2023 IOQM/Problem 16

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Problem

The sides of a convex hexagon $A_1A_2A_3A_4A_5A_6$ are coloured red. Each of the diagonal of the hexagon is coloured red or blue. If N is the number of colourings suhch that every triangle $A_iA_jA_k$, where $1\le i<j<k\le 6$ has at least one red side, find the sum if the squares of digits of N.

Solution

Two triangle can be formed: $A_1A_3A_5$ and $A_2A_4A_6$, which might or might not have red colouring, rest of the triangle will have at least 1 red colouring because they will be a part of the hexagon, eg: $A_1A_2A_6$. \textbf{I}: $A_1A_3A_5$