Difference between revisions of "2025 AIME I Problems"

Revision as of 19:34, 13 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.
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Problem 1

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

Solution

Problem 2

In $\triangle ABC$ points $D$ and $E$ lie on $\overline{AB}$ so that $AD < AE < AB$ , while points $F$ and $G$ lie on $\overline{AC}$ so that $AF < AG < AC$ . Suppose $AD = 4$ , $DE = 16$ , $EB = 8$ , $AF = 13$ , $FG = 52$ , and $GC = 26$ . Let $M$ be the reflection of $D$ through $F$ , and let $N$ be the reflection of $G$ through $E$ . The area of quadrilateral $DEGF$ is $288$ . Find the area of heptagon $AFNBCEM$ , as shown in the figure below.

$[asy] unitsize(14); pair A = (0, 9), B = (-6, 0), C = (12, 0), D = (5A + 2B)/7, E = (2A + 5B)/7, F = (5A + 2C)/7, G = (2A + 5C)/7, M = 2F - D, N = 2E - G; filldraw(A--F--N--B--C--E--M--cycle, lightgray); draw(A--B--C--cycle); draw(D--M); draw(N--G); dot(A); dot(B); dot(C); dot(D); dot(E); dot(F); dot(G); dot(M); dot(N); label("$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 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 $r$ and $s$ , with $r \neq s$ . Find $r^2 + s^2$ .

Solution

Problem 7

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

Solution

Problem 11

A piecewise linear function is defined by $f(x) = \begin{cases} x & \operatorname{if} ~ -1 \leq x < 1 \\ 2 - x & \operatorname{if} ~ 1 \leq x < 3\end{cases}$ and $f(x + 4) = f(x)$ for all real numbers $x$ . The graph of $f(x)$ has the sawtooth pattern depicted below.

The parabola $x^{2} = 34y$ intersects the graph of $f(x)$ at finitely many points. The sum of the $y$ -coordinates of all these intersection points can be expressed in the form $\tfrac{a + b\sqrt{c}}{d}$ , where $a$ , $b$ , $c$ , and $d$ are positive integers such that $a$ , $b$ , $d$ have 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

Solution

Problem 13

Solution

Problem 14

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 42: / Line 42: @@
 ==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==

2025 AIME I (Problems • Answer Key • Resources)
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Difference between revisions of "2025 AIME I Problems"

Revision as of 19:34, 13 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