▪ IMO Shortlist 2011 – Number Theory

N1: For any integer let be the smallest possible integer that has exactly positive divisors (so for example we have and ). Prove that for every integer the number divides

N2: Consider a polynomial

   

where are nine distinct integers. Prove that there exists an integer such that for all integers the number is divisible by a prime number greater than 20.

N3: Let be an odd integer. Determine all functions from the set of integers to itself, such that for all integers and the difference divides

N4: For each positive integer let be the largest odd divisor of Determine all positive integers for which there exists a positive integer such that all the differences

are divisible by 4.

N6: Let and be two polynomials with integer coefficients, such that no nonconstant polynomial with rational coefficients divides both and Suppose that for every positive integer the integers and are positive, and divides Prove that is a constant polynomial.

N7: Let be an odd prime number. For every integer define the number

Let such that

   

Prove that divides

N8: Let and set Prove that is a prime number if and only if the following holds: there is a permutation of the numbers and a sequence of integers such that divides for every where we set