IMO Shortlist 1985 problem 7


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The positive integers x_1, \cdots , x_n, n \geq 3, satisfy x_1 < x_2 <\cdots< x_n < 2x_1. Set P = x_1x_2 \cdots x_n. Prove that if p is a prime number, k a positive integer, and P is divisible by pk, then \frac{P}{p^k} \geq n!.
Izvor: Međunarodna matematička olimpijada, shortlist 1985