ΒΆ 2009 USAMO Day 2 Problems and Solutions
Problem Set Workbook
Access the downloadable workbook for 2009 USAMO Day 2 problems here.
Discussion Forum
Engage in discussion about the 2009 USAMO Day 2 math contest by visiting [Random Math USAMO Day 2 2009 Forum](https://forums.randommath.com/c/tournaments/MAA/2009-usamo-day 2)
Individual Problems and Solutions
For problems and detailed solutions to each of the 2009 USAMO Day 2 problems, please refer below:
Problem 4: For nβ₯2 let a1β,a2β,β¦,anβ be positive real numbers such that
(a1β+a2β+β―+anβ)(a1β1β+a2β1β+β―+anβ1β)β€(n+21β)2.
Prove that max(a1β,a2β,β¦,anβ)β€4min(a1β,a2β,β¦,anβ).
Solution:
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Problem 5: Trapezoid ABCD, with ABβ₯CD, is inscribed in circle Ο and point G lies inside triangle BCD. Rays AG and BG meet Ο again at points P and Q, respectively. Let the line through G parallel to AB intersect BD and BC at points R and S, respectively. Prove that quadrilateral PQRS is cyclic if and only if BG bisects β CBD.
Solution:
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Problem 6: Let s1β,s2β,s3β,β¦ be an infinite, nonconstant sequence of rational numbers, meaning it is not the case that s1β=s2β=s3β=β―. Suppose that t1β,t2β,t3β,β¦ is also an infinite, nonconstant sequence of rational numbers with the property that (siββsjβ)(tiββtjβ) is an integer for all i and j. Prove that there exists a rational number r such that (siββsjβ)r and (tiββtjβ)/r are integers for all i and j.
Solution:
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The problems on this page are the property of the MAA's American Mathematics Competitions