MathDB

3

Part of 2018 Caucasus Mathematical Olympiad

Problems(2)

Classical number theory

Source: III Caucasus Mathematical Olympiad

3/17/2018
Suppose that a,b,ca,b,c are positive integers such that aba^b divides bcb^c, and aca^c divides cbc^b. Prove that a2a^2 divides bcbc.
number theory
very nice combinatorics with NT

Source: III Caucasus Mathematical Olympiad

3/17/2018
For 2n2n positive integers a matching (i.e. dividing them into nn pairs) is called {\it non-square} if the product of two numbers in each pair is not a perfect square. Prove that if there is a non-square matching, then there are at least n!n! non-square matchings. (By n!n! denote the product 1⋅2⋅3⋅…⋅n1\cdot 2\cdot 3\cdot \ldots \cdot n.)
combinatoricsnumber theory