Difference between revisions of "2023 AIME I Problems"

Revision as of 14:35, 8 February 2023

2023 AIME I (Answer Key) Printable version \| AoPS Contest Collections • PDF
Instructions 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. No aids other than scratch paper, graph paper, ruler, compass, and protractor are permitted. In particular, calculators and computers are not permitted.
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Problem 1

Five men and nine women stand equally spaced around a circle in random order. The probability that every man stands diametrically opposite a woman is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Solution

Problem 2

Positive real numbers $b \not= 1$ and $n$ satisfy the equations $\sqrt{\log_b n} = \log_b \sqrt{n} \qquad \text{and} \qquad b \cdot \log_b n = \log_b (bn).$ The value of $n$ is $\frac{j}{k},$ where $j$ and $k$ are relatively prime positive integers. Find $j+k.$

Solution

Problem 3

These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM. Unofficial problem statement has been posted.

Solution

Problem 4

These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.

Solution

Problem 5

Let $P$ be a point on the circle circumscribing square $ABCD$ that satisfies $PA\cdot PC=56$ and $PB\cdot PD=90$ . Find the area of $ABCD$ .

Solution

Problem 6

These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.

Solution

Problem 7

These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM. Unofficial problem statement has been posted.

Solution

Problem 8

These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.

Solution

Problem 9

These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM. Unofficial problem statement has been posted.

Solution

Problem 10

These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.

Solution

Problem 11

These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM. Unofficial problem statement has been posted.

Solution

Problem 12

These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.

Solution

Problem 13

These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.

Solution

Problem 14

These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.

Solution

Problem 15

These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.

Solution

@@ Line 22: / Line 22: @@
 ==Problem 5==
-These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.
+Let <math>P</math> be a point on the circle circumscribing square <math>ABCD</math> that satisfies <math>PA\cdot PC=56</math> and <math>PB\cdot PD=90</math>. Find the area of <math>ABCD</math>.
-Unofficial problem statement has been posted.
 [[2023 AIME I Problems/Problem 5|Solution]]

2023 AIME I (Problems • Answer Key • Resources)
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Difference between revisions of "2023 AIME I Problems"

Revision as of 14:35, 8 February 2023

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

See also