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The hexagon and the inequality

Source: IMO LongList 1982 - P21

March 18, 2011
inequalitiescombinatorics unsolvedcombinatoricsIMO Longlist

Problem Statement

All edges and all diagonals of regular hexagon A1A2A3A4A5A6A_1A_2A_3A_4A_5A_6 are colored blue or red such that each triangle AjAkAm,1≤j<k<m≤6A_jA_kA_m, 1 \leq j < k < m\leq 6 has at least one red edge. Let RkR_k be the number of red segments AkAj,(j≠k)A_kA_j, (j \neq k). Prove the inequality ∑k=16(2Rk−7)2≤54.\sum_{k=1}^6 (2R_k-7)^2 \leq 54.
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