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
美国数学竞赛试题-部分
2000AMC12ProblemsProblem1In the year,the United States will host the International Mathematical Olympiad.Let and be distinct positive integers such that the product.What is the largest possible value of the sum?Problem2Problem3Each day,Jenny ate of the jellybeans that were in her jar at the beginning of that day. At the end of the second day,remained.How many jellybeans were in the jar originally?Problem4The Fibonacci sequence starts with two1s,and each term afterwards is the sum of its two predecessors.Which one of the ten digits is the last to appear in the units position of a number in the Fibonacci sequence?Problem5If where thenProblem6Two different prime numbers between and are chosen.When their sum is subtracted from their product,which of the following numbers could be obtained?Problem7How many positive integers have the property that is a positive integer?Problem8Figures,,,and consist of,,,and non-overlapping squares.If the pattern continued,how many non-overlapping squares would there be in figure?Problem9Mrs.Walter gave an exam in a mathematics class of five students.She entered the scores in random order into a spreadsheet,which recalculated the class average after each score was entered.Mrs. Walter noticed that after each score was entered,the average was always an integer.The scores (listed in ascending order)were71,76,80,82,and91.What was the last score Mrs.Walters entered?Problem10The point is reflected in the-plane,then its image is rotatedby about the-axis to produce,and finally,is translated by5units in the positive-direction to produce.What are the coordinates of?Problem11Two non-zero real numbers,and satisfy.Which of the following is a possible value of?Problem12Let A,M,and C be nonnegative integers such that.What is the maximum value of+++?Problem13One morning each member of Angela’s family drank an8-ounce mixture of coffee with milk.The amounts of coffee and milk varied from cup to cup,but were never zero.Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee.How many people are in the family?Problem14When the mean,median,and modeof the listare arranged in increasing order,they form a non-constant arithmetic progression.What is the sum of all possible real values of?Problem15Let be a function for which.Find the sum of all values of for which.Problem16A checkerboard of rows and columns has a number written in each square,beginning in the upper left corner,so that the first row is numbered,the second row, and so on down the board.If the board is renumbered so that the left column,top to bottom, is,the second column and so on across the board,some squares have the same numbers in both numbering systems.Find the sum of the numbers in these squares (under either system).Problem17A centered at has radius and contains the point.The segment is tangent to the circle at and.If point lies on and bisects,thenProblem18In year,the day of the year is a Tuesday.In year,the day is also a Tuesday.On what day of the week did th day of year occur?Problem19triangle,,,.Let denote the midpointof and let denote the intersection of with the bisector of angle.Which of the following is closest to the area of the triangle?Problem20If and are positive numbers satisfyingThen what is the value of latex?Problem21Through a point on the hypotenuse of right triangle,lines are drawn parallel to the legs of the triangle so that the triangle is divided into asquare and two smaller right triangles.The area of one of the two small right triangles times the area of the square.The ratio of the area of the other small right triangle to the area of the square isProblem22The graph below shows a portion of the curve defined by the quarticpolynomial.Which of the following is the smallest?Problem23Professor Gamble buys a lottery ticket,which requires that he pick six different integers from through,inclusive.He chooses his numbers so that the sum of the base-ten logarithms of his six numbers is an integer.It so happens that the integers on the winning ticket have the same property—the sum of the base-ten logarithms is an integer.What is the probability that Professor Gamble holds the winning ticket?Problem24If circular arcs and centers at and,respectively,then there exists a circletangent to both and,and to.If the length of is,then the circumference of the circle isProblem25Eight congruent Equilateral triangle each of a different color,are used to construct a regular octahedron.How many distinguishable ways are there to construct the octahedron?(Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.)Problem6For how many positive integers does there exist at least one positive integer n such that?infinitely manyProblem7A arc of circle A is equal in length to a arc of circle B.What is the ratio of circle A's area and circle B's area?Problem8Betsy designed a flag using blue triangles,small white squares,and a red center square,as shown. Let be the total area of the blue triangles,the total area of the white squares,and the area of the red square.Which of the following is correct?Problem9Jamal wants to save30files onto disks,each with1.44MB space.3of the files take up0.8MB, 12of the files take up0.7MB,and the rest take up0.4MB.It is not possible to split a file onto2different disks.What is the smallest number of disks needed to store all30files?Problem10Sarah places four ounces of coffee into an eight-ounce cup and four ounces of cream into a second cup of the same size.She then pours half the coffee from the first cup to the second and,after stirring thoroughly,pours half the liquid in the second cup back to the first.What fraction of the liquid in the first cup is now cream?Problem11Mr.Earl E.Bird gets up every day at8:00AM to go to work.If he drives at an average speed of40miles per hour,he will be late by3minutes.If he drives at an average speed of60milesper hour,he will be early by3minutes.How many miles per hour does Mr.Bird need to drive to get to work exactly on time?Problem12Both roots of the quadratic equation are prime numbers.The number of possible values of isProblem13Two different positive numbers and each differ from their reciprocals by.What is?Problem14For all positive integers,let.Let. Which of the following relations is true?Problem15The mean,median,unique mode,and range of a collection of eight integers are all equal to8. The largest integer that can be an element of this collection isProblem16Tina randomly selects two distinct numbers from the set{1,2,3,4,5},and Sergio randomly selects a number from the set{1,2,...,10}.What is the probability that Sergio's number is larger than the sum of the two numbers chosen by Tina?Problem17Several sets of prime numbers,such as use each of the nine nonzero digits exactly once.What is the smallest possible sum such a set of primes could have?Problem18Let and be circles definedby and respectively.What is the length of the shortest line segment that is tangent to at and to at?Problem19The graph of the function is shown below.How many solutions does theequation have?Problem20Suppose that and are digits,not both nine and not both zero,and the repeatingdecimal is expressed as a fraction in lowest terms.How many different denominators are possible?Problem21Consider the sequence of numbers:For,the-th term of the sequence is the units digit of the sum of the two previous terms.Let denote the sum of the first terms of this sequence.The smallest value of for which is:Problem22Triangle is a right triangle with as its right angle,, and.Let be randomly chosen inside,and extend to meet at. What is the probability that?Problem23In triangle,side and the perpendicular bisector of meet in point,and bisects.If and,what is the area of triangle?SAT II数学词汇表代数部分1.基础add，plus加subtract减difference差multiply times乘product积divide除divisible可被整除的divided evenly被整除dividend被除数divisor因子，除数quotient商remainder余数factorial阶乘power乘方radical sign,root sign根号round to四舍五入to the nearest四舍五入2.有关集合union并集proper subset真子集solution set解集3.有关代数式、方程和不等式algebraic term代数项like terms,similar terms同类项numerical coefficient数字系数literal coefficient字母系数inequality不等式triangle inequality三角不等式range值域original equation原方程equivalent equation同解方程等价方程linear equation线性方程(e.g.5x+6=22)4.有关分数和小数proper fraction真分数improper fraction假分数mixed number带分数vulgar fraction，common fraction普通分数simple fraction简分数complex fraction繁分数numerator分子denominator分母(least)common denominator(最小)公分母quarter四分之一decimal fraction纯小数infinite decimal无穷小数recurring decimal循环小数tenths unit十分位5.基本数学概念arithmetic mean算术平均值weighted average加权平均值geometric mean几何平均数exponent指数，幂base乘幂的底数,底边cube立方数，立方体square root平方根cube root立方根common logarithm常用对数digit数字constant常数variable变量inverse function反函数complementary function余函数linear一次的，线性的factorization因式分解absolute value绝对值，e.g.｜-32｜=32round off四舍五入6.有关数论natural number自然数positive number正数negative number负数odd integer奇整数,odd number奇数even integer,even number偶数integer,whole number整数positive whole number正整数negative whole number负整数consecutive number连续整数real number,rational number实数,有理数irrational(number)无理数inverse倒数composite number合数e.g.4,6,8,9,10,12,14,15……reciprocal 倒数common divisor公约数multiple倍数(least)common multiple(最小)公倍数(prime)factor(质)因子common factor公因子prime number质数e.g.2,3,5,7,11,13,15……ordinary scale,decimal scale十进制nonnegative非负的tens十位units个位mode众数median中数common ratio公比7.数列arithmetic progression(sequence)等差数列geometric progression(sequence)等比数列8.其它approximate近似(anti)clockwise(逆)顺时针方向cardinal基数ordinal序数direct proportion正比distinct不同的estimation估计，近似parentheses括号proportion比例permutation排列combination组合table表格trigonometric function三角函数unit单位,位几何部分1.所有的角alternate angle内错角corresponding angle同位角vertical angle对顶角central angle圆心角interior angle内角exterior angle外角supplementary angles补角complementary angle余角adjacent angle邻角acute angle锐角obtuse angle钝角right angle直角round angle周角straight angle平角included angle夹角2.所有的三角形equilateral triangle等边三角形scalene triangle不等边三角形isosceles triangle等腰三角形right triangle直角三角形oblique斜三角形inscribed triangle内接三角形3.有关收敛的平面图形，除三角形外semicircle半圆concentric circles同心圆quadrilateral四边形pentagon五边形hexagon六边形heptagon七边形octagon八边形nonagon九边形decagon十边形polygon多边形parallelogram平行四边形equilateral等边形plane平面square正方形，平方rectangle长方形regular polygon正多边形rhombus菱形trapezoid梯形4.其它平面图形arc弧line,straight line直线line segment线段parallel lines平行线segment of a circle弧形5.有关立体图形cube立方体，立方数rectangular solid长方体regular solid/regular polyhedron正多面体circular cylinder圆柱体cone圆锥sphere球体solid立体的6.有关图形上的附属物altitude高depth深度side边长circumference,perimeter周长radian弧度surface area表面积volume体积arm直角三角形的股cross section横截面center of acircle圆心chord弦radius半径angle bisector角平分线diagonal对角线diameter直径edge棱face of a solid立体的面hypotenuse斜边included side夹边leg三角形的直角边median of a triangle三角形的中线base底边，底数(e.g.2的5次方，2就是底数) opposite直角三角形中的对边midpoint中点endpoint端点vertex(复数形式vertices)顶点tangent切线的transversal截线intercept截距7.有关坐标coordinate system坐标系rectangular coordinate直角坐标系origin原点abscissa横坐标ordinate纵坐标Number line数轴quadrant象限slope斜率complex plane复平面8.其它plane geometry平面几何trigonometry三角学bisect平分circumscribe外切inscribe内切intersect相交perpendicular垂直Pythagorean theorem勾股定理congruent全等的multilateral多边的其它相关词汇cent美分penny一美分硬币nickel5美分硬币dime一角硬币dozen打(12个)score廿(20个)Centigrade摄氏Fahrenheit华氏quart夸脱gallon加仑(1gallon=4quart)yard码meter米micron微米inch英寸foot英尺minute分(角度的度量单位，60分=1度) square measure平方单位制cubic meter立方米pint品脱(干量或液量的单位)。
普特兰数学竞赛题
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
第68美国大学生数学竞赛
第68美国大学生数学竞赛A-1 求出所有使曲线 y =a 2x ++124和曲线x =a 2y +a y +124相切的a 的值.解 仅有a 是23，32，（13±）/12.解法 设1c 和2c分别是曲线y =a 2x ++124和x =a 2y +a y +124，并且设l 是直线y=x .我们考虑三种情况. i 如果1c和l 相切，则切点（x ，x ）满足2ax +a =1，x =a 2y +a y +124，由对称性，2c也和l 在那相切，因此1c 和2c相切.在上面的第一个方程写a =1(21)x +并带入第二个方程我们有x =221x xx +++124它可简化为 0=242x-2x -1=（61x +）-（41x -）或{11,46x ∈-这就得出1(21)a x =+}{32,32∈ ii 如果1c 不和l 相交，那么1c 和2c被l 分开，因此它们不可能相切.Iii 如果1c 和l 交于两个不同的点12,p p ,那么它不可能和l 在其他点相切.假设在这两点之一,比如说11,p c 的切线是垂直于l 的,则由对称性,2c 也是这样，一次12,c c 将在1p 处相切.在这种情况,点1(,)p x x =满足2121,24ax a x ax ax +=-=++在上面的第一个方程写1(21)a x -=+并代入第二个方程,我们有 x =-221x xx +++124或(23/72x =-,这导致1(13/12(21)a x -==+如果1c 在12,p p 的切线不与l 垂直，那么我们断言1c 和2c 不可能有任何切点。
确实，如果我们去数1c 和2c 的交点个数（通过1c 把带入到2c 中的y ，然后对y 求解），我们算上重数至多得出4个解。
其中两个是1p 和2p ，而任意切点就是多出来的两个。
然而除了和l 的交点外，任意切点都有一个镜像，它也是切点。
但是我们不可能有6个解，这样，我们就求出了所有可能的a 。
美赛历年题目_pdf
马剑整理历年美国大学生数学建模赛题目录MCM85问题-A 动物群体的管理 (3)MCM85问题-B 战购物资储备的管理 (3)MCM86问题-A 水道测量数据 (4)MCM86问题-B 应急设施的位置 (4)MCM87问题-A 盐的存贮 (5)MCM87问题-B 停车场 (5)MCM88问题-A 确定毒品走私船的位置 (5)MCM88问题-B 两辆铁路平板车的装货问题 (6)MCM89问题-A 蠓的分类 (6)MCM89问题-B 飞机排队 (6)MCM90-A 药物在脑内的分布 (6)MCM90问题-B 扫雪问题 (7)MCM91问题-B 通讯网络的极小生成树 (7)MCM 91问题-A 估计水塔的水流量 (7)MCM92问题-A 空中交通控制雷达的功率问题 (7)MCM 92问题-B 应急电力修复系统的修复计划 (7)MCM93问题-A 加速餐厅剩菜堆肥的生成 (8)MCM93问题-B 倒煤台的操作方案 (8)MCM94问题-A 住宅的保温 (9)MCM 94问题-B 计算机网络的最短传输时间 (9)MCM-95问题-A 单一螺旋线 (10)MCM95题-B A1uacha Balaclava学院 (10)MCM96问题-A 噪音场中潜艇的探测 (11)MCM96问题-B 竞赛评判问题 (11)MCM97问题-A Velociraptor(疾走龙属)问题 (11)MCM97问题-B为取得富有成果的讨论怎样搭配与会成员 (12)MCM98问题-A 磁共振成像扫描仪 (12)MCM98问题-B 成绩给分的通胀 (13)MCM99问题-A 大碰撞 (13)MCM99问题-B “非法”聚会 (14)MCM2000问题-A空间交通管制 (14)MCM2000问题-B: 无线电信道分配 (14)MCM2001问题- A: 选择自行车车轮 (15)MCM2001问题-B 逃避飓风怒吼（一场恶风...） .. (15)MCM2001问题-C我们的水系-不确定的前景 (16)MCM2002问题-A风和喷水池 (16)MCM2002问题-B航空公司超员订票 (16)MCM2002问题-C (16)MCM2003问题-A: 特技演员 (18)MCM2003问题-B: Gamma刀治疗方案 (18)MCM2003问题-C航空行李的扫描对策 (19)MCM2004问题-A：指纹是独一无二的吗？ (19)MCM2004问题-B：更快的快通系统 (19)MCM2004问题-C安全与否？ (19)MCM2005问题A.水灾计划 (19)MCM2005B.Tollbooths (19)MCM2005问题C：不可再生的资源 (20)MCM2006问题A: 用于灌溉的自动洒水器的安置和移动调度 (20)MCM2006问题B: 通过机场的轮椅 (20)MCM2006问题C : 抗击艾滋病的协调 (21)MCM2007问题B ：飞机就座问题 (24)MCM2007问题C：器官移植：肾交换问题 (24)MCM2008问题A:给大陆洗个澡 (28)MCM2008问题B：建立数独拼图游戏 (28)MCM85问题-A 动物群体的管理在一个资源有限，即有限的食物、空间、水等等的环境里发现天然存在的动物群体。
美国imo数学竞赛试题及答案
美国imo数学竞赛试题及答案问题1：代数问题设\( a, b, c \) 是正实数，满足 \( a + b + c = 1 \)。
证明：\[ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \geq 9 \]问题2：几何问题在三角形 \( ABC \) 中，点 \( D \) 和 \( E \) 分别是边 \( BC \) 和 \( AC \) 上的点，使得 \( AD \) 平行于 \( BE \)。
如果\( \angle A = 60^\circ \)，证明 \( \angle ADB = \angle BEC \)。
问题3：数论问题给定一个正整数 \( n \)，证明对于所有 \( n \) 的倍数 \( k \)，\( k \) 除以 \( n \) 的余数等于 \( k \) 除以 \( n+1 \) 的余数。
问题4：组合问题有 \( 2n \) 个不同的球和 \( n \) 个相同的盒子。
证明至少有一个盒子包含至少 \( 3 \) 个球。
问题5：不等式问题证明对于所有正实数 \( x \) 和 \( y \)，以下不等式成立：\[ \sqrt{x^2 + y^2} + \sqrt{2xy} \geq x + y \]答案问题1：代数问题由柯西不等式，我们知道：\[ (a + b + c)\left(\frac{1}{a} + \frac{1}{b} +\frac{1}{c}\right) \geq (1 + 1 + 1)^2 \]因为 \( a + b + c = 1 \)，所以：\[ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \geq 9 \]问题2：几何问题由于 \( AD \) 平行于 \( BE \)，根据相似三角形的性质，我们有\( \triangle ABD \sim \triangle CBE \)。
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。
第63届美国大学生数学竞赛
第63届美国大学生数学竞赛A-1 令k 是一个固定的正整数，)1(1-k x 的n 阶导数有形式1)1()(+-n k n x x P ，其中)(x P n 是一个多项式。
求)1(n P 。
解 求导数给出)()1()(')1()(11x P kx n x P x x P n k n k n -++--= 以致 )1()1()1(1n n P n k P +-=+ 因为1)1(0=P ，因而有归纳法得 !)()1(n k P n n -=A-2 在一个球面上任意给定5个点，证明：其中必有4个点位于一个闭的半球面上。
解 考虑通过给定五个点中的任意一对点的一个大圆，则球面被此大圆分成两个闭半球面，它们在此大圆上重叠，剩下来的三个点中至少有两个点必定位于这两个半径面中的一个上。
这个半球面至少包含了五个点中的四个点。
A-3 令2≥n 是一个整数，n T 是},,3,2,1{n 的某些非空子集S 的个数，这种子集的元素的平均值是整数。
证明：n T n -总是偶数。
解 如果a 是S 中元素的平均值，则a n -+1即是集合}|1{*S s s n S ∈-+=元素的平均值。
这样，n T 等于集合对},{*S S （其中*S S ≠）的数目与n U 之和，n U 是满足条件*S S =的非空集合S 的数目，因而，2mod n n U T =。
这样，如果n U S ∈并且S s ∈，则S a n ∈-+1，以致S 元素的平均值是2)1(+n 。
因而如果n 是偶的，则0=n U ，即得结论。
如果n 是奇数，则S 是在}2)1({},2)3(,2)1({,},2,3{},1,2{},,1{++---n n n n n n的一个非空子集中元素的并。
这样 2(mod)1122)1(=-=+n n U 亦可得结论。
A-4 在行列式游戏中，游戏者1在一个空的33⨯矩阵中填入一个1.游戏者0在某个位置填入一个0，游戏这样继续下去，直到33⨯矩阵被填入5个1和4个0,。
美国普特南数学竞赛试题2015(题目)
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.
普特南数学竞赛试题解答
f (x) − f (x + 1) dx = f (x) =
b−1 0
1
k =a
f (x + k ) − f (x + k + 1) dx f (x + k ) f (x + k ) − f (x + k + 1) dx f (x + k ) f (x + k ) − f (x + k + 1) dx f (x + a)
Define the function g : R → R by g (x) = f (x + 1) − f (x), and put c = g (0), d = f (0). For all x ∈ R, g ′ (x) = f ′ (x + 1) − f ′ (x) = 0, so g (x) = c identically, and f ′ (x) = f (x + 1) − f (x) = g (x) = c, so f (x) = cx + d identically as desired. A3 If a = b = 0, then the desired result holds trivially, so we assume that at least one of a, b is nonzero. Pick any point (a0 , b0 ) ∈ R2 , and let L be the line given by the parametric equation L(t) = (a0 , b0 ) + (a, b)t for t ∈ R. By the chain rule and the given equation, we d have dt (h ◦ L) = h ◦ L. If we write f = h ◦ L : R → R, then f ′ (t) = f (t) for all t. It follows that f (t) = Cet for some constant C . Since |f (t)| ≤ M for all t, we must have C = 0. It follows that h(a0 , b0 ) = 0; since (a0 , b0 ) was an arbitrary point, h is identically 0 over all of R2 . A4 Put N = 1010
国际数学竞赛题02
国际数学竞赛题021. 对任何自然数 n ，试计算下式的值[n+12]+[n+24]+⋯+[n+2k2k+1]+... 其中[x]表示不超过 x 的最大整数.2. 对任意正整数 n ，求证有无穷多个正整数 m 使得m + n 4 不是质数.3. 令 f(x) = cos(a 1 + x) + 1/2 cos(a 2 + x) + 1/4 cos(a 3 + x) + ... + 1/2n-1 cos(a n + x), 其中 a i 是实数常量，x 是实数变量. 现已知 f(x 1) = f(x 2) = 0，求证 x 1 - x 2 是 π 的整数倍.4. 对每一个k = 1, 2, 3, 4, 5，试找出 a>0 应满足的充要条件使得存在一个四面体，其中 k 个边长均为 a ，其余 6-k 个边的长度均为 1.5. 以AB 为直径的半圆弧，C 是其上不同于A 、B 的一点，D 是C 向AB 作垂线的垂足. K 1 是三角形ABC 的内切圆， 圆K 2 与CD 、DA 以及半圆都相切，圆K 3 与CD 、DB 及半圆相切. 求证：圆K 1、 K 2 、 K 3 除AB 外还有一条公切线.6. 平面上已给定了 n>4个点，无三点共线. 求证至少有 (n-3)(n-4)/2 个凸四边形，其顶点都是已给点集中的点.7. 给定实数x 1, x 2, y 1, y 2, z 1, z 2, 满足 x 1 > 0, x 2 > 0, x 1y 1 > z 12, x 2y 2 > z 22，求证： 8≤ 1 + 1(x 1 + x 2)(y 1 + y 2) - (z 1 + z 2)2x 1y 1 - z 12 x 2y 2 - z并给出等号成立的充分必要条件.8. M 是三角形ABC 的边AB 上的任何一点，r 、r 1、r 2 分别是三角形ABC 、AMC 、BMC 的内切圆的半径，q 是AB 外旁切圆的半径（即与AB 边相切，与CA 、CB 的延长线上相切的圆），类似的， q 1、q 2分别是AC 、BC 外旁切圆的圆心. 求证： r 1r 2q = rq 1q 2.9.已知0 ≤ x i < b，i = 0, 1, ... ,n 并且 x n > 0, x n-1 > 0. 如果 a>b，x n x n-1 (x0)是数A在a进制下的表示、也是B在b进制下的表示，则 x n-1x n-2...x0表示了 A'在a 进制下的表示、B'在b进制下的表示. 求证：A'B<AB'.10.实数 a0, a1, a2, ...满足 1 = a0 <= a1 <= a2 <= ...，并定义b n=∑(1 - a k-1/a k)/√a k其中求和是k从1到n.a.求证0≤ b n<2；b.设c满足0≤c <2，求证可找到a n使得当n足够大时b n >c成立.11.试找出所有的正整数 n 使得集合 {n, n+1, n+2, n+3, n+4, n+5} 可被分拆成两个子集合，每个子集合的元素的乘积相等.12.四面体ABCD，角BDC是直角，D向平面ABC作垂线的垂足恰好是三角形ABC的垂心. 求证：(AB + BC + CA)2≤ 6(AD2 + BD2 + CD2).并问何时等号成立？13.平面上给定100个点，无三点共线，求证：这些点构成的三角形中至多70% 是锐角三角形.14.令 E n = (a1 - a2)(a1 - a3) ... (a1 - a n) + (a2 - a1)(a2 - a3) ... (a2 - a n) + ... + (a n - a1)(a n - a2) ... (a n - a n-1). 求证 E n >= 0 对于n=3或5成立，而对于其他自然数n>2不成立.15.凸多边形 P1的顶点是 A1, A2, ... , A9，若将顶点 A1 平移至A i 时则 P1平移成了多边形 P i ，求证 P1, P2, ... , P9 之中至少有两个具有一共同内点.16.求证能够找到一个由形式 2n - 3 （n是正整数）的整数构成的集合并满足任何两个元素互质.17.四面体ABCD的所有面都是锐角三角形，在线段AB上取一内点X，现在BC上取内点Y，CD上取内点Z，AD上内点T. 求证：a.如果∠DAB+∠BCD ≠ ∠CDA+∠ABC，则没有一条闭路径XYZTX具有最小值；b.如果∠DAB+∠BCD ＝∠CDA+∠ABC，则有无穷多最短路径XYZTX，它们的长度是 2AC sin(k/2)，其中k=∠BAC+∠CAD+∠DAB.18.对任何自然数 m ，求证存在平面上一有限点集 S，满足：对S中的每一个点 A，存在S中的恰好 m 个点与 A的距离为单位长.19.设 A = (a ij),其中 i, j = 1, 2, ... , n，是一个方阵，元素 a ij都是非负整数. 若 i、j使得a ij = 0，则第i行和第j列的元素之和大于或等于 n. 求证：该方阵中所有元素之和大于或等于n2/2.20.有十个互不相同的二位数，求证必可从中选出两个不相交的子集，使得这两个子集中的元素之和相等.21.设 n>4，求证每一个圆内接四边形都可以分割成 n 个圆内接四边形.22. m,n是任意非负整数，求证下式是一整数.(2m)!(2n)!m!n!(m+n)!23.试找出下述方程组的所有正实数解：(x12 - x3x5)(x22 - x3x5) <= 0(x22 - x4x1)(x32 - x4x1) <= 0(x32 - x5x2)(x42 - x5x2) <= 0(x42 - x1x3)(x52 - x1x3) <= 0(x52 - x2x4)(x12 - x2x4) <= 024. f、g都是定义在实数上并取值实数的函数，并且满足方程f(x + y) + f(x - y) = 2f(x)g(y)，又已知 f 不恒等于0且 |f(x)| <= 1 . 求证对所有x同样有 |g(x)| <= 1 .25.给定四个不相同的平行平面，求证存在一个正四面体，它的四个定点分别在这四个平面上.26. OP1, OP2, ... , OP2n+1 是平面上的单位向量，其中点 P1, P2, ... , P2n+1都是位于通过点O的一条直线的同一侧，求证|OP1 + ... + OP2n+1| >= 1.27.问能否在空间中找到一个不共面的有限点集M使得，对M中的任何两点A、B，都可以再在M中寻找到两点C、D，而直线AB、CD是不相同的并且是互相平行的.28.考虑所有这样的实数a、b使得方程x4 + ax3 + bx2 + ax + 1 = 0至少有一个实根. 试找出 a2 + b2 的最小值.。
普特南大学数学竞赛1986年试题
(1 − x)n f (x) = 1 +
ai xbi .
i=1
A–4 A transversal of an n × n matrix A consists of n entries of A, no two in the same row or column. Let f (n) be the number of n × n matrices A satisfying the following two conditions: (b) The sum of the n entries of a transversal is the same for all transversals of A. An example of such a matrix A is −1 0 −1 A = 0 1 0 . 0 1 0 Determine with proof a formula for f (n) of the form
n n f (n) = a1 bn 1 + a2 b 2 + a3 b 3 + a4 ,
(a) Each entry αi,j of A is in the set {−1, 0, 1}.
where the ai ’s and bi ’s are rational numbers. A–5 Suppose f1 (x), f2 (x), . . . , fn (x) are functions of n real variables x = (x1 , . . . , xn ) with continuous secondorder partial derivatives everywhere on Rn . Suppose further that there are constants cij such that ∂fi ∂fj − = cij ∂xj ∂xi for all i ቤተ መጻሕፍቲ ባይዱnd j , 1 ≤ i ≤ n, 1 ≤ j ≤ n. Prove that there is a function g (x) on Rn such that fi + ∂g/∂xi is linear for all i, 1 ≤ i ≤ n. (A linear function is one of the form a0 + a1 x1 + a2 x2 + · · · + an xn .) A–6 Let a1 , a2 , . . . , an be real numbers, and let b1 , b2 , . . . , bn be distinct positive integers. Suppose that there is a polynomial f (x) satisfying the identity
2009普特南数学竞赛真题
A1 Let f be a real-valued function on the plane such that for every square ABCD in the plane, f (A) + f (B ) + f (C ) + f (D) = 0. Does it follow that f (P ) = 0 for all points P in the plane? A2 Functions f, g, h are differentiable on some open interval around 0 and satisfy the equations and initial conditions f = 2f 2 gh + 1 , f (0) = 1, gh 4 , g (0) = 1, g = f g2 h + fh 1 h = 3f gh2 + , h(0) = 1. fg
Prove that limx→∞ f (x) = ∞. B6 Prove that for every positive integer n, there is a sequence of integers a0 , a1 , . . . , a2009 with a0 = 0 and a2009 = n such that each term after a0 is either an earlier term plus 2k for some nonnegative integer k , or of the form b mod c for some earlier positive terms b and c. [Here b mod c denotes the remainder when b is divided by c, so 0 ≤ (b mod c) < c.]
全球大学生竞赛试题答案
全球大学生竞赛试题答案一、数学与逻辑1. 证明题：请证明对于任意正整数\( n \)，\( 1^n + 1 = 2 \)。
答案：由于\( n \)是正整数，\( 1^n \)总是等于1。
因此，\( 1^n + 1 = 1 + 1 = 2 \)。
2. 应用题：如果一个圆的半径增加2厘米，那么它的面积增加了多少？答案：设原圆的半径为\( r \)厘米。
原圆的面积为\( \pi r^2 \)。
半径增加2厘米后，新圆的半径为\( r+2 \)厘米，面积为\( \pi(r+2)^2 \)。
面积增加量为：\[\pi (r+2)^2 - \pi r^2 = \pi (r^2 + 4r + 4) - \pi r^2 =4\pi r + 4\pi\]二、物理与工程1. 解答题：一个质量为\( m \)的物体在无摩擦的水平面上，受到一个恒定的力\( F \)作用，求物体的加速度。
答案：根据牛顿第二定律，\( F = ma \)。
因此，物体的加速度\( a \)为：\[a = \frac{F}{m}\]2. 设计题：设计一个能够测量液体密度的装置，并简述其工作原理。
答案：可以使用浮力原理来设计一个测量液体密度的装置。
装置包括一个已知质量的物体和测量物体在液体中浮力的设备。
当物体完全浸入液体中时，根据阿基米德原理，物体受到的浮力等于它排开液体的重量。
通过测量浮力和物体的体积，可以计算出液体的密度。
三、化学与生物1. 实验题：如何通过实验确定一个未知溶液的pH值？答案：可以使用pH试纸或pH计来确定溶液的pH值。
将pH试纸浸入溶液中，然后与颜色对照表比较以确定pH值。
或者，使用pH计，将电极浸入溶液中，读取显示的pH值。
2. 分析题：解释为什么植物在光合作用过程中需要叶绿素。
答案：叶绿素是植物进行光合作用的关键色素，它能够吸收太阳光中的光能。
光合作用是植物将光能转化为化学能，并将二氧化碳和水转化为葡萄糖和氧气的过程。
叶绿素分子中的镁原子能够吸收特定波长的光，从而激发电子，启动光合作用的光依赖反应。
2011年美国大学生数学竞赛试题
1 1 + (x − ai )2
2
dx ≤ An.
Prove there is a constant B > 0 such that for all n,
n i,j =1
(1 + (ai − aj )2 ) ≥ Bn3 .
be a (not necessarily minimal) set of distinct generators of G. A special die, which randomly selects one
m→∞
lim
1 b2m
x∈G
Prob(g = x) −
1 n
2
is positive and finite. B1 Let h and k be positive integers. Prove that for every ǫ > 0, there are positive integers m and n such that √ √ ǫ < |h m − k n| < 2ǫ. B2 Let S be the set of all ordered triples (p, q, r) of prime numbers for which at least one rational number x satisfies px2 + qx + r = 0. Which primes appear in seven or more elements of S ? B3 Let f and g be (real-valued) functions defined on an open interval containing 0, with g nonzero and continuous at 0. If f g and f /g are differentiable at 0, must f be differentiable at 0? B4 In a tournament, 2011 players meet 2011 times to play a multiplayer game. Every game is played by all 2011 players together and ends with each of the players either winning or losing. The standings are kept in two 2011 × 2011 matrices, T = (Thk ) and W = (Whk ). Initially, T = W = 0. After every game, for every (h, k ) (including for h = k ), if players h and k tied (that is, both won or both lost), the entry Thk is increased by 1, while if player h won and player k lost, the entry Whk is increased by 1 and Wkh is decreased by 1. Prove that at the end of the tournament, det(T + iW ) is a non-negative integer divisible by 22010 . B5 Let a1 , a2 , . . . be real numbers. Suppose that there is a constant A such that for av ) ∈ R2 , the vector ∇F (u, v ) is either 0 or parallel to the vector g (u), −g (v ) . Prove that there exists a constant C such that for every n ≥ 2 and any x1 , . . . , xn+1 ∈ R, we have C min |F (xi , xj )| ≤ . i=j n A6 Let G be an abelian group with n elements, and let {g1 = e, g2 , . . . , gk } G
2023年美国第84届普特南数学竞赛试题
2023普特南数学竞赛A组(0)|>题 1.对正整数n,设f n(x)=cos(x)cos(2x)cos(3x)···cos(nx).求最小的n,使得|f′′n 2023.题2.设n是一个偶数.设p是一个首一的实系数多项式,次数为2n；即p(x)=x2n+a2n−1x2n−1+···+a1x+a0,其中系数a0,...,a2n−1是实数.已知对于所有满足1≤|k|≤n的整数k,有p(1/k)= k2.找出所有满足p(1/x)=x2的其他实数x.题3.确定最小的正实数r,使得存在可微函数f:R→R和g:R→R满足以下条件：(a)f(0)>0,(b)g(0)=0,(c)对所有x,|f′(x)|≤|g(x)|,(d)对所有x,|g′(x)|≤|f(x)|,(e)f(r)=0.题4.已知v1,...,v12是单位向量,其方向从原点指向正二十面体的所有顶点.证明对于每个向量v∈R3和每个ε>0,存在整数a1,...,a12满足∥a1v1+···+a12v12−v∥<ε.题5.对于非负整数k,设f(k)为k在3进制表示中的1的个数.找出所有复数z,使得31010−1(−2)f(k)(z+k)2023=0k=0题6.甲和乙进行一个游戏,在游戏中他们轮流选择1到n之间的整数.在选择数字之前,乙选择一个“奇数”或“偶数”的目标.在第一轮中,甲选择n个整数中的一个.在第二轮中,乙选择剩下的整数中的一个.他们继续轮流选择尚未选择过的整数,直到第n轮,也就是最后一轮,游戏结束.如果{k：第k轮选择了数字k}的奇偶性与乙的目标匹配,乙获胜.对于哪些n,乙有获胜策略？B 组题1.考虑一个m ×n 的单位方格网格,其下标为(i,j ),其中1≤i ≤m 且1≤j ≤n .有(m −1)(n −1)个硬币,最初放置在下标为(i,j ),其中1≤i ≤m −1且1≤j ≤n −1的单元格中.如果一个硬币占据方格(i,j ),其中i ≤m −1且j ≤n −1且方格(i +1,j )、(i,j +1)和(i +1,j +1)空闲,那么合法的移动是将硬币从(i,j )滑动到(i +1,j +1).从初始布局出发,通过一系列合法移动（可能为空）可以达到多少种不同的硬币配置？题2.对于每个正整数n ,令k (n )表示二进制表示中2023·n 中的1的个数.求k (n )的最小值.题3.一个由实数y 1,y 2,...,y k 组成的数列被称为“锯齿”序列,如果k =1,或者y 2−y 1,y 3−y 2,...,y k −y k −1非零并且交替改变符号.从均匀分布[0,1]上独立选择X 1,X 2,...,X n .令a (X 1,X 2,...,X n )是最大的k 值,使得存在递增的整数序列i 1,i 2,...,i k ,满足X i 1,X i 2,...X i k 是“锯齿”序列.对于n ≥2,求a (X 1,X 2,...,X n )的期望值.题4.对于非负整数n 和严格递增的实数序列t 0,t 1,...,t n ,令f (t )是一个实值函数.使得对于t ≥t 0,它具有以下性质：(1)对于t ≥t 0,f (t )是连续函数,对于除了t 1,...,t n 外所有的t >t 0,它是二阶可微的；(2)f (t 0)=1/2；(3)当0≤k ≤n 时,lim t →t +kf ′(t )=0；(4)当0≤k ≤n −1时,当t k <t <t k +1时,f ′′(t )=k +1,当t >t n 时,f ′′(t )=n +1.考虑所有满足t k ≥t k −1+1（1≤k ≤n ）的n 和t 0,t 1,...,t n 的选择,求使得f (t 0+T )=2023的T 的最小值.题5.确定哪些正整数n 具有以下性质：对于所有与n 互质的整数m ,存在一个置换π:{1,2,...,n }→{1,2,...,n },满足对于所有k ∈{1,2,...,n },π(π(k ))≡mk (mod n ).题6.令n 为正整数.对于{1,2,...,n }中的i 和j ,令s (i,j )为满足ai +bj =n 的非负整数对(a,b )的数量.令S 为n ×n 矩阵,其(i,j )元素为s (i,j ).例如,当n =5时,我们有S = 6322230101210012000121112 .计算矩阵S 的行列式.。
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=
证
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)
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
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.
for all real numbers x and all positive integers n. A3. Suppose that the function h : R2 → R has continuous partial derivatives and satisfies the equation ∂h ∂h h(x, y) = a (x, y) + b (x, y) ∂x ∂y for some constants a, b. Prove that if there is a constant M such that |h(x, y)| ≤ M for all (x, y) ∈ R2 , then h is identically zero. A4. Prove that for each positive integer n, the number 1010 prime.
SEVENTY-FIRST ANNUAL WILLIAM LOWELL PUTNAM MATHEMATICAL COMPETITION Saturday, December 4, 2010 Examination A
A1. Given a positive integer n, what is the largest k such that the numbers 1, 2, . . . , n can be put into k boxes so that the sum of the numbers in each box is the same? [When n = 8, the example {1, 2, 3, 6}, {4, 8}, {5, 7} shows that the largest k is at least 3.] A2. Find all differentiable functions f : R → R such that f ′ (x) = f (x + n) − f (x) n
10n
+ 1010 + 10n − 1 is not
n
A5. Let G be a group, with operation ∗. Suppose that (i) G is a subset of R3 (but ∗ need not be related to addition of vectors); (ii) For each a, b ∈ G, either a × b = a ∗ b or a × b = 0 (or both), where × is the usual cross product in R3 . Prove that a × b = 0 for all a, b ∈ G. A6. Let f : [0, ∞) → R be a strictly decreasing continuous function such that limx→∞ f (x) = ∞ 0. Prove that 0 f (x)−f (x+1) dx diverges. f (x)
B5. Is there a strictly increasing function f : R → R such that f ′ (x) = f (f (x)) for all x? B6. Let A be an n × n matrix of real numbers for some n ≥ 1. For each positive integer k, let A[k] be the matrix obtained by raising each entry to the k th power. Show that if Ak = A[k] for k = 1, 2, . . . , n + 1, then Ak = A[k] for all k ≥ 1.