Difference between revisions of "2023 AMC 12A Problems/Problem 13"
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− | + | ==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 <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> | ||
+ | |||
+ | ==Solution 1 (3 min solve)== | ||
+ | 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 equal 72 or 96. Clearly <math>72=8*9</math>, so the answer is <math>\boxed{\textbf{(B) }36}</math>. | ||
+ | |||
+ | ~~ Antifreeze5420 | ||
+ | |||
+ | ==Solution 2== | ||
+ | First, every player played the other, so there's <math>n\choose2</math> games. Also, if the right-handed won <math>x</math> games, the left handed won <math>7/5x</math>, meaning that the total amount of games was <math>12/5x</math>, so the total amount of games is divisible by <math>12</math>. | ||
+ | |||
+ | Then, we do something funny and look at the answer choices. Only <math>\boxed{\textbf{(B) }36}</math> satisfies our 2 findings. | ||
+ | |||
+ | ==Solution 3== | ||
+ | Let <math>r</math> be the amount of games the right-handed won. Since the left-handed won <math>1.4r</math> games, the total number of games played can be expressed as <math>(2.4)r</math>, or <math>12/5r</math>, meaning that the answer is divisible by 12. This brings us down to two answer choices, <math>B</math> and <math>D</math>. | ||
+ | We note that the answer is some number <math>x</math> choose <math>2</math>. This means the answer is in the form <math>x(x-1)/2</math>. Since answer choice D gives <math>48 = x(x-1)/2</math>, and <math>96 = x(x-1)</math> has no integer solutions, we know that <math>\boxed{\textbf{(B) }36}</math> is the only possible choice. | ||
+ | |||
+ | == Video Solution 1 by OmegaLearn == | ||
+ | https://youtu.be/BXgQIV2WbOA | ||
+ | |||
+ | ==See Also== | ||
+ | {{AMC10 box|year=2023|ab=A|num-b=15|num-a=17}} | ||
+ | {{AMC12 box|year=2023|ab=A|num-b=12|num-a=14}} | ||
+ | {{MAA Notice}} |
Revision as of 22:48, 9 November 2023
Contents
[hide]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 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?
Solution 1 (3 min solve)
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 , and since , . Given that and are both integers, also must be an integer. From here we can see that 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 , the sum of the first triangular numbers. Now setting 36 and 48 equal to the equation will show that two consecutive numbers must equal 72 or 96. Clearly , so the answer is .
~~ Antifreeze5420
Solution 2
First, every player played the other, so there's games. Also, if the right-handed won games, the left handed won , meaning that the total amount of games was , so the total amount of games is divisible by .
Then, we do something funny and look at the answer choices. Only satisfies our 2 findings.
Solution 3
Let be the amount of games the right-handed won. Since the left-handed won games, the total number of games played can be expressed as , or , meaning that the answer is divisible by 12. This brings us down to two answer choices, and . We note that the answer is some number choose . This means the answer is in the form . Since answer choice D gives , and has no integer solutions, we know that is the only possible choice.
Video Solution 1 by OmegaLearn
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.