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Spanish Mathematical Olympiad Day 2 Problem 6

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April 4, 2016
national olympiadinequalitiesJensenrearrangement inequalitySpainESP

Problem Statement

Let n≥2n\geq 2 an integer. Find the least value of γ\gamma such that for any positive real numbers x1,x2,...,xnx_1,x_2,...,x_n with x1+x2+...+xn=1x_1+x_2+...+x_n=1 and any real y1+y2+...+yn=1y_1+y_2+...+y_n=1 and 0≤y1,y2,...,yn≤120\leq y_1,y_2,...,y_n\leq \frac{1}{2} the following inequality holds: x1x2...xn≤γ(x1y1+x2y2+...+xnyn)x_1x_2...x_n\leq \gamma \left(x_1y_1+x_2y_2+...+x_ny_n\right)
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