An algebra problem by Archit Tripathi

Note that the roots are in $(0,1)$ , and are distinct. Hence, the conditions that we get from the equation are: $a+c>b,\quad a>c,\quad b^2>4ac$ Note that $c$ should be the minimum, since otherwise, $c>b$ and that would imply $b^2>4ba\implies b>4a$ which is a contradiction in the light of the first two conditions. Also, note that $b$ cannot be equal to $c$ in the light of second and third conditions. Thus the possible ordering of the integers $a,b,c$ are $a\ge b>c$ or $b\ge a>c$ . In any case the product $abc$ will be minimized if we take the ordering $a=b>c$ . In that case to make $c$ as small as possible, take $c$ , and take $b=a=5$ , which is the smallest positive integer to satisfy the third condition. Thus the smallest value of the product is $\boxed{25}$ .