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Difference between revisions of "2023 IMO Problems/Problem 5"
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==Problem==
 
==Problem==
 
Let <math>n</math> be a positive integer. A Japanese triangle consists of <math>1 + 2 + \dots + n</math> circles arranged in an equilateral triangular shape such that for each <math>i = 1</math>, <math>2</math>, <math>\dots</math>, <math>n</math>, the <math>i^{th}</math> row contains exactly <math>i</math> circles, exactly one of which is coloured red. A ninja path in a Japanese triangle is a sequence of <math>n</math> circles obtained by starting in the top row, then repeatedly going from a circle to one of the two circles immediately below it and finishing in the bottom row. Here is an example of a Japanese triangle with <math>n = 6</math>, along with a ninja path in that triangle containing two red circles.
 
Let <math>n</math> be a positive integer. A Japanese triangle consists of <math>1 + 2 + \dots + n</math> circles arranged in an equilateral triangular shape such that for each <math>i = 1</math>, <math>2</math>, <math>\dots</math>, <math>n</math>, the <math>i^{th}</math> row contains exactly <math>i</math> circles, exactly one of which is coloured red. A ninja path in a Japanese triangle is a sequence of <math>n</math> circles obtained by starting in the top row, then repeatedly going from a circle to one of the two circles immediately below it and finishing in the bottom row. Here is an example of a Japanese triangle with <math>n = 6</math>, along with a ninja path in that triangle containing two red circles.
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[asy]
 
−
unitsize(7mm);
 
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path q=(3,-3sqrt(3))--(-3,-3sqrt(3));
 
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filldraw(shift(0*dir(240)+0*dir(0))*scale(0.5)*unitcircle,lightred);
 
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filldraw(shift(1*dir(240)+0*dir(0))*scale(0.5)*unitcircle,lightred);
 
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draw(shift(1*dir(240)+1*dir(0))*scale(0.5)*unitcircle);
 
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draw(shift(2*dir(240)+0*dir(0))*scale(0.5)*unitcircle);
 
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draw(shift(2*dir(240)+1*dir(0))*scale(0.5)*unitcircle);
 
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filldraw(shift(2*dir(240)+2*dir(0))*scale(0.5)*unitcircle,lightred);
 
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draw(shift(3*dir(240)+0*dir(0))*scale(0.5)*unitcircle);
 
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draw(shift(3*dir(240)+1*dir(0))*scale(0.5)*unitcircle);
 
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filldraw(shift(3*dir(240)+2*dir(0))*scale(0.5)*unitcircle,lightred);
 
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draw(shift(3*dir(240)+3*dir(0))*scale(0.5)*unitcircle);
 
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draw(shift(4*dir(240)+0*dir(0))*scale(0.5)*unitcircle);
 
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draw(shift(4*dir(240)+1*dir(0))*scale(0.5)*unitcircle);
 
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draw(shift(4*dir(240)+2*dir(0))*scale(0.5)*unitcircle);
 
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filldraw(shift(4*dir(240)+3*dir(0))*scale(0.5)*unitcircle,lightred);
 
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draw(shift(4*dir(240)+4*dir(0))*scale(0.5)*unitcircle);
 
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filldraw(shift(5*dir(240)+0*dir(0))*scale(0.5)*unitcircle,lightred);
 
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draw(shift(5*dir(240)+1*dir(0))*scale(0.5)*unitcircle);
 
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draw(shift(5*dir(240)+2*dir(0))*scale(0.5)*unitcircle);
 
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draw(shift(5*dir(240)+3*dir(0))*scale(0.5)*unitcircle);
 
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draw(shift(5*dir(240)+4*dir(0))*scale(0.5)*unitcircle);
 
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draw(shift(5*dir(240)+5*dir(0))*scale(0.5)*unitcircle);
 
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draw((0,0)--(1/2,-sqrt(3)/2)--(0,-sqrt(3))--(1/2,-3sqrt(3)/2)--(0,-2sqrt(3))--(-1/2, -5sqrt(3)/2),linewidth(1.5));
 
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draw(q,Arrows(TeXHead, 1));
 
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label("<math>n = 6</math>", q, S);
 
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[/asy]
 
  
 
==Solution==
 
==Solution==
 
https://www.youtube.com/watch?v=jZNIpapyGJQ [Video contains solutions to all day 2 problems]
 
https://www.youtube.com/watch?v=jZNIpapyGJQ [Video contains solutions to all day 2 problems]

Revision as of 07:12, 14 July 2023

Problem

Let $n$ be a positive integer. A Japanese triangle consists of $1 + 2 + \dots + n$ circles arranged in an equilateral triangular shape such that for each $i = 1$, $2$, $\dots$, $n$, the $i^{th}$ row contains exactly $i$ circles, exactly one of which is coloured red. A ninja path in a Japanese triangle is a sequence of $n$ circles obtained by starting in the top row, then repeatedly going from a circle to one of the two circles immediately below it and finishing in the bottom row. Here is an example of a Japanese triangle with $n = 6$, along with a ninja path in that triangle containing two red circles.

Solution

https://www.youtube.com/watch?v=jZNIpapyGJQ [Video contains solutions to all day 2 problems]

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