These are the two problems I assigned to the Senior students in the Malaysia IMO 2010 training camp. They are given two weeks to find the solutions.
1. Let be distinct positive integers not exceeding such that does not divide for all . Denote by the number of primes not exceeding . Prove that .
2. Let be a polynomial with rational coefficients such that for all . Prove that for any pair of distinct integers and , the number
is an integer.
( denotes the set of all integers, denotes the least common multiple, denotes the degree of polynomial .)
Source: T. Andreescu & G. Dospinescu, Problems from the Book, pp. 69 & 228, with minor rewording.