This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 138

2021 AIME Problems, 13
Tags: AMC , AIME , AIME II
Find the least positive integer $n$ for which $2^n + 5^n - n$ is a multiple of $1000$.

2018 AIME Problems, 8
Tags: AMC , AIME , AIME II
A frog is positioned at the origin in the coordinate plane. From the point $(x,y)$, the frog can jump to any of the points $(x+1, y), (x+2, y), (x, y+1),$ or $(x, y+2)$. Find the number of distinct sequences of jumps in which the frog begins at $(0,0)$ and ends at $(4,4)$.

2021 AIME Problems, 11
Tags: AIME , AIME II
A teacher was leading a class of four perfectly logical students. The teacher chose a set $S$ of four integers and gave a different number in $S$ to each student. Then the teacher announced to the class that the numbers in $S$ were four consecutive two-digit positive integers, that some number in $S$ was divisible by $6$, and a different number in $S$ was divisible by $7$. The teacher then asked if any of the students could deduce what $S$ is, but in unison, all of the students replied no. However, upon hearing that all four students replied no, each student was able to determine the elements of $S$. Find the sum of all possible values of the greatest element of $S$.

2016 AIME Problems, 2
Tags: AMC , AIME , AIME II
There is a $40\%$ chance of rain on Saturday and a $30\%$ of rain on Sunday. However, it is twice as likely to rain on Sunday if it rains on Saturday than if it does not rain on Saturday. The probability that it rains at least one day this weekend is $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a+b$.

2024 AIME, 4
Tags: AMC , AIME , algebra , system of equations , number theory , relatively prime , AIME II
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations: $$\log_2\left({x \over yz}\right) = {1 \over 2}$$ $$\log_2\left({y \over xz}\right) = {1 \over 3}$$ $$\log_2\left({z \over xy}\right) = {1 \over 4}$$ Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is ${m \over n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$

2025 AIME, 11
Tags: AMC , AIME , AIME II , 2025 AIME II , geometry , regular polygon
Let $S$ be the set of vertices of a regular $24$-gon. Find the number of ways to draw $12$ segments of equal lengths so that each vertex in $S$ is an endpoint of exactly one of the $12$ segments.

2022 AIME Problems, 14
Tags: AMC , AIME , AIME II
For positive integers $a$, $b$, and $c$ with $a < b < c$, consider collections of postage stamps in denominations $a$, $b$, and $c$ cents that contain at least one stamp of each denomination. If there exists such a collection that contains sub-collections worth every whole number of cents up to $1000$ cents, let $f(a, b, c)$ be the minimum number of stamps in such a collection. Find the sum of the three least values of $c$ such that $f(a, b, c) = 97$ for some choice of $a$ and $b$.

2017 AIME Problems, 14
Tags: AMC , AIME , AIME II , Casework
A $10\times 10\times 10$ grid of points consists of all points in space of the form $(i,j,k)$, where $i$, $j$, and $k$ are integers between $1$ and $10$, inclusive. Find the number of different lines that contain exactly $8$ of these points.

CIME II 2018, 2
Tags: AIME II
Garfield and Odie are situated at $(0,0)$ and $(25,0)$, respectively. Suddenly, Garfield and Odie dash in the direction of the point $(9, 12)$ at speeds of $7$ and $10$ units per minute, respectively. During this chase, the minimum distance between Garfield and Odie can be written as $\frac{m}{\sqrt{n}}$ for relatively prime positive integers $m$ and $n$. Find $m+n$. [i]Proposed by [b] Th3Numb3rThr33 [/b][/i]

2017 AIME Problems, 10
Tags: AIME , AIME I , AIME II
Rectangle $ABCD$ has side lengths $AB=84$ and $AD=42$. Point $M$ is the midpoint of $\overline{AD}$, point $N$ is the trisection point of $\overline{AB}$ closer to $A$, and point $O$ is the intersection of $\overline{CM}$ and $\overline{DN}$. Point $P$ lies on the quadrilateral $BCON$, and $\overline{BP}$ bisects the area of $BCON$. Find the area of $\triangle{CDP}$.

CIME II 2018, 7
Tags: AIME II
Consider all the positive integers $N$ with the property that all of the divisors of $N$ can be written as $p-2$ for some prime number $p$. Then, there exists an integer $m$ such that $m$ is the maximum possible number of divisors of all numbers $N$ with such property. Find the sum of all possible values of $N$ such that $N$ has $m$ divisors. [i]Proposed by [b]FedeX333X[/b][/i]

2017 AIME Problems, 8
Tags: AIME , AIME I , AIME II , algebra , polynomial
Find the number of positive integers $n$ less than $2017$ such that \[ 1+n+\frac{n^2}{2!}+\frac{n^3}{3!}+\frac{n^4}{4!}+\frac{n^5}{5!}+\frac{n^6}{6!} \] is an integer.

2017 AIME Problems, 9
Tags: AMC , AIME , AIME II
A special deck of cards contains $49$ cards, each labeled with a number from $1$ to $7$ and colored with one of seven colors. Each number-color combination appears on exactly one card. Sharon will select a set of eight cards from the deck at random. Given that she gets at least one card of each color and at least one card with each number, the probability that Sharon can discard one of her cards and [i]still[/i] have at least one card of each color and at least one card with each number is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.