## IMC 2016 – Day 1 – Problem 3

**Problem 3.** Let be a positive integer. Also let and be reap numbers such that for . Prove that

*Sketch of proof:* The strange condition that is too familiar to the hypothesis needed for the following variant of the Cauchy-Schwarz inequality to work: let be real numbers and be positive real numbers. Then

Subtract from each side of the members of the above inequalities. We obtain

Now we are done, since the resulting inequality is just an application of the above variant of the Cauchy-Schwarz inequality for and .

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Categories: Inequalities, Olympiad, Problem Solving, Uncategorized
Algebra, cauchy, Inequalities

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