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Algebra functions

Source: First JBMO TST of France 2020, Problem 3

March 4, 2020
functionalgebra

Problem Statement

Let n be a nonzero natural number. We say about a function f ∶ R ⟶ R that is n-positive if, for any real numbers x1,x2,...,xnx_1, x_2,...,x_n with the property that x1+x2+...+xn=0x_1+x_2+...+x_n = 0, the inequality f(x1)+f(x2)+...+f(xn)=>0f(x_1)+f(x_2)+...+f(x_n)=>0 is true a) Is it true that any 2020-positive function is also 1010-positive? b) Is it true that any 1010-positive function is 2020-positive?
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