Difference between revisions of "2023 IOQM/Problem 16"
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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. | 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== | ==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>. | 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>. | ||
| − | \ | + | *No. of ways to colour the diagonals <math>A_1A_4</math>, <math>A_2A_5</math> and <math>A_3A_6</math> is <math>2^3</math>. |
| + | *Number of ways that atleast one side of triangle <math>A_1A_3A_5</math> is coloured red is <math>^3C_1 \cdot2^2- ^3C_2\cdot2+^3C_3\cdot2^0=7</math> | ||
| + | *Number of ways that at least one side of triangle <math>A_2A_4A_6</math> is coloured red is <math>^3C_1 \cdot2^2- ^3C_2\cdot2+^3C_3\cdot2^0=7</math> | ||
| + | So number of colourings such that at least one side in triangles is red is <math>8\cdot7\cdot7=392.</math> | ||
| + | |||
| + | Answer: <math>3^2+9^2+2^2=\boxed{92}</math>. | ||
| + | |||
| + | ~Lakshya Pamecha (Inspired by A Mahajan Sir) | ||
Latest revision as of 01:56, 4 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: 
.
- 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
, 
and 
is 
.
So number of colourings such that at least one side in triangles is red is 
Answer: 
.
~Lakshya Pamecha (Inspired by A Mahajan Sir)
