Fix integers a and b greater than 1. For any positive integer n, let r_n be the (non-negative) remainder that bⁿ leaves upon division by aⁿ. Assume there exists a positive integer N such that r_n<2ⁿ/n for all integers n≥N.Prove that a divides b.

给定大于1的整数a和b.对任意的正整数n，记r_n为bⁿ除以aⁿ的非负余数.若存在正整数N，使得对任意的n≥N，都有r_n<2ⁿ/n.证明：a整除b.

Consider an odd prime p and a positive integer N<50p. Let a₁,a₂,⋯,a_N be a list of positive integers less than p such that any specific value occurs at most 51/100 N times and a₁,a₂,⋯,a_N is not divisible by p. Prove that there exists a permutation b₁,b₂,⋯,b_N of the a_i such that, for all k=1,2,⋯,N, the sum b₁+b₂+⋯+b_k is not divisible by p.

【译】已知奇素数p和正整数N<50p.设a₁,a₂,⋯,a_N是一些小于p的正整数，同一数值至多出现51/100 N次，且a₁+a₂+⋯+a_N不能被p整除.

证明：存在a_i的一个排列：b₁,b₂,⋯,b_N，使得对任意的k=1,2,⋯,N，都有b₁+b₂+⋯+b_k不能被p整除.

Let m<n be positive integers. Start with n piles, each of m objects. Repeatedly carry out the following operation: choose two piles and remove n objects in total from the two piles. For which (m ,n) is it possible to empty all the piles?

【译】设正整数m<n.起初一共有n 堆石子，每堆有 m块石子. 重复执行以下操作: 选择两堆石子，从这两堆中移除共n 块石子.问：对于怎样的 (m , n)，可以移除所有石子?

In the sequence 7,76,769,7692,76923,769230,… ,the nth term is given by the first n digits after the decimal point in the expansion of 10/13=0.7692307692⋯.

Prove that of the first 60 terms of the sequence, at least 49 have three or more prime factors (repeated prime factors are allowed; for example, 76=2×2×19 has three prime factors).

【译】在10/13=0.7692307692⋯的十进制表示中，由小数点后的前n位数构成数列：

7,76,769,7692,76923,769230,… ,

求证：在该数列的前60项中，至少有49项有三个或以上的素因子(包含重复的素因子，例如76=2×2×19有三个素因子).

设整数n≥4.证明：若n整除2ⁿ-2，则(2ⁿ-2)/n是合数.