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USAMO 2022 — Days 1 and 2 Question 4

USAMO · USAMO 2022 — Days 1 and 2

Question 4

Find all pairs of primes $(p,q)$ for which $p-q$ and $pq-q$ are both perfect squares.

Proposed by Holden Mui. Source: Evan Chen’s third-party solution notes; https://aops.com/community/p24774670. One point is a nonofficial editorial normalization; this essay requires manual grading.