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Difference between revisions of "2024 AIME II Problems"
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==Problem 1==
 
==Problem 1==
  
 +
Among the <math>900</math> residents of Aimeville, there are <math>195</math> who own a diamond ring, <math>367</math> who own a set of golf clubs, and <math>562</math> who own a garden spade. In addition, each of the <math>900</math> residents owns a bag of candy hearts. There are <math>437</math> residents who own exactly two of these things, and <math>234</math> residents who own exactly three of these things. Find the number of residents of Aimeville who own all four of these things.
  
 
[[2024 AIME II Problems/Problem 1|Solution]]
 
[[2024 AIME II Problems/Problem 1|Solution]]
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==Problem 2==
 
==Problem 2==
  
 +
A list of positive integers has the following properties:
 +
 +
<math>\bullet</math> The sum of the items in the list is <math>30</math>.
 +
 +
<math>\bullet</math> The unique mode of the list is <math>9</math>.
 +
 +
<math>\bullet</math> The median of the list is a positive integer that does not appear in the list itself.
 +
 +
Find the sum of the squares of all the items in the list.
  
 
[[2024 AIME II Problems/Problem 2|Solution]]
 
[[2024 AIME II Problems/Problem 2|Solution]]
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==Problem 3==
 
==Problem 3==
  
 +
Find the number of ways to place a digit in each cell of a 2x3 grid so that the sum of the two numbers formed by reading left to right is <math>999</math>, and the sum of the three numbers formed by reading top to bottom is <math>99</math>. The grid below is an example of such an arrangement because <math>8+991=999</math> and <math>9+9+81=99</math>.
 +
 +
<cmath>
 +
\begin{array}{|c|c|c|} \hline
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0 & 0 & 8 \\ \hline
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9 & 9 & 1 \\ \hline
 +
\end{array}
 +
</cmath>
  
 
[[2024 AIME II Problems/Problem 3|Solution]]
 
[[2024 AIME II Problems/Problem 3|Solution]]

Revision as of 20:18, 8 February 2024

2024 AIME II (Answer Key)
Printable version | AoPS Contest Collections • PDF

Instructions

  1. This is a 15-question, 3-hour examination. All answers are integers ranging from $000$ to $999$, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.
  2. No aids other than scratch paper, graph paper, ruler, compass, and protractor are permitted. In particular, calculators and computers are not permitted.
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15

Problem 1

Among the $900$ residents of Aimeville, there are $195$ who own a diamond ring, $367$ who own a set of golf clubs, and $562$ who own a garden spade. In addition, each of the $900$ residents owns a bag of candy hearts. There are $437$ residents who own exactly two of these things, and $234$ residents who own exactly three of these things. Find the number of residents of Aimeville who own all four of these things.

Solution

Problem 2

A list of positive integers has the following properties:

$\bullet$ The sum of the items in the list is $30$.

$\bullet$ The unique mode of the list is $9$.

$\bullet$ The median of the list is a positive integer that does not appear in the list itself.

Find the sum of the squares of all the items in the list.

Solution

Problem 3

Find the number of ways to place a digit in each cell of a 2x3 grid so that the sum of the two numbers formed by reading left to right is $999$, and the sum of the three numbers formed by reading top to bottom is $99$. The grid below is an example of such an arrangement because $8+991=999$ and $9+9+81=99$.

\[\begin{array}{|c|c|c|} \hline 0 & 0 & 8 \\ \hline 9 & 9 & 1 \\ \hline \end{array}\]

Solution

Problem 4

Solution

Problem 5

Solution

Problem 6

Solution

Problem 7

Let $N$ be the greatest four-digit integer with the property that whenever one of its digits is changed to $1$, the resulting number is divisible by $7$. Let $Q$ and $R$ be the quotient and remainder, respectively, when $N$ is divided by $1000$. Find $Q+R$.

Solution

Problem 8

Solution

Problem 9

Solution

Problem 10

Solution

Problem 11

Find the number of triples of nonnegative integers $(a, b, c)$ satisfying $a + b + c = 300$ and

\[a^2 b + a^2 c + b^2 a + b^2 c + c^2 a + c^2 b = 6,000,000.\]

Solution

Problem 12

Solution

Problem 13

Solution

Problem 14

Solution

Problem 15

Solution

See also

2024 AIME II (Problems • Answer Key • Resources)
Preceded by
2024 AIME I 		Followed by
2025 AIME I
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

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