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Romanian District Olympiad 2015, Grade VII, Problem 2

Source:

September 25, 2018
number theory

Problem Statement

a) Show that if two non-negative integers p,q p,q satisfy the property that both 2p−q \sqrt{2p-q} and 2p+q \sqrt{2p+q} are non-negative integers, then q q is even.
b) Determine how many natural numbers m m are there such that 2m−4030 \sqrt{2m-4030} and 2m+4030 \sqrt{2m+4030} are both natural.
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