ISL 1982 Problem 21

All edges and all diagonals of regular hexagon $A_1A_2A_3A_4A_5A_6$ are colored blue or red such that each triangle $A_jA_kA_m, 1 \leq j < k < m\leq 6$ has at least one red edge. Let $R_k$ be the number of red segments $A_kA_j, (j \neq k)$ . Prove the inequality $$\sum_{k=1}^6 (2R_k-7)^2 \leq 54.$$