【数学】美国普特南数学竞赛题(1)

Putnam 2002

A1. Let k be a positive integer. The n-th derivative of 1/(x^k-1)
has the form P_n(x) / (x^k-1)^{n+1} where P_n(x) is a polynomial.
Find P_n(1).

A1. k是正整数,设1/(x^k-1) 的n阶导数等于P_n(x) / (x^k-1)^{n+1},
试求P_n(1)的值。

A2. Given any five points on a sphere, show that some four of them
must lie on a closed hemisphere.

A2. 证明:球面上任意五个点,其中必有四个点在同一个闭半球面上。

A3. Let n>=2 be an integer and T_n be the number of non-empty subsets
S of {1, 2, 3, …, n} with the property that the average of the
elements of S is an integer. Prove that T_n – n is always even.

A3. 集合{1,2,3,…,n},其某非空子集的全部元素平均值是整数,Tn是
这种所有满足上述条件的子集个数。证明:Tn-n是偶数。

A4. In Determinant Tic-Tac-Toe, Player 1 enters a 1 in an empty 3×3 matrix. Player 0 counters with a 0 in a vacant position and play continues in turn until the 3×3 matrix is completed with five 1’s and four 0’s. Player 0 wins if the determinant is 0 and player 1
wins otherwise. Assuming both players pursue optimal strategies,
who will win and how?

A4. 一个3×3的矩阵由两个人轮流往其中添数字:
甲只能添1,乙只能添0,甲先添,
两个人交替添入数字一直到矩阵被添满,
如果最后矩阵的值(就是行列式了)为1,则甲胜,
如果最后矩阵的值为0,则乙胜,
请问最后是那一个人胜?即谁有必胜策略?

A5. Define a sequence by a_0 = 1, together with the rules a_{2n+1} = a_n
and a_{2n+2} = a_n + a_{n+1} for each integer n >= 0. Prove that
every positive rational number appears in the set
{ a_{n-1}/a_n : n >= 1 } = { 1/1, 1/2, 2/1, 1/3, 3/2, …}

A5. 某数列满足:a(0)=1,a(2n+1)=a(n),a(2n+2)=a(n)+a(n+1)
试证明集合{a(n-1)/a(n):n>=1}是正有理数集,即它包括所有的正有理数。

A6. Fix an integer b >= 2. Let f(1) = 1, f(2) = 2, and for each
n >=3, define f(n) = n f(d), where d is the number of base-b
digits of n. For which values of b does
∑{n=1}^{∞} 1 / f(n)
converge?

A6. 整数b>=2,函数f(1)=1, f(2)=2, f(n)=n*f(d),d是n在b进制中的值,求b
使得∑{n=1}^{∞} 1 / f(n) 的极限存在。

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