Difference between revisions of "2023 AIME I Problems"

Latest revision as of 11:54, 20 February 2024

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.
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

Rhombus $ABCD$ has $\angle BAD < 90^\circ.$ There is a point $P$ on the incircle of the rhombus such that the distances from $P$ to the lines $DA,AB,$ and $BC$ are $9,5,$ and $16,$ respectively. Find the perimeter of $ABCD.$

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 $\{-20,-19,-18,\ldots,18,19,20\},$ such that there is a unique integer $m \not= 2$ 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 $\{1,2,3,\ldots,10\}$ that contain exactly one pair of consecutive integers. Examples of such subsets are $\{\mathbf{1},\mathbf{2},5\}$ and $\{1,3,\mathbf{6},\mathbf{7},10\}.$

Solution

Problem 12

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

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.

$[asy] unitsize(2cm); pair o = (0, 0), u = (1, 0), v = 0.8*dir(40), w = dir(70); draw(o--u--(u+v)); draw(o--v--(u+v), dotted); draw(shift(w)*(o--u--(u+v)--v--cycle)); draw(o--w); draw(u--(u+w)); draw(v--(v+w), dotted); draw((u+v)--(u+v+w)); [/asy]$

Solution

Problem 14

The following analog clock has two hands that can move independently of each other. $[asy] unitsize(2cm); draw(unitcircle,black+linewidth(2)); for (int i = 0; i < 12; ++i) { draw(0.9*dir(30*i)--dir(30*i)); } for (int i = 0; i < 4; ++i) { draw(0.85*dir(90*i)--dir(90*i),black+linewidth(2)); } for (int i = 1; i < 13; ++i) { label("\small" + (string) i, dir(90 - i * 30) * 0.75); } draw((0,0)--0.6*dir(90),black+linewidth(2),Arrow(TeXHead,2bp)); draw((0,0)--0.4*dir(90),black+linewidth(2),Arrow(TeXHead,2bp)); [/asy]$ Initially, both hands point to the number $12$ . The clock performs a sequence of hand movements so that on each movement, one of the two hands moves clockwise to the next number on the clock face while the other hand does not move.

Let $N$ be the number of sequences of $144$ hand movements such that during the sequence, every possible positioning of the hands appears exactly once, and at the end of the $144$ movements, the hands have returned to their initial position. Find the remainder when $N$ is divided by $1000$ .

Solution

Problem 15

Find the largest prime number $p<1000$ for which there exists a complex number $z$ satisfying

the real and imaginary part of $z$ are both integers;

$|z|=\sqrt{p},$ and

there exists a triangle whose three side lengths are $p,$ the real part of $z^{3},$ and the imaginary part of $z^{3}.$

Solution

@@ Line 12: / Line 12: @@
 ==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.
+A plane contains <math>40</math> lines, no <math>2</math> of which are parallel. Suppose that there are <math>3</math> points where exactly <math>3</math> lines intersect, <math>4</math> points where exactly <math>4</math> lines intersect, <math>5</math> points where exactly <math>5</math> lines intersect, <math>6</math> points where exactly <math>6</math> lines intersect, and no points where more than <math>6</math> lines intersect. Find the number of points where exactly <math>2</math> lines intersect.
 [[2023 AIME I Problems/Problem 3|Solution]]
 ==Problem 4==
-These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.
+The sum of all positive integers <math>m</math> such that <math>\frac{13!}{m}</math> is a perfect square can be written as <math>2^a3^b5^c7^d11^e13^f,</math> where <math>a,b,c,d,e,</math> and <math>f</math> are positive integers. Find <math>a+b+c+d+e+f.</math>
 [[2023 AIME I Problems/Problem 4|Solution]]
 ==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]]
 ==Problem 6==
-These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.
+Alice knows that <math>3</math> red cards and <math>3</math> 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 <math>\frac{m}{n},</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n.</math>
 [[2023 AIME I Problems/Problem 6|Solution]]
 ==Problem 7==
-These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.
+Call a positive integer <math>n</math> extra-distinct if the remainders when <math>n</math> is divided by <math>2, 3, 4, 5,</math> and <math>6</math> are distinct. Find the number of extra-distinct positive integers less than <math>1000</math>.
-Unofficial problem statement has been posted.
 [[2023 AIME I Problems/Problem 7|Solution]]
 ==Problem 8==
-These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.
+Rhombus <math>ABCD</math> has <math>\angle BAD < 90^\circ.</math> There is a point <math>P</math> on the incircle of the rhombus such that the distances from <math>P</math> to the lines <math>DA,AB,</math> and <math>BC</math> are <math>9,5,</math> and <math>16,</math> respectively. Find the perimeter of <math>ABCD.</math>
 [[2023 AIME I Problems/Problem 8|Solution]]
 ==Problem 9==
-These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.
+Find the number of cubic polynomials <math>p(x) = x^3 + ax^2 + bx + c,</math> where <math>a, b,</math> and <math>c</math> are integers in <math>\{-20,-19,-18,\ldots,18,19,20\},</math> such that there is a unique integer <math>m \not= 2</math> with <math>p(m) = p(2).</math>
-Unofficial problem statement has been posted.
 [[2023 AIME I Problems/Problem 9|Solution]]
 ==Problem 10==
-These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.
+There exists a unique positive integer <math>a</math> for which the sum <cmath>U=\sum_{n=1}^{2023}\left\lfloor\dfrac{n^{2}-na}{5}\right\rfloor</cmath> is an integer strictly between <math>-1000</math> and <math>1000</math>. For that unique <math>a</math>, find <math>a+U</math>.
+(Note that <math>\lfloor x\rfloor</math> denotes the greatest integer that is less than or equal to <math>x</math>.)
 [[2023 AIME I Problems/Problem 10|Solution]]
 ==Problem 11==
-These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.
+Find the number of subsets of <math>\{1,2,3,\ldots,10\}</math> that contain exactly one pair of consecutive integers. Examples of such subsets are <math>\{\mathbf{1},\mathbf{2},5\}</math> and <math>\{1,3,\mathbf{6},\mathbf{7},10\}.</math>
-Unofficial problem statement has been posted.
 [[2023 AIME I Problems/Problem 11|Solution]]
 ==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>\triangle 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 <math>\overline{BC},</math> <math>\overline{CA},</math> and <math>\overline{AB},</math> respectively, with <math>BD = 7,</math> <math>CE=30,</math> and <math>AF=40.</math> Point <math>P</math> inside <math>\triangle ABC</math> has the property that <cmath>\angle AEP = \angle BFP = \angle CDP.</cmath> Find <math>\tan^2(\angle AEP).</math>
 [[2023 AIME I Problems/Problem 12|Solution]]
 ==Problem 13==
-These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.
+Each face of two noncongruent parallelepipeds is a rhombus whose diagonals have lengths <math>\sqrt{21}</math> and <math>\sqrt{31}</math>. The ratio of the volume of the larger of the two polyhedra to the volume of the smaller is <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>. A parallelepiped is a solid with six parallelogram faces such as the one shown below.
+<asy>
+unitsize(2cm);
+pair o = (0, 0), u = (1, 0), v = 0.8*dir(40), w = dir(70);
+draw(o--u--(u+v));
+draw(o--v--(u+v), dotted);
+draw(shift(w)*(o--u--(u+v)--v--cycle));
+draw(o--w);
+draw(u--(u+w));
+draw(v--(v+w), dotted);
+draw((u+v)--(u+v+w));
+</asy>
 [[2023 AIME I Problems/Problem 13|Solution]]
 ==Problem 14==
-These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.
+The following analog clock has two hands that can move independently of each other.
+<asy>
+            unitsize(2cm);
+            draw(unitcircle,black+linewidth(2));
+            for (int i = 0; i < 12; ++i) {
+                draw(0.9*dir(30*i)--dir(30*i));
+            }
+            for (int i = 0; i < 4; ++i) {
+                draw(0.85*dir(90*i)--dir(90*i),black+linewidth(2));
+            }
+            for (int i = 1; i < 13; ++i) {
+                label("\small" + (string) i, dir(90 - i * 30) * 0.75);
+            }
+            draw((0,0)--0.6*dir(90),black+linewidth(2),Arrow(TeXHead,2bp));
+            draw((0,0)--0.4*dir(90),black+linewidth(2),Arrow(TeXHead,2bp));
+</asy>
+Initially, both hands point to the number <math>12</math>. The clock performs a sequence of hand movements so that on each movement, one of the two hands moves clockwise to the next number on the clock face while the other hand does not move.
+Let <math>N</math> be the number of sequences of <math>144</math> hand movements such that during the sequence, every possible positioning of the hands appears exactly once, and at the end of the <math>144</math> movements, the hands have returned to their initial position. Find the remainder when <math>N</math> is divided by <math>1000</math>.
 [[2023 AIME I Problems/Problem 14|Solution]]
 ==Problem 15==
-These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.
+Find the largest prime number <math>p<1000</math> for which there exists a complex number <math>z</math> satisfying
+* the real and imaginary part of <math>z</math> are both integers;
+* <math>|z|=\sqrt{p},</math> and
+* there exists a triangle whose three side lengths are <math>p,</math> the real part of <math>z^{3},</math> and the imaginary part of <math>z^{3}.</math>
 [[2023 AIME I Problems/Problem 15|Solution]]

2023 AIME I (Problems • Answer Key • Resources)
Preceded by 2022 AIME II	Followed by 2023 AIME II
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

Page

Toolbox

Search

Difference between revisions of "2023 AIME I Problems"

Latest revision as of 11:54, 20 February 2024

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