ISL 1989 Problem 41

Let $f(x) = a \sin^2x + b \sin x + c,$ where $a, b,$ and $c$ are real numbers. Find all values of $a, b$ and $c$ such that the following three conditions are satisfied simultaneously:

(i) $f(x) = 381$ if $\sin x = \frac{1}{2}.$ (ii) The absolute maximum of $f(x)$ is $444.$ (iii) The absolute minimum of $f(x)$ is $364.$