数”; ④函数ln y x =是“依赖函数”; ⑤若函数(),()y f x y g x ==都是“依赖函数”,且定义域相同,则函数()()y f x g x =也是“依赖函数”.其中所有真命题的序号是(填上你认为正确的所有命题的序号)58函数()1)f x =的值域为73已知函数121(),14()log ,1xx f x x x ⎧<⎪=⎨≥⎪⎩,则[(1)]f f -等于A.2B.2-C.14D.12-75若方程()2f x =在(,0)-∞内有解,则函数()y f x =的图象可能是A. B. C. D.77已知函数()||f x x x =,若对任意的1x ≥都有()()0f x m mf x ++<恒成立,则实数m 的取值范围是A.(,1)-∞-B.(,1]-∞-C.(,2)-∞-D.(,2]-∞- 78若函数3()43f x x x =-在(,2)a a +上存在最大值,则实数a 的取值范围是 A.5(,1)2-- B.5(,1]2-- C.51(,)22-- D.51(,]22-- 85已知点(4,1)P 在函数()log ()(0)a f x x b b =->的图象上,则ab 的最大值是96函数y =3.已知函数22()53196|53196|f x x x x x =-++-+,则(1)(2)(50)f f f +++= A.660 B.664 C.668 D.672 8.已知实数,a b 满足:lg 10,1010b a a b +=+=,则lg()a b +=3.函数2y x =+的值域是4.函数()f x =M ,最小值为m ,则Mm的值为D.27.已知复数41z i=-(其中i 为虚数单位,满足21)i =-,则||z = 8.已知469a b c==,则121a b c-+=2.方程||1y = A.一个圆 B.两个半圆 C.一个椭圆 D.两个圆3.用[]x 表示不大于x 的最大整数,方程2[]20x x --=共有( )个不同的实数根 A.1 B.2 C.3 D.44.方程35711x x x x++=共有( )个不同的实数根A.0B.1C.2D.3 1.对于任意实数x ,符号[]x 表示不超过x 的最大整数;符号{}x 表示x 的小数部分,即{}[]x x x =-,则2320142014201420142014++++=2015201520152015⎧⎫⎧⎫⎧⎫⎧⎫⎨⎬⎨⎬⎨⎬⎨⎬⎩⎭⎩⎭⎩⎭⎩⎭4.设,x y 是实数,且满足:33(1)2015(1)1(1)2015(1)1x x y y ⎧-+-=-⎪⎨-+-=⎪⎩,则x y += 7.函数3sin 2cos xy x-=-的最大值是微分、积分 36.设0(sin cos )k x x dx π=-⎰,若88018(1)kx a a x a x -=+++ ,则128a a a +++ 等于A.1-B.0C.1D.238.已知函数5()sin 1f x x =+,根据函数的性质,积分的性质和积分的几何意义,探求22()f x dx ππ-⎰的值,结果是A.162π+ B.0 C.1 D.π 数列4.在数列{}n a 中,122,10,a a ==对所有的正整数n 都有21n n n a a a ++=-,则2015a = 13.已知复数数列{}n z 满足111,1(1,2,)n n z z z ni n +==++= ,其中i 为虚数单位, n z 表示n z 的共轭复数,则2015z 的值是39.设正项等比数列{}n a 的前n 项和为n S ,且10103010202(21)S S S +=+,则数列{}n a 的公比为 A.18 B.14 C.12D 。

2015AMC12A试题及详解(繁体中文)

2015AMC12A试题及详解(繁体中文)
2015 第 66 屆 AMC12A 試題
0 2 1. 算出 2 1 5 0 1
5 之值為何? (C)
(A) 125
答: (C)
(B) 120
1 5
(D)
5 24
(E) 25 。
【2015AMC12A】
解: 所求 1 1 25 0 1 5
解: 外離( k 4 ),外切( k 3 ),相交( k 2 ),內切( k 1 ),內離 k 0 共五種可能值
12. 兩拋物線 y ax 2 及 y 4 bx 與坐標軸恰交於四點, 若這四點為一個面積為 12 的鳶形的頂點,則 a b 之值為何? (A) 1 (B) 1.5 (C) 2 (D) 2.5 (E) 3 。
2
x
2 2
y
2
4 x 1 為完全平方,令 x t
t t 1 1006
2
1, y 2 t , t N
則P 2 x y 2 2t
2 t 3 2015
t 1 ~ 31
20. 設 T 與 T ' 是兩個不全等的等腰三角形,它們有相同的面積與相同的周長。 若三角形 T 的三邊長分別為 5 、 5 與 8 ;而三角形 T ' 的三邊長分別為 a 、 a 與 b , 則下列哪一個數最接近 b ? (A) 3 (B) 4 (C) 5 (D) 6 (E) 8 。 【2015AMC12A】
15. 若將
(A) 4
答: (C)
123456789 2
26
5
4
用小數表示,則此數在小數點後最少有幾位數?
(B) 22
(C) 26

美国数学大联盟2015年复赛试题

美国数学大联盟2015年复赛试题

Descriptive Statistics, (the supplementary materials, see separate document).
2
For all the questions below, login to your account at / , and enter your answers. Answers written on this document will NOT be credited.
Hint: Q3 is the third quartile. For the definition of Q3 , see Section 14.3, of Chapter 14, “Descriptive Statistics.”
Question 3: (credit: 4)
Note: Q1 is the first quartile. Q3 is the third quartile. For the definition of Q1 and Q3 , see Section 14.3, of Chapter 14, “Descriptive Statistics.” Note: For this question, please write your answer on file “high-school-answersheet.doc”, downloadable together with this document at , and submit file “high-school-answersheet.doc” at after you are done.
Understanding the above Chapter on “Normal Distributions” requires familiarity with basic statistics terms such as mean, median, standard deviation, percentile, quartile, and etc. It helps to read following Chapter on “Descriptive Statistics”, the supplementary materials, if you need to refresh your memory of these terms.

普特南大学数学竞赛试题及答案

普特南大学数学竞赛试题及答案

普特南大学数学竞赛试题及答案问题一:证明对于任意正整数\( n \),\( 1^2 + 1 + 2^2 + 2 + \ldots +n^2 + n = \frac{n(n+1)(2n+1)}{6} \)。

答案一:我们可以使用数学归纳法来证明这个等式。

首先,当\( n = 1 \)时,左边等于1,右边等于1,等式成立。

假设对于某个正整数\( k \),等式成立,即:\[ 1^2 + 1 + 2^2 + 2 + \ldots + k^2 + k =\frac{k(k+1)(2k+1)}{6} \]我们需要证明对于\( n = k + 1 \)时,等式也成立:\[ 1^2 + 1 + 2^2 + 2 + \ldots + k^2 + k + (k+1)^2 + (k+1) \] \[ = \frac{k(k+1)(2k+1)}{6} + (k+1)^2 + (k+1) \]\[ = \frac{k(k+1)(2k+1) + 6(k+1)^2 + 6(k+1)}{6} \]\[ = \frac{(k+1)(2k^2 + k + 6k + 6 + 6)}{6} \]\[ = \frac{(k+1)(2k^2 + 7k + 12)}{6} \]\[ = \frac{(k+1)(2(k+1)(k+3) + 6)}{6} \]\[ = \frac{(k+1)(k+1)(2(k+1) + 3)}{6} \]\[ = \frac{(k+1)(k+1)(2(k+1) + 1)}{6} \]这证明了当\( n = k + 1 \)时等式也成立。

因此,对于所有正整数\( n \),等式成立。

问题二:给定一个圆的半径为\( r \),求圆内接正六边形的边长。

答案二:圆内接正六边形的边长等于半径\( r \)。

这是因为正六边形可以被划分为六个等边三角形,每个等边三角形的边长都是\( r \)。

问题三:如果\( a \)和\( b \)是两个正整数,且\( a^2 - b^2 = 1 \),证明\( a \)和\( b \)中至少有一个是偶数。

普特南数学竞赛试题 美国大学生数学竞赛试题 国际大学生数学竞赛试题

普特南数学竞赛试题 美国大学生数学竞赛试题 国际大学生数学竞赛试题

SEVENTY-FIRST ANNUAL WILLIAM LOWELL PUTNAM MATHEMATICAL COMPETITION Saturday, December 4, 2010 Examination B
B1. Is there an infinite sequence of real numbers a1 , a2 , a3 , . . . such that am + am + am + · · · = m 1 2 3 for every positive integer m? B2. Given that A, B, and C are noncollinear points in the plane with integer coordinates such that the distances AB, AC, and BC are integers, what is the smallest possible value of AB? B3. There are 2010 boxes labeled B1 , B2 , . . . , B2010 , and 2010n balls have been distributed among them, for some positive integer n. You may redistribute the balls by a sequence of moves, each of which consists of choosing an i and moving exactly i balls from box Bi into any one other box. For which values of n is it possible to reach the distribution with exactly n balls in each box, regardless of the initial distribution of balls? B4. Find all pairs of polynomials p(x) and q(x) with real coefficients for which p(x)q(x + 1) − p(x + 1)q(x) = 1.

(参考资料)美国数学大联盟2015年复赛试题

(参考资料)美国数学大联盟2015年复赛试题

2015-2016 Math League Contests, Grades 9 – 12Second-Round, Jan - Feb 2016Instructions:1.This second-round contest consists of two parts. Part 1 is math questions. Part 2 is essay.2.For all the questions below, login to your account at / , and enter youranswers. Answers written on this document will NOT be credited.3.In Part 1, you are asked to read one math subject, The Mathematics of Normal Distributions, and thesupplementary materials if necessary, Descriptive Statistics. Then you have 28 questions to work on.You will need to give precise, unambiguous answers to Questions 1-19 (The Mathematics of Normal Distributions), and Questions 21-26 (Descriptive Statistics).4.Question 20 (The Mathematics of Normal Distributions) and 27-28 (Descriptive Statistics) areProjects and Papers, which means you need to do your research and write a paper for each question.There is no word limit on each of your papers, but it doesn’t necessary mean the more words thebetter. The best paper is precise and succinct. Please don’t feel frustrated at all if you can’t write apaper, as the topics, Confidence Intervals (Question 20); Lies, Damn Lies, and Statistics (Question27); and Data in Your Daily Life (Question 28), are very hard for a high school student, even for anadult. Please don’t feel frustrated even if you can’t finish all of Questions 1-19, 21-26, as they are not trivial questions and it requires a lot reading and thinking. Students who can work out a few questions should be commended.5.For the subject of Normal Distributions, if you really understand what it is and how it works,then the questions are fairly easy to solve. So our recommendation is don’t rush to solve theproblems. Instead please take your time to read and understand the subject thoroughly. Once you understand how Normal Distributions works, you can solve most of the problems without much difficulty. So this is really a test of your research and analytical skill, your patience, and perseverance.6.In order to understand Normal Distributions, you need to be familiar with basic statistics termssuch as mean, median, standard deviation, percentile, quartile, and etc. It is a good idea to read the subject Descriptive Statistics, the supplementary materials, thoroughly to refresh yourmemory.7.The more questions you answered correctly, the more credit you will get. The total credit, or perfectscore, of Part 1 is 180. The total credit, or perfect score, of Part 2, is 90. The problems are ordered by content, NOT DIFFICULTY. It is to your advantage to attempt problems from throughout the test. 8.You can seek help by reading books, searching the Internet, asking an expert, and etc. But you can’tdelegate this to someone else and turn in whatever he/she wrote for you. To make it clear, the purpose of the second-round contest is to test your ability to read and research. You need to be the one who understand the topics and solve the problems. You will be caught if it is not the case during theinterview.9.For Part 1, you can write in either English or Chinese.10.In Part 2, you are asked to write an essay. You have to write in English in Part 2.11.If you have any questions regarding the contest, please contact us at once atINFO@12.This document contains 16 pages in total, including this page.13.Submission of your answers:a)For all the questions below, login to your account at / , and enter youranswers. Answers written on this document will NOT be credited.b)You need to submit your answers no later than 12:00AM, Feb 7, 2016, Beijing Time. Latersubmission will not be accepted.Part 1 – The Mathematics of Normal DistributionThe following is an excerpt from some math book.The Mathematics of Normal Distributions, (see separate document).Understanding the above Chapter on “Normal Distributions” requires familiarity with basic statistics terms such as mean, median, standard deviation, percentile, quartile, and etc. It helps to read the following Chapter on “Descriptive Statistics”, the supplementary materials, if you need to refresh your memory of these terms.Descriptive Statistics, (the supplementary materials, see separate document).For all the questions below, login to your account at / , and enter your answers. Answers written on this document will NOT be credited.Question 1: (credit: 4)Hint:1Q is the first quartile.3Q is the third quartile. For the definition of 1Q and 3Q , see Section 14.3, of Chapter 14, “Descriptive Statistics.”Question 2: (credit: 4)Estimate the value of the standard deviation (rounded to the nearest inch) of a normal distribution with 81.2μ=inch and 394.7Q =inch.Hint:3Q is the third quartile. For the definition of 3Q , see Section 14.3, of Chapter 14, “Descriptive Statistics.”Question 3: (credit: 4)Note:1Q is the first quartile.3Q is the third quartile. For the definition of 1Q and 3Q , see Section 14.3, of Chapter 14, “Descriptive Statistics.”Note: For this question, please write your answer on file “high-school-answersheet.doc ”, downloadable together with this document at , and submit file “high-school-answersheet.doc ” at after you are done.Question 4: (credit: 4)Question 5: (credit: 4)Question 6: (credit: 4): Question 7: (credit: 4)Question 8: (credit: 4)Question 9: (credit: 4)Questions 10 & 11 refer to the following:Note: For the definition of percentile, see Section 14.3, of Chapter 14, “Descriptive Statistics.”Question 10: (credit: 6)Question 11: (credit: 6)Hint:(c) Rounded to the nearest pound.Question 12: (credit: 6)Hint:(b)Rounded to nearest integer.(c)Enter your answer as a decimal between 0 and 1, rounded to the nearest hundredth. Question 13: (credit: 6)Hint:(b)Rounded to nearest thousandth.(c)Rounded to the nearest hundredth.Question 14: (credit: 4)Questions 15-18:In Questions 15-18, you should use the table above to make your estimates.Note: For the definition of percentile, s ee Section 14.3, of Chapter 14, “Descriptive Statistics.”Question 15: (credit: 4)Question 16: (credit: 4)Consider again the distribution of weights of six-month-old baby boys in Question 15.Question 17: (credit: 6)Question 18: (credit: 8)Question 19: (credit: 4)Question 20: (credit: 20, note: paper with exceptional quality can get up to 40 credits)Note: You can write in either Engish or Chinese.Note: For this question, please write your answer on file “high-school-answersheet.doc”, downloadable together with this document at , and submit file “high-school-answersheet.doc” at after you are done.Question 21: (credit: 4)Question 22: (credit: 4)The two histograms below summarize the team payrolls in Major League Baseball (2008).Using the information in the figure, where did the median payroll of 2008 baseball teams fall?(a)Somewhere between $50 million and $80 million(b)Somewhere between $70 million and $80 million(c)Somewhere between $70 million and $100 million(d)Somewhere between $80 million and $100 millionQuestion 23: (credit: 6)This question refers to histograms with unequal class intervals.Question 24: (credit: 8)Note: For (c) and (d), please write your answer on file “high-school-answersheet.doc ”, downloadable together with this document at , and submit file “high-school-answersheet.doc ” at after you are done.Question 25: (credit: 4)Let A denote the mean of data set 12{,,...,}N x x x . Let B denote the mean of dataset 12{,,...,}N x c x c x c +++.(a) Find the relationship between A and B .a) A B <b) A B c =-c) A B c =+d) Nondeterministic(b) Find the mean of 12{,,...,}N x A x A x A ---.Question 26: (credit: 4)Let 1R and 1σdenote the range and standard deviation of data set 12{,,...,}N x x x , respectively. Let 2R and 2σdenote the range and standard deviation of data set 12{,,...,}N x c x c x c +++, respectively. (a) Find the relationship between 1R and 2R .a) Nondeterministicb) 1R = 2Rc) 1R = 2R + cd) 1R = 2R - c(b) Find the relationship between 1σand 2σ.a) Nondeterministicb) 1σ = 2σc) 1σ = 2σ+ cd) 1σ = 2σ- cQuestion 27: (credit: 20, note: paper with exceptional quality can get up to 40 credits)Note: You can write in either Engish or Chinese.Note: For this question, please write your answer on fi le “high -school-answersheet .doc”, downloadable together with this document at , and submit file“high-school-answersheet.doc” at after you are done.Question 28: (credit: 20, note: paper with exceptional quality can get up to 40 credits)Note: You can write in either Engish or Chinese.Note: For this question, please write your answer on fi le “high-school-answersheet.do c”, downloadable together with this document at , and submit file “high-school-answersheet.doc” at after you are done.Part 2 Essay (Credit: 90)Pros and Cons of Math Competitionsby Richard RusczykMathematics competitions such as MATH LEAGUE, MATHCOUNTS, and the American Mathematics Competitions are probably the extracurricular math programs with the widest participation. The most immediate value of these math contests is obvious –they pique students’ interest in mathematics and encourage them to value intellectual pursuits. Kids love games, and many will turn just about any activity into a contest, or in other words, something to get good at. Math contests thus inspire them to become good at mathematics just like sports encourage physical fitness. Eventually, students put aside the games. By then, hopefully an interest in the underlying activity has developed.Beyond encouraging an interest in mathematics, contests help prepare students for competition. For better or worse, much of life is competition, be it for jobs or resources or whatever. Competition of any sort trains students to deal with success and failure, and teaches them that effective performance requires practice. Moreover, nearly every interesting and worthwhile venture in life comes with some element of pressure; competition teaches students how to handle it.Despite all the benefits of math contests, they are not an unmitigated good. First of all, not all contests are designed well. Students shouldn’t take too seriously contests that greatly emphasize speed or memorization. Curricular contests (particularly calculus contests for high school students) can also be misleading, as they deepen the misconception that there is no more to math than what is in the classroom. Such contests run the risk of encouraging students to overvalue skills that aren’t nearly as valuable as the one asset a contest should help them develop – the ability to think about and solve complex problems.A second danger of contests is extending kids beyond their ability. Students should certainly be challenged with problems they can’t do from time to time, but if this happens consistently, the experien ce goes from humbling and challenging to humiliating and discouraging.A third potential pitfall, burnout, often comes on the heels of the first two. Participants in math contests are just as much at risk of burnout as musicians or athletes. Parents, teachers, and the students themselves should be on the lookout for signs of decreased interest, and they must be willing to back off and allow the student to rediscover an interest in mathematics on his or her own. Burnout is particularly pernicious because th e end result often isn’t a backlash against competition, but against math in general. Indeed, even students not involved in contests have to watch out for burnout, though the pressure of contests tends to encourage burnout more quickly than the classroom.These possible perils are usually more than offset not only by the values we’ve already mentioned, but also by the greatest asset of math contests - cooperation. These competitions bring together students of like interests and abilities, allowing them to form their own community in which they will find friendship, inspiration, and encouragement to a far greater degree than most of these students can find in the typical classroom. Whereas a student may be one of only three or four in her school who pursues math the way others play basketball, she will not find herself so lonely at a math contest, where she’ll find many kindred spirits.In summary, math contests are a tremendous social and intellectual opportunity for students, but exposingstudents to contests must be done wisely, else they become counterproductive to the goal of encouraging a lifelong interest in mathematics and other intellectual pursuits.Direction (Note: For this question, please write your answer on fi le “high-school-answersheet.doc”, downloadable together with this document at , and submit file“high-school-answersheet.doc” at after you are done.):In his article “Pros and Cons of Math Competitions”, Richard Rusczyk lists and explains the advantages and the disadvantages of math competitions. Are math contests beneficial or harmful to students? Everything always has two sides: the good and the bad. This proverb probably also applies to math competitions. Write a response in which you discuss whether students should be encouraged to participate in math competitions. You may draw examples from your reading, studies, experience, observations, and etc.Hint: Here are some questions to think about. You do not have to answer them, but they will help you to craft your response.1. What is the purpose of math contests?2. What are the good and the bad sides of math contests?3. How should you value your scores on math contests?4. What are the most important qualities that a successful mathematician should have?5. To a student who is highly interested in mathematics, what are much more important than math contests?。

数学思维(高中):2015-2016年度美国“数学大联盟”思维探索十至十二年级试卷(含参考答案)

数学思维(高中):2015-2016年度美国“数学大联盟”思维探索十至十二年级试卷(含参考答案)

2015-2016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:2015年11月14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一、选择题(每小题10分,答对加10分,答错不扣分,共100分,请将正确答案A、B、C或者D写在每题后面的圆括号内。

)正确答案填写示例如下:20 − 5 × 2 = 2 ×? ( A )A) 5 B) 15 C) 25 D) 301.If a square has the same area as a circle whose radius is 10, then the side-length of thesquare is ( )A) B) 10πC) D) 100π2.x2–y2 + x + y = ( )A) (x + y– 1)(x–y) B) (x + y)(x–y– 1)C) (x + y + 1)(x–y) D) (x + y)(x–y + 1)3.If x + y = 25 and x2–y2 = 50. What is the value of xy? ( )A) 150.25 B) 155.25 C) 175 D) 12504.Janet picked a number from 1 to 10 and rolled a die. What is the probability that the sumof the number she picked and the outcome on the die is an even number? ( )A) 1/5 B) 1/4 C) 1/3 D) 1/25.Let r be a solution of x2– 7x + 11 = 0. What is the value of (r– 3)(r– 4) + (r– 12)(r + 5)?( )A) -71 B) -70 C) -69 D) 70st month the ratio of males to females in Miss Fox’s company was 3:4. When 9 newmales and 52 new females were employed this month, the new ratio of males to females is now 1/2. How many employees are there now in the company total? ( )A) 68 B) 120 C) 180D) 240第1页,共4页his task, he returned 40 mph from the castle to home. What is his average speed, in mph, of his quest? ( )A) 120/7 B) 240/7 C) 35 D) 70to shoot 3 apples, then when I use up the darts, I will be left with 35apples; if each dart is used to shoot 4 apples, then when I use up the apples,I will be left with 5 darts. I have ? apples at the beginning. ( )A) 51 B) 55 C) 200 D) 2409.x/2 = y/3 = z/4, what is the value of x:y:z? ( )A) 6:4:3 B) 3:4:6 C) 2:3:4 D) 4:3:210.Super Jack and Almighty Jill were doing the 100-mile walk at the same time and samestarting point, at constant speeds. Jack took a 5-minute break at the end of every 10 miles;Jill took a 10-minute break at the end of each 20 miles. Jill’s speed was 5/8 of that of Jack.They finished at the same time. How long, in minutes, does the trip take? ( )A) 53.333 B) 56.667 C) 60.333 D) 60.667二、填空题(每小题10分,答对加10分,答错不扣分,共200分。

2015 AMC 12A 考题及答案

2015 AMC 12A 考题及答案

2015AMC12A考题及答案Problem1What is the value ofProblem2Two of the three sides of a triangle are20and15.Which of the following numbers is not a possible perimeter of the triangle?Problem3Mr.Patrick teaches math to15students.He was grading tests and found that when he graded everyone's test except Payton's,the average grade for the class was80.After he graded Payton's test,the class average became 81.What was Payton's score on the test?Problem4The sum of two positive numbers is5times their difference.What is the ratio of the larger number to the smaller?Problem5Amelia needs to estimate the quantity,where and arelarge positive integers.She rounds each of the integers so that the calculation will be easier to do mentally.In which of these situations will her answer necessarily be greater than the exact value of?Problem6Two years ago Pete was three times as old as his cousin Claire.Two years before that,Pete was four times as old as Claire.In how many years will the ratio of their ages be?Problem7Two right circular cylinders have the same volume.The radius of the second cylinder is more than the radius of the first.What is therelationship between the heights of the two cylinders?Problem8The ratio of the length to the width of a rectangle is:.If therectangle has diagonal of length,then the area may be expressed asfor some constant.What is?Problem9A box contains2red marbles,2green marbles,and2yellow marbles. Carol takes2marbles from the box at random;then Claudia takes2of the remaining marbles at random;and then Cheryl takes the last2marbles. What is the probability that Cheryl gets2marbles of the same color?Problem10Integers and with satisfy.What is?Problem11On a sheet of paper,Isabella draws a circle of radius,a circle of radius ,and all possible lines simultaneously tangent to both circles.Isabella notices that she has drawn exactly lines.How many differentvalues of are possible?Problem12The parabolas and intersect the coordinate axes in exactly four points,and these four points are the vertices of a kite of area.What is?Problem13A league with12teams holds a round-robin tournament,with each team playing every other team exactly once.Games either end with one team victorious or else end in a draw.A team scores2points for every game it wins and1point for every game it draws.Which of the following is NOT a true statement about the list of12scores?Problem14What is the value of for which?Problem15What is the minimum number of digits to the right of the decimal pointneeded to express the fraction as a decimal?Problem16Tetrahedron hasand.What is the volume of the tetrahedron?Problem17Eight people are sitting around a circular table,each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated.What is the probability that no two adjacent people will stand?Problem18The zeros of the function are integers.What is the sum of the possible values of?Problem19For some positive integers,there is a quadrilateral withpositive integer side lengths,perimeter,right angles at and,,and.How many different values of arepossible?Problem20Isosceles triangles and are not congruent but have the same areaand the same perimeter.The sides of have lengths of and,while those of have lengths of and.Which of the followingnumbers is closest to?Problem21A circle of radius passes through both foci of,and exactly four pointson,the ellipse with equation.The set of all possible values of is an interval.What is?Problem22For each positive integer,let be the number of sequences of length consisting solely of the letters and,with no more thanthree s in a row and no more than three s in a row.What is theremainder when is divided by12?Problem23Let be a square of side length1.Two points are chosen independentlyat random on the sides of.The probability that the straight-linedistance between the points is at least is,where andare positive integers and.What is?Problem24Rational numbers and are chosen at random among all rationalnumbers in the interval that can be written as fractions whereand are integers with.What is the probabilitythat is a real number?Problem25A collection of circles in the upper half-plane,all tangent to the-axis,is constructed in layers as yer consists of two circles of radiiand that are externally tangent.For,the circles inare ordered according to their points of tangency with the-axis.For every pair of consecutive circles in this order,a new circle is constructed externally tangent to each of the two circles in the yerconsists of the circles constructed in this way.Let,andfor every circle denote by its radius.What is2015AMC12A Answer Key1.C2.E3.E4.B5.D6.B7.D8.C9.C10.E11.D12.B13.E14.D15.C16.C17.A18.C19.B20.A21.D22.D23.A24.D25.D。

1940-1959普特南大学数学竞赛试题

1940-1959普特南大学数学竞赛试题

A-2. 有一个浮标由三部分组成 一个圆筒与两个相等圆锥 其中每个圆锥的高 等于圆筒的高 问当表面积一定时 什么样的形状会有最大的体积 A-3. 如果一个质点在平面内运动 其坐标可表为时间 t 的函数 x=t3-t y=t4+t 证明曲线在 t=0 处有一个拐点 并且质点运动的速度在 t=0 处有一个极大 值 A-4. 伐木工砍一棵树 树干是圆柱形 粗细均匀 他先砍出一道 V 形槽 槽的 两边是平面 两面的交线是通过圆柱的轴的一条水平线 其二面角为θ 如果给定θ 证明平分θ的平面是水平面时 所砍去的材料的体积最小 n2 n →∞ e n 1 x (1 + sin 2t ) t dt 0 x→ 0 x ∫
B-7. (i) 设 u = 1 +
v=
w=

明 u3+v3+w3-3uvw=1 (ii) 设中心锥 (ax2+by2)+2(px+qy)+c=0 (ax2+by2)+2λ(px+qy)+λ2c=0 λ 是已 知的正的常数 证明 如果从原点到第一个锥的所有射线段按λ比 1 改变 新的射线段的端点生成第二个锥 q 2λ p 2λ 设 P 点坐标为 − b 1+ λ a 1+ λ 线段按λ比 1 呈反向改变 情形 证明 如果从 P 到第一个锥的所有射 注意λ=1 的
2 v0 − 2 gh (忽略大气阻力)
A-7. (i) 求与曲线族(y-k2)2=x2(k2-x2)的所有曲线切触的曲线 两条曲线的草图 (ii) 如果函数
作这条曲线及族中
1 展开为 x 的幂级数 c0+c1x+c2x2+ (1 − ax)(1 − bx)
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A2 Let a0 = 1, a1 = 2, and an = 4an−1 − an−2 for n ≥ 2. Find an odd prime factor of a2015.
A3 Compute
2015 2015
∏ ∏ log2
(1 + e2πiab/2015)
a=1 b=1
Here i is the imaginary unit (that is, i2 = −1).
B3 Let S be the set of all 2 × 2 real matrices
M=
ab cd
whose entries a, b, c, d (in that order) form an arithmetic progression. Find all matrices M in S for which there is some integer k > 1 such that Mk is also in S.
B2 Given a list of the positive integers 1, 2, 3, 4, . . . , take the first three numbers 1, 2, 3 and their sum 6 and cross all
four numbers off the list. Repeat with the three smallest remaining numbers 4, 5, 7 and their sum 16. Continue in this way, crossing off the three smallest remaining numbers and their sum, and consider the sequence of sums produced: 6, 16, 27, 36, . . . . Prove or disprove that there is some number in the sequence whose base 10 representation ends with 2015.
as a rational number in lowest terms.
B5 Let Pn be the number of permutations π of {1, 2, . . . , n} such that
|i − j| = 1 implies |π(i) − π( j)| ≤ 2
for all i, j in {1, 2, . . . , n}. Show that for n ≥ 2, the quantity
A5 Let q be an odd positive integer, and let Nq denote the number of integers a such that 0 < a < q/4 and gcd(a, q) = 1. Show that Nq is odd if and only if q is of the form pk with k a positive integer and p a prime
The 76th William Lowell Putnam Mathematical Competition Saturday, December 5, 2015
A1 Let A and B be points on the same branch of the hyperbola xy = 1. Suppose that P is a point lying between A and B on this hyperbola, such that the area of the triangle APB is as large as possible. Show that the region bounded by the hyperbola and the chord AP has the same area as the region bounded by the hyperbola and the chord PB.
B1 Let f be a three times differentiable function (defined on R and real-valued) such that f has at least five distinct real zeros. Prove that f + 6 f + 12 f + 8 f has at least two distinct real zeros.
Pn+5 − Pn+4 − Pn+3 + Pn does not depend on n, and find its value. B6 For each positive integer k, let A(k√) be the number of odd divisors of k in the interval [1, 2k). Evaluate
∑∞ (−1)k−1 A(to 5 or 7 modulo 8.
A6 Let n be a positive integer. Suppose that A, B, and M are n×n matrices with real entries such that AM = MB, and such that A and B have the same characteristic polynomial. Prove that det(A − MX) = det(B − XM) for every n × n matrix X with real entries.
B4 Let T be the set of all triples (a, b, c) of positive integers for which there exist triangles with side lengths a, b, c. Express
2a
∑ (a,b,c)∈T 3b5c
A4 For each real number x, let
1
∑ f (x) = n∈Sx 2n ,
where Sx is the set of positive integers n for which nx is even. What is the largest real number L such that f (x) ≥ L for all x ∈ [0, 1)? (As usual, z denotes the greatest integer less than or equal to z.)
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