Classic olympiad problem · Algebra
Let a,b,c>0 with abc=1. Prove a+b+c ≥ tfrac1a+tfrac1b+tfrac1c is…
Let with . Prove is FALSE in general, and determine when equality holds.
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 levelMore Algebra problems
- IMO 1964, Problem 2IMO 1964, Problem 2 · difficulty 5/6
- Functional equationClassic (functional equation) · difficulty 4/6
- AM–GMClassic (AM–GM) · difficulty 3/6
- All 81 problems →