Problem

Source: Iran 2024 3rd round algebra exam P3

Tags: algebra



An integer number $n\geq 2$ and real numbers $x_1<x_2<\cdots < x_n$ are given. $f: \mathbb R \to \mathbb R$ is a function defined as $$ f(x) = \left | \dfrac{(x-x_2)(x-x_3)\cdots (x-x_n)}{(x_1-x_2)(x_1-x_3)\cdots (x_1-x_n)} \right | + \cdots + \left | \dfrac{(x-x_1)(x-x_2)\cdots (x-x_{n-1})}{(x_n-x_1)(x_n-x_2)\cdots (x_n-x_{n-1})} \right |. $$Prove that there exists $i\in \{1,2,\cdots,n-1\}$ such that for all $x\in (x_i,x_{i+1})$ one has $f(x)< \sqrt n$. Proposed by Navid Safaei