# 2006 UNCO Math Contest II Problems/Problem 9

## Problem

Determine three positive integers $a,b$ and $c$ that simultaneously satisfy the following three conditions:

(i) $a

(ii) Each of $a+b,a+c$ and $b+c$ is the square of an integer, and

(iii) $c$ is as small as is possible.

## Solution

$a=6;b=19,c=30$ (also $2-34-47$)

Let $a+b = x^2$, $a + c = y^2$, $b + c = z^2$. We can easily find that $a = \dfrac{x^2+y^2-z^2}{2}$, $b = \dfrac{x^2+z^2-y^2}{2}$, $c = \dfrac{y^2+z^2-x^2}{2}$. Taking mod 2, we find that $(x, y, z)$ must be either $(0, 0, 0)$, $(0, 1, 1)$, $(1, 0, 1)$, $(1, 1, 0)$. For the first case, we check $(2n, 2n + 2, 2n + 4)$ until $(a, b, c)$ is positive. We get $n = 4$, and $(a, b, c) = (10, 54, 90)$. For the second case, we check $(2n, 2n+1, 2n+3)$, getting $n = 3$, and $(a, b, c) = (2, 34, 47)$. For the third case, we check $(2n + 1, 2n + 2, 2n + 3)$, getting $n = 2$ and $(a, b, c) = (6, 19, 30)$. For the last case, we check $(2n + 1, 2n + 3, 2n + 4)$, getting $n = 2$, and $(a, b, c) = (5, 20, 44)$. Clearly, our triple with the minimum $c$ value is $(6, 19, 30)$. ~Puck_0