Question 133
A triple of positive integers $(a, b, c)$ is called *Quadratic* if $a, b, c$ is a nondecreasing arithmetic progression, $\gcd(b, a) = \gcd(b, c) = 1$, and $abc$ is a perfect square. Prove that for any quadratic triple, there exists some other quadratic triple containing at least one number from the first triple. Let's call the triple of natural numbers $(a, b, c)$ square if they form a non-decreasing arithmetic progression (in that order), the number $b$ is mutually simple with each of the numbers $a$ and $c$, and the $abc$ number is the exact square. Prove that for any square three, there is another square three that has at least one total number with it.