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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+ | [Image to be inserted; also available in solution video] | ||
==Solution== | ==Solution== | ||
+ | <math>k=\lfloor\log_2(n)\rfloor+1</math> | ||
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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] | ||
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+ | ==See Also== | ||
+ | |||
+ | {{IMO box|year=2023|num-b=4|num-a=6}} |
Latest revision as of 17:38, 30 April 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.
[Image to be inserted; also available in solution video]
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 |