¶ 2012 USAJMO Day 1 Problems and Solutions
Individual Problems and Solutions
For problems and detailed solutions to each of the 2012 USAJMO Day 1 problems, please refer below:
Problem 1:Given a triangle ABC, let P and Q be points on segments AB and AC, respectively, such that AP=AQ. Let S and R be distinct points on segment BC such that S lies between B and R,∠BPS=∠PRS, and ∠CQR=∠QSR. Prove that P,Q,R,S are concyclic (in other words, these four points lie on a circle).
Solution:
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Problem 2:Find all integers n≥3 such that among any n positive real numbers a1,a2,…,an with
max(a1,a2,…,an)≤n⋅min(a1,a2,…,an)
there exist three that are the side lengths of an acute triangle.
Solution:
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Problem 3: Let a,b,c be positive real numbers. Prove that
5a+ba3+3b3+5b+cb3+3c3+5c+ac3+3a3≥32(a2+b2+c2)
Solution:
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The problems on this page are the property of the MAA's American Mathematics Competitions