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Difference between revisions of "2023 AMC 12A Problems/Problem 13"

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==Problem==
 
==Problem==
  
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In a table tennis tournament every participant played every other participant exactly once. Although there were twice as many right-handed players as left-handed players, the number of games won by left-handed players was <math>40\%</math> more than the number of games won by right-handed players. (There were no ties and no ambidextrous players.) What is the total number of games played?
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In a table tennis tournament, every participant played every other participant exactly once. Although there were twice as many right-handed players as left-handed players, the number of games won by left-handed players was <math>40\%</math> more than the number of games won by right-handed players. (There were no ties and no ambidextrous players.) What is the total number of games played?
  
 
<math>\textbf{(A) }15\qquad\textbf{(B) }36\qquad\textbf{(C) }45\qquad\textbf{(D) }48\qquad\textbf{(E) }66</math>
 
<math>\textbf{(A) }15\qquad\textbf{(B) }36\qquad\textbf{(C) }45\qquad\textbf{(D) }48\qquad\textbf{(E) }66</math>
  
 
==Solution 1==
 
==Solution 1==
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We know that the total amount of games must be the sum of games won by left and right handed players. Then, we can write <math>g = l + r</math>, and since <math>l = 1.4r</math>, <math>g = 2.4r</math>. Given that <math>r</math> and <math>g</math> are both integers, <math>g/2.4</math> also must be an integer. From here we can see that <math>g</math> must be divisible by 12, leaving only answers B and D. Now we know the formula for how many games are played in this tournament is <math>n(n-1)/2</math>, the sum of the first <math>n-1</math> triangular numbers. Now, setting 36 and 48 equal to the equation will show that two consecutive numbers must have a product of 72 or 96. Clearly, <math>72=8*9</math>, so the answer is <math>\boxed{\textbf{(B) }36}</math>.
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We know that the total amount of games must be the sum of games won by left and right handed players. Then, we can write <math>g = l + r</math>, and since <math>l = 1.4r</math>, <math>g = 2.4r</math>. Given that <math>r</math> and <math>g</math> are both integers, <math>g/2.4</math> also must be an integer. From here we can see that <math>g</math> must be divisible by 12, leaving only answers B and D. Now we know the formula for how many games are played in this tournament is <math>n(n-1)/2</math>, the sum of the first <math>n-1</math> numbers. Now, setting 36 and 48 equal to the equation will show that two consecutive numbers must have a product of 72 or 96. Clearly, <math>72=8*9</math>, so the answer is <math>\boxed{\textbf{(B) }36}</math>.
  
 
~~ Antifreeze5420
 
~~ Antifreeze5420
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==Solution 2==
 
==Solution 2==
 
First, we know that every player played every other player, so there's a total of <math>\dbinom{n}{2}</math> games since each pair of players forms a bijection to a game. Therefore, that rules out D. Also, if we assume the right-handed players won a total of <math>x</math> games, the left-handed players must have won a total of <math>\dfrac{7}{5}x</math> games, meaning that the total number of games played was <math>\dfrac{12}{5}x</math>. Thus, the total number of games must be divisible by <math>12</math>. Therefore leaving only answer choices B and D. Since answer choice D doesn't satisfy the first condition, the only answer that satisfies both conditions is <math>\boxed{\textbf{(B) }36}</math>
 
First, we know that every player played every other player, so there's a total of <math>\dbinom{n}{2}</math> games since each pair of players forms a bijection to a game. Therefore, that rules out D. Also, if we assume the right-handed players won a total of <math>x</math> games, the left-handed players must have won a total of <math>\dfrac{7}{5}x</math> games, meaning that the total number of games played was <math>\dfrac{12}{5}x</math>. Thus, the total number of games must be divisible by <math>12</math>. Therefore leaving only answer choices B and D. Since answer choice D doesn't satisfy the first condition, the only answer that satisfies both conditions is <math>\boxed{\textbf{(B) }36}</math>
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A simpler way is to know that the result must be a number in the form <math>\dbinom{n}{2}</math> games. This eliminates D. Then write a expression that equals the answer, 1.4x+x=n games total. Only 48 and 36 satisfy so 36 is the answer which is <math>\boxed{\textbf{(B) }36}</math>
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~breakingbread
  
 
==Solution 3==
 
==Solution 3==
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~MathFun1000
 
~MathFun1000
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==Solution 6 (Cheese 2)==
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Call the number of games x, the number of games won by right handed player R, and by left handed players L. L+R=x and L=1.4 R. Therefore, x/2.4 must be an integer, which leaves only 36 and 48. Then, note x must be half the product of two consecutive numbers (the number of possible combinations of two people) and we can eliminate 48.
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~meikh_neiht
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How is this cheese... it is literally the first solution but scaled down
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-Shreyansh
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==Solution 7==
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Let the number of people be <math>n</math>. This means that there will be <math>\frac{n(n-1)}{2}</math> games played. We can now eliminate <math>\textbf{(D)}</math> because it's not a triangular number. The problem states that there are twice as many right handed people as left handed people with no ambidextrous people. Based on this, we can say that the number of people (or <math>n</math>) has to be divisible by 3. Thus, the only answer choice that fits the conditions above is answer <math>\boxed{\textbf{(B) }36}</math>.
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~ROGER8432V3
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==Video Solution by Little Fermat==
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https://youtu.be/h2Pf2hvF1wE?si=W4Ad9Bm3vhcTBB4G&t=3440
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~little-fermat
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==Video Solution by Math-X ==
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https://youtu.be/GP-DYudh5qU?si=DCXVk-iVlqWT-6bS&t=4158
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~Math-X
  
 
== Video Solution by Power Solve ==
 
== Video Solution by Power Solve ==
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~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
 
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
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==Video Solution by Math-X ==
 
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https://youtu.be/N2lyYRMuZuk?si=7IEmYDGBHi5i_XKt&t=1452
 
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~Math-X
 
  
 
==Video Solution by SpreadTheMathLove==
 
==Video Solution by SpreadTheMathLove==

Latest revision as of 10:12, 23 September 2025

The following problem is from both the 2023 AMC 10A #16 and 2023 AMC 12A #13, so both problems redirect to this page.

Problem

In a table tennis tournament, every participant played every other participant exactly once. Although there were twice as many right-handed players as left-handed players, the number of games won by left-handed players was $40\%$ more than the number of games won by right-handed players. (There were no ties and no ambidextrous players.) What is the total number of games played?

$\textbf{(A) }15\qquad\textbf{(B) }36\qquad\textbf{(C) }45\qquad\textbf{(D) }48\qquad\textbf{(E) }66$

Solution 1

We know that the total amount of games must be the sum of games won by left and right handed players. Then, we can write $g = l + r$, and since $l = 1.4r$, $g = 2.4r$. Given that $r$ and $g$ are both integers, $g/2.4$ also must be an integer. From here we can see that $g$ must be divisible by 12, leaving only answers B and D. Now we know the formula for how many games are played in this tournament is $n(n-1)/2$, the sum of the first $n-1$ numbers. Now, setting 36 and 48 equal to the equation will show that two consecutive numbers must have a product of 72 or 96. Clearly, $72=8*9$, so the answer is $\boxed{\textbf{(B) }36}$.

~~ Antifreeze5420

Solution 2

First, we know that every player played every other player, so there's a total of $\dbinom{n}{2}$ games since each pair of players forms a bijection to a game. Therefore, that rules out D. Also, if we assume the right-handed players won a total of $x$ games, the left-handed players must have won a total of $\dfrac{7}{5}x$ games, meaning that the total number of games played was $\dfrac{12}{5}x$. Thus, the total number of games must be divisible by $12$. Therefore leaving only answer choices B and D. Since answer choice D doesn't satisfy the first condition, the only answer that satisfies both conditions is $\boxed{\textbf{(B) }36}$


A simpler way is to know that the result must be a number in the form $\dbinom{n}{2}$ games. This eliminates D. Then write a expression that equals the answer, 1.4x+x=n games total. Only 48 and 36 satisfy so 36 is the answer which is $\boxed{\textbf{(B) }36}$

~breakingbread

Solution 3

Let $r$ be the amount of games the right-handed won. Since the left-handed won $1.4r$ games, the total number of games played can be expressed as $(2.4)r$, or $12/5r$, meaning that the answer is divisible by 12. This brings us down to two answer choices, $B$ and $D$. We note that the answer is some number $x$ choose $2$. This means the answer is in the form $x(x-1)/2$. Since answer choice D gives $48 = x(x-1)/2$, and $96 = x(x-1)$ has no integer solutions, we know that $\boxed{\textbf{(B) }36}$ is the only possible choice.

Solution 4

Here is the rigid way to prove that $36$ is the only answer. Let the number of left-handed players be $n$, so the number of right-handed players is $2n$. The number of games won by the left-handed players comes in two ways:

  • The games played by two left-left pairs, which is $\frac{n(n-1)}{2}$, and
  • The games played by left-right pairs, which we'll call $x$.

Note that $x\leq 2n^2,$ which is the total number of games played by left-right pairs. Using the same logic for right-right pairs and right-left pairs, we have that \[\frac{\frac{n(n-1)}{2}+x}{\frac{2n(2n-1)}{2}+2n^2-x}=1.4,\] which gives \[x=\frac{17n^2}{8}-\frac{3n}{8}.\] We know that $x\leq 2n^2$, applying that becomes \begin{align*} \frac{17n^2}{8}-\frac{3n}{8} &\le 2n^2 \\ 17n^2 - 3n &\le 16n^2 \\ n^2 - 3n &\le 0 \\ n &\le 3. \end{align*} (We can safely divide by $n$ because it must be positive). So the total number of players $p$ can only be $3$, $6$, and $9$.

Since the total number of games $p(p-1)/2$ is $(1.4+1 = 12/5)$ times a non-negative integer number of games won by righties, $p(p-1)$ must be a multiple of $12(2) = 24$. Among $\{3,6,9\}$, only $p = 9$ satisfies this condition, so the total number of games is $(9)(8)/2 = \boxed{\textbf{(B)}\ 36}.$

~ggao5uiuc, oinava, yingkai_0_ (minor edits)

Solution 5 (🧀Cheese🧀)

If there are $n$ players, the total number of games played must be $\binom{n}{2}$, so it has to be a triangular number. The ratio of games won by left-handed to right-handed players is $7:5$, so the number of games played must also be divisible by $12$. Finally, we notice that only $\boxed{\textbf{(B) }36}$ satisfies both of these conditions.

~MathFun1000

Solution 6 (Cheese 2)

Call the number of games x, the number of games won by right handed player R, and by left handed players L. L+R=x and L=1.4 R. Therefore, x/2.4 must be an integer, which leaves only 36 and 48. Then, note x must be half the product of two consecutive numbers (the number of possible combinations of two people) and we can eliminate 48.

~meikh_neiht


How is this cheese... it is literally the first solution but scaled down

-Shreyansh

Solution 7

Let the number of people be $n$. This means that there will be $\frac{n(n-1)}{2}$ games played. We can now eliminate $\textbf{(D)}$ because it's not a triangular number. The problem states that there are twice as many right handed people as left handed people with no ambidextrous people. Based on this, we can say that the number of people (or $n$) has to be divisible by 3. Thus, the only answer choice that fits the conditions above is answer $\boxed{\textbf{(B) }36}$.


~ROGER8432V3

Video Solution by Little Fermat

https://youtu.be/h2Pf2hvF1wE?si=W4Ad9Bm3vhcTBB4G&t=3440 ~little-fermat

Video Solution by Math-X

https://youtu.be/GP-DYudh5qU?si=DCXVk-iVlqWT-6bS&t=4158 ~Math-X

Video Solution by Power Solve

https://youtu.be/2_ytwADrIto

Video Solution ⚡️ Under 2 min ⚡️

https://youtu.be/wSeSYxDCOZ8

~Education, the Study of Everything

Video Solution 1 by OmegaLearn

https://youtu.be/BXgQIV2WbOA

Video Solution by CosineMethod

https://www.youtube.com/watch?v=N-eZMOv_ZjY

Video Solution 2 by TheBeautyofMath

https://www.youtube.com/watch?v=sLtsF1k9Fx8&t=227s

Video Solution

https://youtu.be/S9l1C6pjXWI

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

Video Solution by SpreadTheMathLove

https://www.youtube.com/watch?v=sypOvNiR3sw

See Also

2023 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 15 		Followed by
Problem 17
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions
2023 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 12 		Followed by
Problem 14
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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