Lemma › Problems › Prove that for every real number x, x^2+1 ≥ 2x, with equality if and…
Classic olympiad problem · Algebra

Prove that for every real number x, x^2+1 ≥ 2x, with equality if and…

Prove that for every real number xx, x2+1≥2xx^2+1 \ge 2x, with equality if and only if x=1x=1.
Topic: Algebra Difficulty: 1/6 Expected time: ~10 min Progressive hints on Lemma: 2

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 2 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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