Difference between revisions of "2023 IMO Problems/Problem 5"
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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> |
| + | // credit to vEnhance for the diagram (which was better than my original asy): | ||
| + | size(4cm); | ||
| + | pair X = dir(240); pair Y = dir(0); | ||
| + | path c = scale(0.5)*unitcircle; | ||
| + | int[] t = {0,0,2,2,3,0}; | ||
| + | for (int i=0; i<=5; ++i) { | ||
| + | for (int j=0; j<=i; ++j) { | ||
| + | filldraw(shift(i*X+j*Y)*c, (t[i]==j) ? lightred : white); | ||
| + | draw(shift(i*X+j*Y)*c); | ||
| + | } | ||
| + | } | ||
| + | draw((0,0)--(X+Y)--(2*X+Y)--(3*X+2*Y)--(4*X+2*Y)--(5*X+2*Y),linewidth(1.5)); | ||
| + | path q = (3,-3sqrt(3))--(-3,-3sqrt(3)); | ||
| + | draw(q,Arrows(TeXHead, 1)); | ||
| + | label("$n = 6$", q, S); | ||
| + | label("$n = 6$", q, S); | ||
| + | </asy> | ||
| + | |||
| + | In terms of <math>n</math>, find the greatest <math>k</math> such that in each Japanese triangle there is a ninja path containing at least <math>k</math> red circles. | ||
==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] | ||
Latest revision as of 08:46, 5 July 2024
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] // credit to vEnhance for the diagram (which was better than my original asy): size(4cm); pair X = dir(240); pair Y = dir(0); path c = scale(0.5)*unitcircle; int[] t = {0,0,2,2,3,0}; for (int i=0; i<=5; ++i) { for (int j=0; j<=i; ++j) { filldraw(shift(i*X+j*Y)*c, (t[i]==j) ? lightred : white); draw(shift(i*X+j*Y)*c); } } draw((0,0)--(X+Y)--(2*X+Y)--(3*X+2*Y)--(4*X+2*Y)--(5*X+2*Y),linewidth(1.5)); path q = (3,-3sqrt(3))--(-3,-3sqrt(3)); draw(q,Arrows(TeXHead, 1)); label("$n = 6$", q, S); label("$n = 6$", q, S); [/asy]](http://latex.artofproblemsolving.com/f/b/a/fba6b5fd48bca54a3103c12a3fd27b464a32abf2.png)
In terms of , find the greatest 
such that in each Japanese triangle there is a ninja path containing at least red circles.
Solution
https://www.youtube.com/watch?v=jZNIpapyGJQ [Video contains solutions to all day 2 problems]
See Also
| 2023 IMO (Problems) • Resources | ||
| Preceded by Problem 4 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 6 |
| All IMO Problems and Solutions | ||
