Difference between revisions of "2023 AIME I Problems/Problem 1"

(Solution 1)
(Solution 2)
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Use combinatorics
 
Use combinatorics
  
===Solution 2===
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===Solution 2 (constructive)===
Something else
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This problem is equivalent to solving for the probability that no man is sitting diametrically opposite to another man. We can simply just construct this.
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We first place the <math>1</math>st man anywhere on the table, now we have to place the <math>2</math>nd man somewhere around the table such that he is not diametrically opposite to the first man. This can happen with a probability of <math>\frac{12}{13}</math> because there are <math>13</math> available seats, and <math>12</math> of them are not opposite to the first man.
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We do the same thing for the <math>3</math>rd man, finding a spot for him such that he is not opposite to the other <math>2</math> men, which would happen with a probability of <math>\frac{10}{12}</math> using similar logic. Doing this for the <math>4</math>th and <math>5</math>th men, we get probabilities of <math>\frac{8}{11}</math> and <math>\frac{6}{10}</math> respectively.
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Multiplying these probabilities, we get, <cmath>\frac{12}{13}\cdot\frac{10}{12}\cdot\frac{8}{11}\cdot\frac{6}{10}=\frac{48}{143}\longrightarrow\boxed{191}.</cmath>
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~s214425
  
 
==See also==
 
==See also==
 
{{AIME box|year=2023|before=First Question|num-a=2|n=I}}
 
{{AIME box|year=2023|before=First Question|num-a=2|n=I}}

Revision as of 12:29, 8 February 2023

Problem

Note: This is not official (I am posting this because the AIME I is officially over). Please post official problem statement after it is released.

There are five men and nine women randomly arranged in a circle. Let $\frac{x}{y}$ be the probability that every man stands diametrically opposite from a woman.

Find $x+y$.

Solutions

Solution 1

Use combinatorics

Solution 2 (constructive)

This problem is equivalent to solving for the probability that no man is sitting diametrically opposite to another man. We can simply just construct this.

We first place the $1$st man anywhere on the table, now we have to place the $2$nd man somewhere around the table such that he is not diametrically opposite to the first man. This can happen with a probability of $\frac{12}{13}$ because there are $13$ available seats, and $12$ of them are not opposite to the first man.

We do the same thing for the $3$rd man, finding a spot for him such that he is not opposite to the other $2$ men, which would happen with a probability of $\frac{10}{12}$ using similar logic. Doing this for the $4$th and $5$th men, we get probabilities of $\frac{8}{11}$ and $\frac{6}{10}$ respectively.

Multiplying these probabilities, we get, \[\frac{12}{13}\cdot\frac{10}{12}\cdot\frac{8}{11}\cdot\frac{6}{10}=\frac{48}{143}\longrightarrow\boxed{191}.\]

~s214425

See also

2023 AIME I (ProblemsAnswer KeyResources)
Preceded by
First Question
Followed by
Problem 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions