HMMT 二月 2004 · 冲刺赛 · 第 38 题

英文解析

38. Let S = { p p · · · p | p , p , . . . , p are distinct primes and p , . . . , p < 30 } . Assume
    1 2 n 1 2 n 1 n
    1 is in S . Let a be an element of S . We define, for all positive integers n :
    a = a / ( n + 1) if a is divisible by n + 1;1
    n +1 n na = ( n + 2) a if a is not divisible by n + 1 .
    n +1 n n
    How many distinct possible values of a are there such that a = a for infinitely many
    1 j 1
    j ’s?
    Solution: 512
    If a is odd, then we can see by induction that a = ( j +1) a when j is even and a = a
    1 j 1 j 1
    when j is odd (using the fact that no even j can divide a ). So we have infinitely manyj ’s for which a = a .1
    j 1
    If a > 2 is even, then a is odd, since a = a / 2, and a may have only one factor of
    1 2 2 1 1