HMMT 二月 2005 · 冲刺赛 · 第 44 题
HMMT February 2005 — Guts Round — Problem 44
题目详情
英文原题
- Write down a set S of positive integers, all greater than 1, whose product is P , suchthat for each x ∈ S , x is a proper divisor of ( P/x ) + 1. Your score is 2 n , where n = | S | .
解析
英文解析
- Write down a set S of positive integers, all greater than 1, such that for each x ∈ S , xis a proper divisor of ( P/x ) + 1, where P is the product of all the elements of S . Yourscore is 2 n , where n = | S | .
Remark: This question was posed by Stefan Zn´ am in 1972. Solutions exist for | S | ≥ 5,
but an explicit solution has been found only for | S | ≤ 13. The simplest solution is S =
{ 2 , 3 , 11 , 23 , 31 } . A solution for | S | = 7 is given by S = { 2 , 3 , 11 , 17 , 101 , 149 , 3109 } .
For more information, see http://mathworld.wolfram.com/Znams problem.html .