Difference between revisions of "2023 AIME I Problems"

Revision as of 20:10, 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, rulers and compasses are permitted. In particular, graph paper, protractors, calculators and computers are not permitted.
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15

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

A plane contains $40$ lines, no $2$ of which are parallel. Suppose that there are $3$ points where exactly $3$ lines intersect, $4$ points where exactly $4$ lines intersect, $5$ points where exactly $5$ lines intersect, $6$ points where exactly $6$ lines intersect, and no points where more than $6$ lines intersect. Find the number of points where exactly $2$ lines intersect.

Solution

Problem 4

The sum of all positive integers $m$ such that $\frac{13!}{m}$ is a perfect square can be written as $2^a3^b5^c7^d11^e13^f,$ where $a,b,c,d,e,$ and $f$ are positive integers. Find $a+b+c+d+e+f.$

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

Alice knows that $3$ red cards and $3$ black cards will be revealed to her one at a time in random order. Before each card is revealed, Alice must guess its color. If Alice plays optimally, the expected number of cards she will guess correctly is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Solution

Problem 7

Call a positive integer $n$ extra-distinct if the remainders when $n$ is divided by $2, 3, 4, 5,$ and $6$ are distinct. Find the number of extra-distinct positive integers less than $1000$ .

Solution

Problem 8

There is a rhombus $ABCD$ in which $m\angle{B}<90^{\circ}$ . A point $P$ is chosen somewhere on the incircle of $ABCD$ , and the distances from $P$ to sides $AB$ , $CD$ , and $BC$ , are $9$ , $16$ , and $5$ , respectively. Evaluate the perimeter of the rhombus.

Solution

Problem 9

Find the number of cubic polynomials $p(x) = x^3 + ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are integers in $\{-20, -19,-18, \dots , 18, 19, 20\}$ , such that there is a unique integer $m \neq 2$ with $p(m) = p(2)$ .

Solution

Problem 10

There exists a unique positive integer $a$ for which the sum $U=\sum_{n=1}^{2023}\left\lfloor\dfrac{n^{2}-na}{5}\right\rfloor$ is an integer strictly between $-1000$ and $1000$ . For that unique $a$ , find $a+U$ .

(Note that $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x$ .)

Solution

Problem 11

Find the number of subsets of ${1,2,3,...,10}$ that contain exactly one pair of consecutive integers. Examples of such subsets are ${1,2,5}$ and ${1,3,6,7,10}$ .

Solution

Problem 12

Let $ABC$ be an equilateral triangle with side length $55$ . Points $D$ , $E$ , and $F$ lie on sides $\overline{BC}$ , $\overline{CA}$ , and $\overline{AB}$ , respectively, such that $BD=7$ , $CE=30$ , and $AF=40$ . A unique point $P$ inside $\triangle ABC$ has the property that $\measuredangle AEP=\measuredangle BFP=\measuredangle CDP.$ Find $\tan^{2}\measuredangle AEP$ .

Solution

Problem 13

Each face of two noncongruent parallelepipeds is a rhombus whose diagonals have lengths $\sqrt{21}$ and $\sqrt{31}$ . The ratio of the volume of the larger of the two polyhedra to the volume of the smaller is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ . A parallelepiped is a solid with six parallelogram faces such as the one shown below.

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 59: / Line 59: @@
 ==Problem 12==
-These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.
+Let <math>ABC</math> be an equilateral triangle with side length <math>55</math>. Points <math>D</math>, <math>E</math>, and <math>F</math> lie on sides <math>\overline{BC}</math>, <math>\overline{CA}</math>, and <math>\overline{AB}</math>, respectively, such that <math>BD=7</math>, <math>CE=30</math>, and <math>AF=40</math>. A unique point <math>P</math> inside <math>\triangle ABC</math> has the property that<cmath>\measuredangle AEP=\measuredangle BFP=\measuredangle CDP.</cmath>Find <math>\tan^{2}\measuredangle AEP</math>.
 [[2023 AIME I Problems/Problem 12|Solution]]

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

Revision as of 20:10, 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