MathDB
VMO 2015 Problem 6

Source: VMO 2015

July 31, 2016
number theory

Problem Statement

For a,n∈Z+a,n\in\mathbb{Z}^+, consider the following equation: a^2x+6ay+36z=n  (1) where x,y,z∈Nx,y,z\in\mathbb{N}.
a) Find all aa such that for all n≥250n\geq 250, (1)(1) always has natural roots (x,y,z)(x,y,z).
b) Given that a>1a>1 and gcd⁡(a,6)=1\gcd (a,6)=1. Find the greatest value of nn in terms of aa such that (1)(1) doesn't have natural root (x,y,z)(x,y,z).
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