|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| − | ==Problem==
| + | #redirect[[2023 AMC 12A Problems/Problem 13]] |
| − | | |
| − | 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 r be the amount of games the right-handed won. If the left-handed won 1.4r games, then the total amount of games is (2.4)r games, or 12/5r games, meaning that the answer is divisible by 12. This brings us down to two answer choices, B and D. Lastly, we note that the answer is some number <math>x</math> choose 2. 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}}
| |
| − | {{MAA Notice}}
| |
