Problem

Source: ELMO 2013/1, by Ray Li; also Shortlist C3

Tags: pigeonhole principle, probability, expected value, combinatorics, Elmo, 2013



Let $a_1,a_2,...,a_9$ be nine real numbers, not necessarily distinct, with average $m$. Let $A$ denote the number of triples $1 \le i < j < k \le 9$ for which $a_i + a_j + a_k \ge 3m$. What is the minimum possible value of $A$? Proposed by Ray Li