MathDB

A5

Part of 2008 Balkan MO Shortlist

Problems(1)

Find f and x so that the equality holds

Source: Balkan MO ShortList 2008 A5

4/6/2020
Consider an integer n≥1n \geq 1, a1,a2,…,ana_1,a_2, \ldots , a_n real numbers in [−1,1][-1,1] satisfying \begin{align*}a_1+a_2+\ldots +a_n=0 \end{align*} and a function f:[−1,1]↦Rf: [-1,1] \mapsto \mathbb{R} such \begin{align*} \mid f(x)-f(y) \mid \le \mid x-y \mid \end{align*} for every x,y∈[−1,1]x,y \in [-1,1]. Prove \begin{align*} \left| f(x) - \frac{f(a_1) +f(a_2) + \ldots + f(a_n)}{n} \right| \le 1 \end{align*} for every xx ∈[−1,1]\in [-1,1]. For a given sequence a1,a2,…,ana_1,a_2, \ldots ,a_n, Find ff and xx so hat the equality holds.