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. | ||
| + | |||
| + | [asy] | ||
| + | unitsize(7mm); | ||
| + | path q=(3,-3sqrt(3))--(-3,-3sqrt(3)); | ||
| + | filldraw(shift(0*dir(240)+0*dir(0))*scale(0.5)*unitcircle,lightred); | ||
| + | filldraw(shift(1*dir(240)+0*dir(0))*scale(0.5)*unitcircle,lightred); | ||
| + | draw(shift(1*dir(240)+1*dir(0))*scale(0.5)*unitcircle); | ||
| + | draw(shift(2*dir(240)+0*dir(0))*scale(0.5)*unitcircle); | ||
| + | draw(shift(2*dir(240)+1*dir(0))*scale(0.5)*unitcircle); | ||
| + | filldraw(shift(2*dir(240)+2*dir(0))*scale(0.5)*unitcircle,lightred); | ||
| + | draw(shift(3*dir(240)+0*dir(0))*scale(0.5)*unitcircle); | ||
| + | draw(shift(3*dir(240)+1*dir(0))*scale(0.5)*unitcircle); | ||
| + | filldraw(shift(3*dir(240)+2*dir(0))*scale(0.5)*unitcircle,lightred); | ||
| + | draw(shift(3*dir(240)+3*dir(0))*scale(0.5)*unitcircle); | ||
| + | draw(shift(4*dir(240)+0*dir(0))*scale(0.5)*unitcircle); | ||
| + | draw(shift(4*dir(240)+1*dir(0))*scale(0.5)*unitcircle); | ||
| + | draw(shift(4*dir(240)+2*dir(0))*scale(0.5)*unitcircle); | ||
| + | filldraw(shift(4*dir(240)+3*dir(0))*scale(0.5)*unitcircle,lightred); | ||
| + | draw(shift(4*dir(240)+4*dir(0))*scale(0.5)*unitcircle); | ||
| + | filldraw(shift(5*dir(240)+0*dir(0))*scale(0.5)*unitcircle,lightred); | ||
| + | draw(shift(5*dir(240)+1*dir(0))*scale(0.5)*unitcircle); | ||
| + | draw(shift(5*dir(240)+2*dir(0))*scale(0.5)*unitcircle); | ||
| + | draw(shift(5*dir(240)+3*dir(0))*scale(0.5)*unitcircle); | ||
| + | draw(shift(5*dir(240)+4*dir(0))*scale(0.5)*unitcircle); | ||
| + | draw(shift(5*dir(240)+5*dir(0))*scale(0.5)*unitcircle); | ||
| + | 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)); | ||
| + | draw(q,Arrows(TeXHead, 1)); | ||
| + | label("<math>n = 6</math>", q, S); | ||
| + | [/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 
be a positive integer. A Japanese triangle consists of 
circles arranged in an equilateral triangular shape such that for each 
, 
, 
, , the 
row contains exactly 
circles, exactly one of which is coloured red. A ninja path in a Japanese triangle is a sequence of 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 
, along with a ninja path in that triangle containing two red circles.
[asy]
unitsize(7mm);
path q=(3,-3sqrt(3))--(-3,-3sqrt(3));
filldraw(shift(0*dir(240)+0*dir(0))*scale(0.5)*unitcircle,lightred);
filldraw(shift(1*dir(240)+0*dir(0))*scale(0.5)*unitcircle,lightred);
draw(shift(1*dir(240)+1*dir(0))*scale(0.5)*unitcircle);
draw(shift(2*dir(240)+0*dir(0))*scale(0.5)*unitcircle);
draw(shift(2*dir(240)+1*dir(0))*scale(0.5)*unitcircle);
filldraw(shift(2*dir(240)+2*dir(0))*scale(0.5)*unitcircle,lightred);
draw(shift(3*dir(240)+0*dir(0))*scale(0.5)*unitcircle);
draw(shift(3*dir(240)+1*dir(0))*scale(0.5)*unitcircle);
filldraw(shift(3*dir(240)+2*dir(0))*scale(0.5)*unitcircle,lightred);
draw(shift(3*dir(240)+3*dir(0))*scale(0.5)*unitcircle);
draw(shift(4*dir(240)+0*dir(0))*scale(0.5)*unitcircle);
draw(shift(4*dir(240)+1*dir(0))*scale(0.5)*unitcircle);
draw(shift(4*dir(240)+2*dir(0))*scale(0.5)*unitcircle);
filldraw(shift(4*dir(240)+3*dir(0))*scale(0.5)*unitcircle,lightred);
draw(shift(4*dir(240)+4*dir(0))*scale(0.5)*unitcircle);
filldraw(shift(5*dir(240)+0*dir(0))*scale(0.5)*unitcircle,lightred);
draw(shift(5*dir(240)+1*dir(0))*scale(0.5)*unitcircle);
draw(shift(5*dir(240)+2*dir(0))*scale(0.5)*unitcircle);
draw(shift(5*dir(240)+3*dir(0))*scale(0.5)*unitcircle);
draw(shift(5*dir(240)+4*dir(0))*scale(0.5)*unitcircle);
draw(shift(5*dir(240)+5*dir(0))*scale(0.5)*unitcircle);
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));
draw(q,Arrows(TeXHead, 1));
label("", q, S);
[/asy]
Solution
https://www.youtube.com/watch?v=jZNIpapyGJQ [Video contains solutions to all day 2 problems]
