25. It is given that $a, b, c$ are positive integers such that the roots of the three quadratic equations
$$
x^{2}-2 a x+b=0, \quad x^{2}-2 b x+c=0, \quad x^{2}-2 c x+a=0
$$
are all positive integers. Determine the maximum value of the product $a b c$.
25. Answer: 1
We shall show that the only possible values of $a, b, c$ are $a=b=c=1$ so that $a b c=1$. From the first equation, we note that $a^{2}-b=(x-a)^{2}$ is a perfect square less than $a^{2}$. Thus $a^{2}-b \leq(a-1)^{2}$. That is $b \geq 2 a-1$. Likewise $c \geq 2 b-1$ and $a \geq 2 c-1$. Combining these inequalities, we have $a \geq 8 a-7$ or $a \leq 1$. Thus $a=1$. Similarly $b=c=1$.