Problem

Source: RMO 2024 Q4

Tags: algebra, inequalities, Real



Let $a_1,a_2,a_3,a_4$ be real numbers such that $a_1^2 + a_2^2 + a_3^2 + a_4^2 = 1$. Show that there exist $i,j$ with $ 1 \leq i < j \leq 4$, such that $(a_i - a_j)^2 \leq \frac{1}{5}$.