$x,y,z,a,b,c$ are positive integers that satisfy $xy \equiv a \pmod z$, $yz \equiv b \pmod x$, $zx \equiv c \pmod y$. Prove that $\min{\{x,y,z\}} \le ab+bc+ca$.
Source: Taiwan NMO 2006
Tags: modular arithmetic, number theory unsolved, number theory
$x,y,z,a,b,c$ are positive integers that satisfy $xy \equiv a \pmod z$, $yz \equiv b \pmod x$, $zx \equiv c \pmod y$. Prove that $\min{\{x,y,z\}} \le ab+bc+ca$.