IMO 1959, Problem 2

For what real values of

x

\sqrt{x+\sqrt{2x-1}} + \sqrt{x-\sqrt{2x-1}} = A

, where (a)

A=\sqrt{2}

, (b)

A=1

, (c)

A=2

? (Only non-negative real numbers are admitted under the square roots, and only non-negative values are taken for the roots themselves.)

Topic: Algebra Difficulty: 3/6 Expected time: ~30 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 1959, Problem 2

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