2023 IOQM/Problem 16

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: $A_1A_2A_6$ .

No. of ways to colour the diagonals $A_1A_4$ , $A_2A_5$ and $A_3A_6$ is $2^3$ .
Number of ways that atleast one side of triangle $A_1A_3A_5$ is coloured red is $^3C_1 \cdot2^2- ^3C_2\cdot2+^3C_3\cdot2^0=7$
Number of ways that at least one side of triangle $A_2A_4A_6$ is coloured red is $^3C_1 \cdot2^2- ^3C_2\cdot2+^3C_3\cdot2^0=7$

So number of colourings such that at least one side in triangles is red is $8\cdot7\cdot7=392.$

Answer: $3^2+9^2+2^2=\boxed{92}$ .

~Lakshya Pamecha (Inspired by A Mahajan Sir)

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2023 IOQM/Problem 16

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