ΒΆ 2012 USAMO Day 2 Problems and Solutions
Individual Problems and Solutions
For problems and detailed solutions to each of the 2012 USAMO Day 2 problems, please refer below:
Problem 4: Find all functions f:Z+βZ+(where Z+is the set of positive integers) such that f(n!)=f(n) ! for all positive integers n and such that mβn divides f(m)βf(n) for all distinct positive integers m,n.
Solution:
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Problem 5: Let P be a point in the plane of β³ABC, and Ξ³ a line passing through P. Let Aβ²,Bβ²,Cβ² be the points where the reflections of lines PA,PB,PC with respect to Ξ³ intersect lines BC,AC,AB, respectively. Prove that Aβ²,Bβ²,Cβ² are collinear.
Solution:
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Problem 6: For integer nβ₯2, let x1β,x2β,β¦,xnβ be real numbers satisfying
x1β+x2β+β―+xnβ=0, and x12β+x22β+β―+xn2β=1
For each subset Aβ{1,2,β¦,n}, define
SAβ=iβAββxiβ
(If A is the empty set, then SAβ=0.)
Prove that for any positive number Ξ», the number of sets A satisfying SAββ₯Ξ» is at most 2nβ3/Ξ»2. For what choices of x1β,x2β,β¦,xnβ,Ξ» does equality hold?
Solution:
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The problems on this page are the property of the MAA's American Mathematics Competitions