Difference between revisions of "2023 IOQM/Problem 16"
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==Problem== | ==Problem== | ||
| − | The sides of a convex hexagon <math>A_1A_2A_3A_4A_5A_6</math> 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 <math>A_iA_jA_k</math>, where <math>1\ | + | The sides of a convex hexagon <math>A_1A_2A_3A_4A_5A_6</math> 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 <math>A_iA_jA_k</math>, where <math>1\le i<j<k\le 6</math> has at least one red side, find the sum if the squares of digits of N. |
| + | ==Solution== | ||
| + | Two triangle can be formed: <math>A_1A_3A_5</math> and <math>A_2A_4A_6</math>, 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: <math>A_1A_2A_6</math>. | ||
| + | \textbf{I}: <math>A_1A_3A_5</math> | ||
Revision as of 15:28, 1 May 2024
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: 
.
\textbf{I}:
