Question 58
On the board 777 pairwise distinct *complex* numbers are written. It turned out that there are exactly 760 ways to choose a pair of numbers $a$ and $b$ written on the board so that
$$
a^2 + b^2 + 1 = 2ab.
$$
(Here pairs considered unordered, i.e., $(a, b)$ and $(b, a)$ is the same pair.) Prove that one can choose numbers $c$ and $d$ written on the board such that
$$
c^2 + d^2 + 2025 = 2cd.
$$
$$
a^2 + b^2 + 1 = 2ab.
$$
(Here pairs considered unordered, i.e., $(a, b)$ and $(b, a)$ is the same pair.) Prove that one can choose numbers $c$ and $d$ written on the board such that
$$
c^2 + d^2 + 2025 = 2cd.
$$