【数学】美国普特南数学竞赛题(2)

Putnam 2001

A1. Consider a set S and a binary operation * on S (that is, for
each a, b in S, a*b is in S). Assume that (a*b)*a = b for all a, b in S. Prove that a*(b*a) =b for all a, b in S.

A1.考虑一个集合S和关于这个集合的一个二元操作 * (指若a,b ∈S,则a*b ∈ S)。假设 (a*b)*a=b,对于a,b ∈ S都成立。
证明:对于任意a,b ∈ S,a*(b*a)=b成立。

A2. You have coins C_1, C_2, …, C_n. For each k, coin C_k is biased
so that, when tossed, it has probability \frac 1{2k+1} of falling heads.
If the n coins are tossed, what is the probability that the
number of heads is odd? Express the answer as a rational function of n.

A2.有n个硬币 C_1, C_2, …, C_n ,对于每个k,硬币C_k都是偏心的,抛掷的时候,落地正面朝上的概率为 \frac 1{2k+1} ,同时抛掷这n枚硬币,
问题:正面朝上的硬币数量是奇数的概率是多少?

A3. For each integer m, consider the polynomial
P_m(x) = x^4 - (2m+4) x^2 + (m-2)^2
For what values of m is P_m(x) the product of two nonconstant
polynomials with integer coefficients?

A3. 对每个整数 m, 设多项式 P_m(x) = x^4 - (2m+4) x^2 + (m-2)^2
问题:对哪些m,P_m(x)可以分解成两个整系数非常数多项式的乘积?

A4. Triangle ABC has area 1. Points E,F,G lie, respectively, on
sides BC, CA, AB such that AE bisects BF at point R,
BF bisects CG at point S, and CG bisects AE at point T.
Find the area of triangle RST.

A4. 三角形 ABC,面积等于 1。三点 E,F,G 非别在边 BC, CA, AB上,并且AE 交 BF于R,BF交CG于S,CG交AE于T。求三角形RST的面积?

A5. Prove that there are unique positive integers a, n such that
a^{n+1} - (a+1)^n = 2001.

A5. 证明:存在唯一的正整数 a, n,满足 a^{n+1} - (a+1)^n = 2001

A6. Can an arc of a parabola inside a circle of radius 1 have length
greater than 4 ?

A6. 在半径为1的圆内,一个抛物线的弧长能否大于4?

B1. Let n be an even positive integer. Write the numbers 1, 2, …, n^2
in the squares of an n x n grid so that the k-th row, from left to
right, is (k-1)n + 1, (k-1)n + 2, …, (k-1)n + n.
Color the squares of the grid so that half of the squares in each row
and in each column are red and the other half are black (a checkerboard
coloring is one possibility). Prove that for each such coloring, the
sum of the numbers on the red squares is equal to the sum of the numbers
on the black squares.

B1. 假设n是一个正偶数,依次将1, 2, …, n^2写到n x n的表格内,使的在第k行,从左到右是(k-1)n + 1, (k-1)n + 2, …, (k-1)n + n
将每个格子染色,满足每一行和每一列都有一半的格子是红色,一半的格子是黑色 (例如国际跳棋的棋盘染色就是一种)。
证明:对于任何一种染色方案,红色格子中的数的和等于黑色格子中的数的和。

B2. Find all pairs of real numbers (x,y) satisfying the system of equations
\frac 1x + \frac 1{2y} = (x^2 + 3 y^2) ( 3 x^2 + y^2 )
\frac 1x - \frac 1{2y} = 2(y^4 - x^4)

B2. 找到全部的实数对 (x,y),满足方程组:
\frac 1x + \frac 1{2y} = (x^2 + 3 y^2) ( 3 x^2 + y^2 )
\frac 1x - \frac 1{2y} = 2(y^4 - x^4)

B3. For any positive integer n let <n> denote the closest integer to \sqrt{n}. Evaluate \sum_{n=1}^{\infty} \frac{2^{<n>}  + 2^{- <n>}}{2^n}

B3. 对任意的正整数n,用 <n>表示与 \sqrt{n}最接近的整数。
计算\sum_{n=1}^{\infty} \frac{2^{<n>}  + 2^{- <n>}}{2^n}的值?

B4. Let S denote the set of rational numbers different from -1, 0, and 1.
Definef : S \to S by f(x) = x - \frac 1x . Prove or disprove that
\bigcap_{n=1}^{\infty} f^n(S) = \emptyset,
where f^n = f(f(…(f(n times)))).
(Note: f(S) denotes the set of all values f(s) for s \in S. )

B4. 设S是不包含 -1, 0, 和 1的全部有理数的集合。
定义函数 f : S \to S 满足 f(x) = x - \frac 1x。证明或否定:
\bigcap_{n=1}^{\infty} f^n(S) = \emptyset,
其中f^n = f(f(…(f(n次))))

B5. Let a and b be real numbers in the interval (0, \frac 12) and
let g be a continuous real-valued function such that
g(g(x)) = a g(x) + b x for all real x. Prove that g(x) = c x for
some constant c.

B5. 假设实数 a和b位于区间 (0, \frac 12) 内,令 g是一个连续实值函数,满足
g(g(x)) = ag(x) + bx ,证明:存在常数c, 使得 g(x) = c x

B6. Assume that(a_n) \{n \ge 1\} is an increasing sequence of positive real numbers such that \lim_{n->\infty} \frac {a_n}n = 0. Must there
exist infinitely many positive integers n such that
a_{n-i} + a_{n+i} < 2 a_n for i = 1, 2, …, n-1 ?

B6. 假设数列 (a_n)\{n \ge 1\} 是一个递增正实数列,
满足\lim_{n->\infty} \frac {a_n}n = 0, 是否存在无限个正整数n,满足
a_{n-i} + a_{n+i} < 2 a_n 其中 i = 1, 2, …, n-1 ?

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