Problem - 3663
Find a quadratic polynomial $f(x)=x^2 + mx +n$ such that $$f(a)=bc,\quad f(b) = ca,\quad f(c) = ab$$ where $a$, $b$, $c$ are three distinct real numbers.
Consider 3-degree polynomial $g(x) = xf(x)-abc$. It is clear that $a$, $b$ and $c$ are three roots of $g(x)$. As such, we must have $$g(x)=(x-a)(x-b)(x-c)$$
$$\therefore\quad xf(x)-abc=(x-a)(x-b)(x-c)$$
$$\implies f(x)=x^2 - (a+b+c)x+(ab+bc+ca)$$