The 16th Japanese Mathematical Olympiad - The First Round

Let $a$, $b$ and $c$ be distinct positive integers with sum of any two of them being squares. We may assume that $a<b<c$. Write $a+b=x^2$, $b+c=y^2$, $c+a=z^2$. Then we shall minimize $x^2+y^2+z^2$ under the conditions $x<y<z$, $z^2<x^2+y^2$, and $x^2+y^2+z^2$ even. $z>5$, since if otherwise $z^2\ge x^2+y^2$. If $z=6$, $x$ and $y$ must be both even or both odd, but neither is appropriate since $6^2>5^2+3^2>4^2+2^2$. If $z=7$, only $(x,y,z)=(5,6,7)$ satisfies the required conditions. If $z\ge 8$, $x^2+y^2+z^2>2z^2\ge 2\times 8^2>7^2+6^2+5^2$. Therefore $(x,y,z)=(5,6,7)$ satisfies the required conditions and minimizes $x^2+y^2+z^2$. So the answer is $(a,b,c)=(6,19,30)$.