IMO 1986, Problem 1

Let

d

be any positive integer not equal to

2

5

13

. Show that one can find distinct

a

b

in the set

\{2, 5, 13, d\}

such that

ab - 1

is not a perfect square.

Topic: Number Theory Difficulty: 5/6 Expected time: ~45 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.

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IMO 1986, Problem 1

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