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Romanian Masters in mathematics 2010 Day 1 Problem 2

Source:

April 25, 2010
functioninequalitiesCauchy Inequalityabsolute valuealgebra proposedalgebra

Problem Statement

For each positive integer nn, find the largest real number CnC_n with the following property. Given any nn real-valued functions f1(x),f2(x),⋯ ,fn(x)f_1(x), f_2(x), \cdots, f_n(x) defined on the closed interval 0≤x≤10 \le x \le 1, one can find numbers x1,x2,⋯xnx_1, x_2, \cdots x_n, such that 0≤xi≤10 \le x_i \le 1 satisfying ∣f1(x1)+f2(x2)+⋯fn(xn)−x1x2⋯xn∣≥Cn|f_1(x_1)+f_2(x_2)+\cdots f_n(x_n)-x_1x_2\cdots x_n| \ge C_n
Marko Radovanović, Serbia
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