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Belarusian Mathematical Olympiad

Number theory Difficulty 5.5 AIME, harder Prove it Belarus

a) Find at least one positive integer $n$ , which can be represented both as $n = a^2 - b$ and $n = c^2 - d$ , where $a, b, c, d$ are divisors of $n$ and $a \neq c$ .
b) Prove that there exist infinitely many positive integers $n$ satisfying a).

Solution

a) $n = 72$ or $n = 88200$ .

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