MathDB
Essentially Increasing Functions

Source: USAMO 2022/5

March 24, 2022
combinatoricsalgebrafunctionUSAMO

Problem Statement

A function f:R→Rf: \mathbb{R}\to \mathbb{R} is essentially increasing if f(s)≤f(t)f(s)\leq f(t) holds whenever s≤ts\leq t are real numbers such that f(s)≠0f(s)\neq 0 and f(t)≠0f(t)\neq 0.
Find the smallest integer kk such that for any 2022 real numbers x1,x2,…,x2022,x_1,x_2,\ldots , x_{2022}, there exist kk essentially increasing functions f1,…,fkf_1,\ldots, f_k such that f1(n)+f2(n)+⋯+fk(n)=xnfor every n=1,2,…2022.f_1(n) + f_2(n) + \cdots + f_k(n) = x_n\qquad \text{for every } n= 1,2,\ldots 2022.
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