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Romanian Masters in mathematics 2010 Day 1 Problem 1

Source:

April 25, 2010
inductionmodular arithmeticcombinatorics proposedcombinatorics

Problem Statement

For a finite non empty set of primes PP, let m(P)m(P) denote the largest possible number of consecutive positive integers, each of which is divisible by at least one member of PP.
(i) Show that ∣P∣≤m(P)|P|\le m(P), with equality if and only if min⁡(P)>∣P∣\min(P)>|P|.
(ii) Show that m(P)<(∣P∣+1)(2∣P∣−1)m(P)<(|P|+1)(2^{|P|}-1).
(The number ∣P∣|P| is the size of set PP)
Dan Schwarz, Romania
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