2023 IOQM/Problem 16
Problem
The sides of a convex hexagon 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 , where has at least one red side, find the sum if the squares of digits of N.
Solution
Two triangle can be formed: and , 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: .
- No. of ways to colour the diagonals , and is .
- Number of ways that atleast one side of triangle is coloured red is
- Number of ways that at least one side of triangle is coloured red is
So number of colourings such that at least one side in triangles is red is
Answer: .
~Lakshya Pamecha (Inspired by A Mahajan Sir)