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Difference between revisions of "2023 IOQM/Problem 16"

(Created page with "==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...")
 
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==Problem==  
 
==Problem==  
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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\ge i\ge j\ge k\ge 6</math> has at least one red side, find the sum if the squares of digits of N.
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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.
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==Solution==
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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>.
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*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>.
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*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>
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*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>
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So number of colourings such that at least one side in triangles is red is <math>8\cdot7\cdot7=392.</math>
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Answer: <math>3^2+9^2+2^2=\boxed{92}</math>.
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~Lakshya Pamecha (Inspired by A Mahajan Sir)

Latest revision as of 02:56, 4 May 2024

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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