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International Mathematics Competition
for University Students
2016

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IMC2016: Day 1, Problem 3

3. Let n be a positive integer. Also let a1,a2,,an and b1,b2,,bn be real numbers such that ai+bi>0 for i=1,2,,n. Prove that ni=1aibib2iai+bini=1aini=1bi(ni=1bi)2ni=1(ai+bi).

Proposed by Daniel Strzelecki, Nicolaus Copernicus University in Torún, Poland

Hint: Use the following variant of the Cauchy-Schwarz inequality: ni=1X2iYi(X1++Xn)2Y1++Yn(Y1,,Yn>0)

    

IMC
2016

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