China Southeastern Mathematical Olympiad

Given $n$ points $P_1,P_2,\dots ,P_n$ on a plane, let $M$ be any point on segment $AB$ on the plane. Denote by $|P_{i}M|$ the distance between $P_i$ and $M$, $i=1,2,3,\dots ,n$. Prove that $$\sum _{i=1}^n|P_{i}M|\le \max \left\{\sum _{i=1}^n|P_{i}A|,\sum _{i=1}^n|P_{i}B|\right\}.$$ (posed by Jin Mengwei)