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Bounded functions on reals

Source: 2015 China TST 2 Day 2 Q1

March 19, 2015
functionalgebraalgebra proposedcombinatorics

Problem Statement

Let nn be a positive integer, let f1(x),…,fn(x)f_1(x),\ldots,f_n(x) be nn bounded real functions, and let a1,…,ana_1,\ldots,a_n be nn distinct reals. Show that there exists a real number xx such that ∑i=1nfi(x)−∑i=1nfi(x−ai)<1\sum^n_{i=1}f_i(x)-\sum^n_{i=1}f_i(x-a_i)<1.
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