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2023 AMC 10A Problem 16

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2023 AMC 10A Problem 16AMC 10 Step by Step
Name:                             Date: September 13, 2026
Games & Processes · difficulty 3 of 5 · about 4 min · position P16-20

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%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?

  1. A)

    1515

  2. B)

    3636

  3. C)

    4545

  4. D)

    4848

  5. E)

    6666

Answer

B

Key insight

With 3L players the game count is C(3L,2), and wins split 5:7, so the total must be a multiple of 12; only 36 fits both.

Solution

Let there be LL left-handed and 2L2L right-handed players, 3L3L in all. Every pair plays once, so the number of games is
(3L2)=3L(3L−1)2.\binom{3L}{2} = \frac{3L(3L-1)}{2}.
Trying L=1,2,3,4L = 1, 2, 3, 4 gives 3,15,36,663, 15, 36, 66 games; only 15,36,6615, 36, 66 appear among the choices.

Each game produces exactly one win. If right-handers won RR games, left-handers won 1.4R1.4R, so the total is 2.4R=12R52.4R = \frac{12R}{5}. For this to be an integer RR must be a multiple of 55, and then the total is a multiple of 1212. Of 15,36,6615, 36, 66, only 3636 is divisible by 1212.

Sanity check for L=3L = 3: nine players, 3636 games, left-handers win 2121 and right-handers win 1515. The three left-handers play 33 games among themselves and 1818 against right-handers; winning all 1818 plus their 33 internal games gives exactly 2121, so the scenario is achievable.

The answer is (B) 36\boxed{\textbf{(B)}\ 36}.

Why this works

Two independent structural facts, "games =(n2)= \binom{n}{2}" and "wins are in ratio 7:57 : 5," each restrict the total; intersecting the restrictions with the answer choices leaves one survivor. In tournament problems, always equate total wins with total games, and use divisibility from any stated ratio.

The trap

Forgetting that every game produces exactly one win, so total wins equal total games, or skipping the check that 3L(3L-1)/2 matches a choice.

Common mistakes

  • Forgetting that every game produces exactly one win, so total wins equal total games, or skipping the check that 3L(3L-1)/2 matches a choice.
  • Interpreting "40%40\% more" as a ratio of 1.41.4 between players rather than between wins, or as left-handers winning 40%40\% of games.

Techniques

Use the answer choices (mod checks, size, form) to eliminate or select · Bound the quantity above/below or estimate to pin it down