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Find the smallest n

Source: VietnamMO2005,problem3

March 10, 2005
combinatorics proposedcombinatorics

Problem Statement

Let A1A2A3A4A5A6A7A8A_1A_2A_3A_4A_5A_6A_7A_8 be convex 8-gon (no three diagonals concruent). The intersection of arbitrary two diagonals will be called "button".Consider the convex quadrilaterals formed by four vertices of A1A2A3A4A5A6A7A8A_1A_2A_3A_4A_5A_6A_7A_8 and such convex quadrilaterals will be called "sub quadrilaterals".Find the smallest nn satisfying: We can color n "button" such that for all i,k∈{1,2,3,4,5,6,7,8},i≠k,s(i,k)i,k \in\{1,2,3,4,5,6,7,8\},i\neq k,s(i,k) are the same where s(i,k)s(i,k) denote the number of the "sub quadrilaterals" has Ai,AkA_i,A_k be the vertices and the intersection of two its diagonals is "button".
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