IMO 1988, Problem 6

Let

a

and

b

be positive integers such that

ab+1

divides

a^2+b^2

. Prove that

\dfrac{a^2+b^2}{ab+1}

is a perfect square.

Topic: Number Theory Difficulty: 6/6 Expected time: ~60 min Progressive hints on Lemma: 3

This is a proof problem — the answer is an argument, not a number. The solution is deliberately not posted here: reading a solution you didn't fight for teaches almost nothing. On Lemma you attempt it cold, take one of the 3 progressive hints only when genuinely stuck, then compare your proof against a full walkthrough and mark yourself with the same rubric a competition coordinator would use.

Train it on Lemma

This problem sits in the Olympiad Set — 81 real competition problems with progressive hints, full walkthroughs and marking rubrics. 78 lessons, 624 curated problems and unlimited generated practice at six difficulties. Free to start — no card, no trial clock.

Find your level

IMO 1988, Problem 6

Train it on Lemma

More Number Theory problems