Putnam 2002

A1. Let k be a positive integer. The n-th derivative of
has the form where
is a polynomial.
Find .
A1. k是正整数,设 的n阶导数等于
,
试求的值。
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 be the number of non-empty subsets
S of with the property that the average of the
elements of S is an integer. Prove that is always even.
A3. 集合,其某非空子集的全部元素平均值是整数,
是
这种所有满足上述条件的子集个数。证明: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 , together with the rules
and
for each integer n >= 0. Prove that every positive rational number appears in the set
A5. 某数列满足: ,
,
试证明集合 是正有理数集,即它包括所有的正有理数。
A6. Fix an integer . Let
, and for each
n >=3, define , where d is the number of base-b
digits of n. For which values of b does
converge?
A6. 整数 ,函数
,
,d是n在b进制中的值,求b
使得 的极限存在。
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