ΒΆ 2011 USAJMO Day 1 Problems and Solutions
Individual Problems and Solutions
For problems and detailed solutions to each of the 2011 USAJMO Day 1 problems, please refer below:
Problem 1:
Find, with proof, all positive integers n for which 2n+12n+2011n is a perfect square.
Solution:
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Problem 2: Let a,b,c be positive real numbers such that a2+b2+c2+(a+b+c)2β€4. Prove that
(a+b)2ab+1β+(b+c)2bc+1β+(c+a)2ca+1ββ₯3
Solution:
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Problem 3: For a point P=(a,a2) in the coordinate plane, let β(P) denote the line passing through P with slope 2a. Consider the set of triangles with vertices of the form P1β=(a1β,a12β), P2β=(a2β,a22β),P3β=(a3β,a32β), such that the intersections of the lines β(P1β),β(P2β),β(P3β) form an equilateral triangle Ξ. Find the locus of the center of Ξ as P1βP2βP3β ranges over all such triangles.
Solution:
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The problems on this page are the property of the MAA's American Mathematics Competitions