2. Let $a, b, c$ be pairwise distinct positive integers such that $a+b, b+c$ and $c+a$ are all square numbers. Find the smallest possible value of $a+b+c$.
(1 mark)
Let $a, b, c$ be pairwise distinct positive integers such that $a+b, b+c$ and $c+a$ are all square numbers. Find the smallest possible value of $a+b+c$.
2. 55
2. Without loss of generality assume \(a < b < c\). Hence \(z\) is at least 6. If \(z=6\), the only possibility is \((x, y) = (4,5)\) in order that \(z^2 < x^2 + y^2\), but this violates the parity constraint. It follows that \(z\) is at least 7, and the bound \(z^2 < x^2 + y^2\) together with the parity constraint force \((x, y) = (5,6)\), and thus we get \(\frac{1}{2} \left(5^2 + 6^2 + 7^2\right) = 55\) as the smallest possible value of \(a + b + c\), which corresponds to \((a, b, c) = (6,19,30)\).