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IMC2016: Day 1, Problem 33. 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 n∑i=1aibi−b2iai+bi≤n∑i=1ai⋅n∑i=1bi−(n∑i=1bi)2n∑i=1(ai+bi). Proposed by Daniel Strzelecki, Nicolaus Copernicus University in Torún, Poland Hint: Use the following variant of the Cauchy-Schwarz inequality: n∑i=1X2iYi≥(X1+…+Xn)2Y1+…+Yn(Y1,…,Yn>0) | |||||||||
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