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Polish 2nd stage 2010, 3rd problem (consecutive numbers)

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November 11, 2010
inequalitiesnumber theoryprime numbersnumber theory proposed

Problem Statement

Positive integer numbers kk and nn satisfy the inequality k>n!k > n!. Prove that there exist pairwisely different prime numbers p1,p2,…,pnp_1, p_2, \ldots, p_n which are divisors of the numbers k+1,k+2,…,k+nk+1, k+2, \ldots, k+n respectively (i.e. pi∣k+ip_i|k+i).
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