Question 64
Let $n$ be a fixed positive integer. The positive integers $a$, $b$, $c$ and $d$ are less than or equal to $n$, $d$ is the largest one and they satisfy the equality
$$
(ab + cd)(bc + ad)(ac + bd) = (d - a)^2 (d - b)^2 (d - c)^2
$$
a) Prove that $d = a + b + c$.
b) Find the number of the quadruples $(a, b, c, d)$ which have the required properties.
$$
(ab + cd)(bc + ad)(ac + bd) = (d - a)^2 (d - b)^2 (d - c)^2
$$
a) Prove that $d = a + b + c$.
b) Find the number of the quadruples $(a, b, c, d)$ which have the required properties.