Difference between revisions of "2025 AIME I Problems"

Latest revision as of 16:12, 23 February 2025

2025 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

Find the sum of all integer bases $b > 9$ for which $17_b$ is a divisor of $97_b$ .

Solution

Problem 2

On $\triangle ABC$ points $A$ , $D$ , $E$ , and $B$ lie in that order on side $\overline{AB}$ with $AD = 4$ , $DE = 16$ , and $EB = 8$ . Points $A$ , $F$ , $G$ , and $C$ lie in that order on side $\overline{AC}$ with $AF = 13$ , $FG = 52$ , and $GC = 26$ . Let $M$ be the reflection of $D$ through $F$ , and let $N$ be the reflection of $G$ through $E$ . Quadrilateral $DEGF$ has area $288$ . Find the area of heptagon $AFNBCEM$ .

$[asy] unitsize(14); pair A = (0, 9), B = (-6, 0), C = (12, 0), D = (5A + 2B)/7, E = (2A + 5B)/7, F = (5A + 2C)/7, G = (2A + 5C)/7, M = 2F - D, N = 2E - G; filldraw(A--F--N--B--C--E--M--cycle, lightgray); draw(A--B--C--cycle); draw(D--M); draw(N--G); dot(A); dot(B); dot(C); dot(D); dot(E); dot(F); dot(G); dot(M); dot(N); label("$A$", A, dir(90)); label("$B$", B, dir(225)); label("$C$", C, dir(315)); label("$D$", D, dir(135)); label("$E$", E, dir(135)); label("$F$", F, dir(45)); label("$G$", G, dir(45)); label("$M$", M, dir(45)); label("$N$", N, dir(135)); [/asy]$

Solution

Problem 3

The $9$ members of a baseball team went to an ice-cream parlor after their game. Each player had a single scoop cone of chocolate, vanilla, or strawberry ice cream. At least one player chose each flavor, and the number of players who chose chocolate was greater than the number of players who chose vanilla, which was greater than the number of players who chose strawberry. Let $N$ be the number of different assignments of flavors to players that meet these conditions. Find the remainder when $N$ is divided by $1000.$

Solution

Problem 4

Find the number of ordered pairs $(x,y)$ , where both $x$ and $y$ are integers between $-100$ and $100$ , inclusive, such that $12x^2-xy-6y^2=0$ .

Solution

Problem 5

There are $8!= 40320$ eight-digit positive integers that use each of the digits $1, 2, 3, 4, 5, 6, 7, 8$ exactly once. Let $N$ be the number of these integers that are divisible by $22$ . Find the difference between $N$ and $2025$ .

Solution

Problem 6

An isosceles trapezoid has an inscribed circle tangent to each of its four sides. The radius of the circle is $3$ , and the area of the trapezoid is $72$ . Let the parallel sides of the trapezoid have lengths $r$ and $s$ , with $r \neq s$ . Find $r^2 + s^2$ .

Solution

Problem 7

The twelve letters $A$ , $B$ , $C$ , $D$ , $E$ , $F$ , $G$ , $H$ , $I$ , $J$ , $K$ , and $L$ are randomly grouped into six pairs of letters. The two letters in each pair are placed next to each other in alphabetical order to form six two-letter words, and then those six words are listed alphabetically. For example, a possible result is $AB$ , $CJ$ , $DG$ , $EK$ , $FL$ , $HI$ . The probability that the last word listed contains $G$ is $\frac mn$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .

Solution

Problem 8

Let $k$ be a real number such that the system \begin{align*} &|25 + 20i - z| = 5 \\ &|z - 4 - k| = |z - 3i - k| \end{align*} has exactly one complex solution $z$ . The sum of all possible values of $k$ can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ . Here $i = \sqrt{-1}$ .

Solution

Problem 9

The parabola with equation $y = x^2 - 4$ is rotated $60^{\circ}$ counterclockwise around the origin. The unique point in the fourth quadrant where the original parabola and its image intersect has $y$ -coordinate $\frac{a - \sqrt{b}}{c}$ , where $a$ , $b$ , and $c$ are positive integers, and $a$ and $c$ are relatively prime. Find $a + b + c$ .

Solution

Problem 10

The $27$ cells of a $3 \times 9$ grid are filled in using the numbers $1$ through $9$ so that each row contains $9$ different numbers, and each of the three $3 \times 3$ blocks heavily outlined in the example below contains $9$ different numbers, as in the first three rows of a Sudoku puzzle.

$[asy] unitsize(20); add(grid(9,3)); draw((0,0)--(9,0)--(9,3)--(0,3)--cycle, linewidth(2)); draw((3,0)--(3,3), linewidth(2)); draw((6,0)--(6,3), linewidth(2)); real a = 0.5; label("5",(a,a)); label("6",(1+a,a)); label("1",(2+a,a)); label("8",(3+a,a)); label("4",(4+a,a)); label("7",(5+a,a)); label("9",(6+a,a)); label("2",(7+a,a)); label("3",(8+a,a)); label("3",(a,1+a)); label("7",(1+a,1+a)); label("9",(2+a,1+a)); label("5",(3+a,1+a)); label("2",(4+a,1+a)); label("1",(5+a,1+a)); label("6",(6+a,1+a)); label("8",(7+a,1+a)); label("4",(8+a,1+a)); label("4",(a,2+a)); label("2",(1+a,2+a)); label("8",(2+a,2+a)); label("9",(3+a,2+a)); label("6",(4+a,2+a)); label("3",(5+a,2+a)); label("1",(6+a,2+a)); label("7",(7+a,2+a)); label("5",(8+a,2+a)); [/asy]$

The number of different ways to fill such a grid can be written as $p^a \cdot q^b \cdot r^c \cdot s^d$ where $p$ , $q$ , $r$ , and $s$ are distinct prime numbers and $a$ , $b$ , $c$ , $d$ are positive integers. Find $p \cdot a + q \cdot b + r \cdot c + s \cdot d$ .

Solution

Problem 11

A piecewise linear function is defined by $f(x) = \begin{cases} x & \text{if } x \in [-1, 1) \\ 2 - x & \text{if } x \in [1, 3)\end{cases}$ and $f(x + 4) = f(x)$ for all real numbers $x.$ The graph of $f(x)$ has the sawtooth pattern depicted below. [color=transparent]Diagram from RandomMath.[/color]

[File:2025-AIME-I-P11.png]

The parabola $x = 34y^2$ intersects the graph of $f(x)$ at finitely many points. The sum of the $y$ -coordinates of these intersection points can be expressed in the form $\tfrac{a + b\sqrt c}d,$ where $a, b, c$ and $d$ are positive integers, $a, b,$ and $d$ has greatest common divisor equal to $1,$ and $c$ is not divisible by the square of any prime. Find $a + b + c + d.$

Solution

Problem 12

The set of points in $3$ -dimensional coordinate space that lie in the plane $x+y+z=75$ whose coordinates satisfy the inequalities $x-yz<y-zx<z-xy$ forms three disjoint convex regions. Exactly one of those regions has finite area. The area of this finite region can be expressed in the form $a\sqrt{b},$ where $a$ and $b$ are positive integers and $b$ is not divisible by the square of any prime. Find $a+b.$

Solution

Problem 13

Alex divides a disk into four quadrants with two perpendicular diameters intersecting at the center of the disk. He draws $25$ more lines segments through the disk, drawing each segment by selecting two points at random on the perimeter of the disk in different quadrants and connecting these two points. Find the expected number of regions into which these $27$ line segments divide the disk.

Solution

Problem 14

Let $ABCDE$ be a convex pentagon with $AB=14,$ $BC=7,$ $CD=24,$ $DE=13,$ $EA=26,$ and $\angle B=\angle E=60^{\circ}.$ For each point $X$ in the plane, define $f(X)=AX+BX+CX+DX+EX.$ The least possible value of $f(X)$ can be expressed as $m+n\sqrt{p},$ where $m$ and $n$ are positive integers and $p$ is not divisible by the square of any prime. Find $m+n+p.$

Solution

Problem 15

Let $N$ denote the number of ordered triples of positive integers $(a, b, c)$ such that $a, b, c \leq 3^6$ and $a^3 + b^3 + c^3$ is a multiple of $3^7$ . Find the remainder when $N$ is divided by $1000$ .

Solution

@@ Line 9: / Line 9: @@
 ==Problem 2==
-In <math>\triangle ABC</math> points <math>D</math> and <math>E</math> lie on <math>\overline{AB}</math> so that <math>AD < AE < AB</math>, while points <math>F</math> and <math>G</math> lie on <math>\overline{AC}</math> so that <math>AF < AG < AC</math>. Suppose <math>AD = 4</math>, <math>DE = 16</math>, <math>EB = 8</math>, <math>AF = 13</math>, <math>FG = 52</math>, and <math>GC = 26</math>. Let <math>M</math> be the reflection of <math>D</math> through <math>F</math>, and let <math>N</math> be the reflection of <math>G</math> through <math>E</math>. The area of quadrilateral <math>DEGF</math> is <math>288</math>. Find the area of heptagon <math>AFNBCEM</math>, as shown in the figure below.
+On <math>\triangle ABC</math> points <math>A</math>, <math>D</math>, <math>E</math>, and <math>B</math> lie in that order on side <math>\overline{AB}</math>
+with <math>AD = 4</math>, <math>DE = 16</math>, and <math>EB = 8</math>. Points <math>A</math>, <math>F</math>, <math>G</math>, and <math>C</math> lie in that order on side <math>\overline{AC}</math> with <math>AF = 13</math>, <math>FG = 52</math>, and <math>GC = 26</math>. Let <math>M</math> be the reflection of <math>D</math> through <math>F</math>, and let <math>N</math> be the reflection of <math>G</math> through <math>E</math>. Quadrilateral <math>DEGF</math> has area <math>288</math>. Find the area of heptagon <math>AFNBCEM</math>.
 <asy>
@@ Line 42: / Line 44: @@
 ==Problem 3==
-[[2025 AIME I Problems/Problem 3|Solution]]
+The <math>9</math> members of a baseball team went to an ice-cream parlor after their game. Each player had a single scoop cone of chocolate, vanilla, or strawberry ice cream. At least one player chose each flavor, and the number of players who chose chocolate was greater than the number of players who chose vanilla, which was greater than the number of players who chose strawberry. Let <math>N</math> be the number of different assignments of flavors to players that meet these conditions. Find the remainder when <math>N</math> is divided by <math>1000.</math>
+[[2025 AIME I Problems/Problem 3|Solution]]
 ==Problem 4==
-Find the number of ordered pairs <math>(x,y)</math>, where both <math>x</math> and <math>y</math> are integers between <math>-100</math> and <math>100</math> inclusive, such that <math>12x^2-xy-6y^2=0</math>.
+Find the number of ordered pairs <math>(x,y)</math>, where both <math>x</math> and <math>y</math> are integers between <math>-100</math> and <math>100</math>, inclusive, such that <math>12x^2-xy-6y^2=0</math>.
 [[2025 AIME I Problems/Problem 4|Solution]]
@@ Line 52: / Line 55: @@
 ==Problem 5==
-[[2025 AIME I Problems/Problem 5|Solution]]
+There are <math>8!= 40320</math> eight-digit positive integers that use each of the digits <math>1, 2, 3, 4, 5, 6, 7, 8</math> exactly once. Let <math>N</math> be the number of these integers that are divisible by <math>22</math>. Find the difference between <math>N</math> and <math>2025</math>.
+[[2025 AIME I Problems/Problem 5|Solution]]
 ==Problem 6==
-An isosceles trapezoid has an inscribed circle tangent to each of it's four sides. The radius of the circle is 3, and the area of the trapezoid is 72. Let the parallel sides of the trapezoid have lengths <math>r</math> and <math>s</math>, with <math>r \neq s</math>. Find <math>r^2 + s^2</math>.
+An isosceles trapezoid has an inscribed circle tangent to each of its four sides. The radius of the circle is <math>3</math>, and the area of the trapezoid is <math>72</math>. Let the parallel sides of the trapezoid have lengths <math>r</math> and <math>s</math>, with <math>r \neq s</math>. Find <math>r^2 + s^2</math>.
 [[2025 AIME I Problems/Problem 6|Solution]]
@@ Line 62: / Line 67: @@
 ==Problem 7==
-[[2025 AIME I Problems/Problem 7|Solution]]
+The twelve letters <math>A</math>,<math>B</math>,<math>C</math>,<math>D</math>,<math>E</math>,<math>F</math>,<math>G</math>,<math>H</math>,<math>I</math>,<math>J</math>,<math>K</math>, and <math>L</math> are randomly grouped into six pairs of letters. The two letters in each pair are placed next to each other in alphabetical order to form six two-letter words, and then those six words are listed alphabetically. For example, a possible result is <math>AB</math>, <math>CJ</math>, <math>DG</math>, <math>EK</math>, <math>FL</math>, <math>HI</math>. The probability that the last word listed contains <math>G</math> is <math>\frac mn</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
+[[2025 AIME I Problems/Problem 7|Solution]]
 ==Problem 8==
@@ Line 83: / Line 90: @@
 ==Problem 10==
-[[2025 AIME I Problems/Problem 10|Solution]]
+The <math>27</math> cells of a <math>3 \times 9</math> grid are filled in using the numbers <math>1</math> through <math>9</math> so that each row contains <math>9</math> different numbers, and each of the three <math>3 \times 3</math> blocks heavily outlined in the example below contains <math>9</math> different numbers, as in the first three rows of a Sudoku puzzle.
+<asy>
+unitsize(20);
+add(grid(9,3));
+draw((0,0)--(9,0)--(9,3)--(0,3)--cycle, linewidth(2));
+draw((3,0)--(3,3), linewidth(2)); draw((6,0)--(6,3), linewidth(2));
+real a = 0.5;
+label("5",(a,a));
+label("6",(1+a,a));
+label("1",(2+a,a));
+label("8",(3+a,a));
+label("4",(4+a,a));
+label("7",(5+a,a));
+label("9",(6+a,a));
+label("2",(7+a,a));
+label("3",(8+a,a));
+label("3",(a,1+a));
+label("7",(1+a,1+a));
+label("9",(2+a,1+a));
+label("5",(3+a,1+a));
+label("2",(4+a,1+a));
+label("1",(5+a,1+a));
+label("6",(6+a,1+a));
+label("8",(7+a,1+a));
+label("4",(8+a,1+a));
+label("4",(a,2+a));
+label("2",(1+a,2+a));
+label("8",(2+a,2+a));
+label("9",(3+a,2+a));
+label("6",(4+a,2+a));
+label("3",(5+a,2+a));
+label("1",(6+a,2+a));
+label("7",(7+a,2+a));
+label("5",(8+a,2+a));
+</asy>
+The number of different ways to fill such a grid can be written as <math>p^a \cdot q^b \cdot r^c \cdot s^d</math> where <math>p</math>, <math>q</math>, <math>r</math>, and <math>s</math> are distinct prime numbers and <math>a</math>, <math>b</math>, <math>c</math>, <math>d</math> are positive integers. Find <math>p \cdot a + q \cdot b + r \cdot c + s \cdot d</math>.
-==Problem 11==
+[[2025 AIME I Problems/Problem 10|Solution]]
-[[2025 AIME I Problems/Problem 11|Solution]]
+==Problem 11==
+A piecewise linear function is defined by <cmath>f(x) = \begin{cases} x & \text{if } x \in [-1, 1) \\ 2 - x & \text{if } x \in [1, 3)\end{cases}</cmath> and <math>f(x + 4) = f(x)</math> for all real numbers <math>x.</math> The graph of <math>f(x)</math> has the sawtooth pattern depicted below. [color=transparent]Diagram from RandomMath.[/color]
+[File:2025-AIME-I-P11.png]
+The parabola <math>x = 34y^2</math> intersects the graph of <math>f(x)</math> at finitely many points. The sum of the <math>y</math>-coordinates of these intersection points can be expressed in the form <math>\tfrac{a + b\sqrt c}d,</math> where <math>a, b, c</math> and <math>d</math> are positive integers, <math>a, b,</math> and <math>d</math> has greatest common divisor equal to <math>1,</math> and <math>c</math> is not divisible by the square of any prime. Find <math>a + b + c + d.</math>
+[[2025 AIME I Problems/Problem 11|Solution]]
 ==Problem 12==
-[[2025 AIME I Problems/Problem 12|Solution]]
+The set of points in <math>3</math>-dimensional coordinate space that lie in the plane <math>x+y+z=75</math> whose coordinates satisfy the inequalities <cmath>x-yz<y-zx<z-xy</cmath>forms three disjoint convex regions. Exactly one of those regions has finite area. The area of this finite region can be expressed in the form <math>a\sqrt{b},</math> where <math>a</math> and <math>b</math> are positive integers and <math>b</math> is not divisible by the square of any prime. Find <math>a+b.</math>
+[[2025 AIME I Problems/Problem 12|Solution]]
 ==Problem 13==
-[[2025 AIME I Problems/Problem 13|Solution]]
+Alex divides a disk into four quadrants with two perpendicular diameters intersecting at the center of the disk. He draws <math>25</math> more lines segments through the disk, drawing each segment by selecting two points at random on the perimeter of the disk in different quadrants and connecting these two points. Find the expected number of regions into which these <math>27</math> line segments divide the disk.
+[[2025 AIME I Problems/Problem 13|Solution]]
 ==Problem 14==
-[[2025 AIME I Problems/Problem 14|Solution]]
+Let <math>ABCDE</math> be a convex pentagon with <math>AB=14,</math> <math>BC=7,</math> <math>CD=24,</math> <math>DE=13,</math> <math>EA=26,</math> and <math>\angle B=\angle E=60^{\circ}.</math> For each point <math>X</math> in the plane, define <math>f(X)=AX+BX+CX+DX+EX.</math> The least possible value of <math>f(X)</math> can be expressed as <math>m+n\sqrt{p},</math> where <math>m</math> and <math>n</math> are positive integers and <math>p</math> is not divisible by the square of any prime. Find <math>m+n+p.</math>
+[[2025 AIME I Problems/Problem 14|Solution]]
 ==Problem 15==

2025 AIME I (Problems • Answer Key • Resources)
Preceded by 2024 AIME II	Followed by 2025 AIME II
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Difference between revisions of "2025 AIME I Problems"

Latest revision as of 16:12, 23 February 2025

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