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P/p^k is not less than n!

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August 29, 2010
number theorySequencePrime numberInequalityIMO Shortlist

Problem Statement

The positive integers x1,⋯ ,xnx_1, \cdots , x_n, n≥3n \geq 3, satisfy x1<x2<⋯<xn<2x1x_1 < x_2 <\cdots< x_n < 2x_1. Set P=x1x2⋯xn.P = x_1x_2 \cdots x_n. Prove that if pp is a prime number, kk a positive integer, and PP is divisible by pkpk, then Ppk≥n!.\frac{P}{p^k} \geq n!.
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