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: 229

2021 MOAA, 10
Tags: team
For how many nonempty subsets $S \subseteq \{1, 2, \ldots , 10\}$ is the sum of all elements in $S$ even? [i]Proposed by Andrew Wen[/i]

2017 CMIMC Team, 9
Tags: team
Circles $\omega_1$ and $\omega_2$ are externally tangent to each other. Circle $\Omega$ is placed such that $\omega_1$ is internally tangent to $\Omega$ at $X$ while $\omega_2$ is internally tangent to $\Omega$ at $Y$. Line $\ell$ is tangent to $\omega_1$ at $P$ and $\omega_2$ at $Q$ and furthermore intersects $\Omega$ at points $A$ and $B$ with $AP<AQ$. Suppose that $AP=2$, $PQ=4$, and $QB=3$. Compute the length of line segment $\overline{XY}$.

2024 LMT Fall, 10
Tags: team
Find the sum of all positive integers $n\le 2024$ such that all pairs of distinct positive integers $(a,b)$ that satisfy $ab=n$ have a sum that is a perfect square.

MOAA Team Rounds, 2021.4
Tags: team
Compute the number of ordered triples $(x,y,z)$ of integers satisfying \[x^2+y^2+z^2=9.\] [i]Proposed by Nathan Xiong[/i]

2022 CMIMC, 6
Tags: team
There are $9$ points arranged in a $3\times 3$ square grid. Let two points be adjacent if the distance between them is half the side length of the grid. (There should be $12$ pairs of adjacent points). Suppose that we wanted to connect $8$ pairs of adjacent points, such that all points are connected to each other. In how many ways is this possible? [i]Proposed by Kevin You[/i]

MOAA Team Rounds, 2021.10
Tags: team
For how many nonempty subsets $S \subseteq \{1, 2, \ldots , 10\}$ is the sum of all elements in $S$ even? [i]Proposed by Andrew Wen[/i]

2024 LMT Fall, 11
Tags: team
Let $\phi=\tfrac{1+\sqrt 5}{2}$. Find \[\left(4+\phi^{\frac12}\right)\left(4-\phi^{\frac12}\right)\left(4+i\phi^{-\frac12}\right)\left(4-i\phi^{-\frac12}\right).\]

2021 MOAA, 17
Tags: team
Compute the remainder when $10^{2021}$ is divided by $10101$. [i]Proposed by Nathan Xiong[/i]

2022 CMIMC, 13
Tags: team
Let $F_n$ denote the $n$th Fibonacci number, with $F_0=0, F_1=1$ and $F_{n}=F_{n-1}+F_{n-2}$ for $n \geq 2$. There exists a unique two digit prime $p$ such that for all $n$, $p | F_{n+100} + F_n$. Find $p$. [i]Proposed by Sam Rosenstrauch[/i]

2019 MOAA, 10
Tags: number theory , team
Let $S$ be the set of all four digit palindromes (a palindrome is a number that reads the same forwards and backwards). The average value of $|m - n|$ over all ordered pairs $(m, n)$, where $m$ and $n$ are (not necessarily distinct) elements of $S$, is equal to $p/q$ , for relatively prime positive integers $p$ and $q$. Find $p + q$.

2024 HMNT, 3
Tags: team
Rectangle $R$ with area $20$ and diagonal of length $7$ is translated $2$ units in some direction to form a new rectangle $R'.$ The vertices of $R$ and $R'$ that are not contained in the other rectangle form a convex hexagon. Compute the maximum possible area of this hexagon.

MOAA Team Rounds, 2019.10
Tags: number theory , team
Let $S$ be the set of all four digit palindromes (a palindrome is a number that reads the same forwards and backwards). The average value of $|m - n|$ over all ordered pairs $(m, n)$, where $m$ and $n$ are (not necessarily distinct) elements of $S$, is equal to $p/q$ , for relatively prime positive integers $p$ and $q$. Find $p + q$.

2021 MOAA, 12
Tags: team
Let $\triangle ABC$ have $AB=9$ and $AC=10$. A semicircle is inscribed in $\triangle ABC$ with its center on segment $BC$ such that it is tangent $AB$ at point $D$ and $AC$ at point $E$. If $AD=2DB$ and $r$ is the radius of the semicircle, $r^2$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Andy Xu[/i]

2019 CMIMC, 15
Tags: team , algebra , polynomial
Call a polynomial $P$ [i]prime-covering[/i] if for every prime $p$, there exists an integer $n$ for which $p$ divides $P(n)$. Determine the number of ordered triples of integers $(a,b,c)$, with $1\leq a < b < c \leq 25$, for which $P(x)=(x^2-a)(x^2-b)(x^2-c)$ is prime-covering.

2023 CMIMC Team, 6
Tags: team
A positive integer $n$ is said to be base-able if there exists positive integers $a$ and $b,$ with $b>1,$ such that $n=a^b.$ How many positive integer divisors of $729000000$ are base-able? [i]Proposed by Kyle Lee[/i]

2025 Harvard-MIT Mathematics Tournament, 7
Tags: team
Determine, with proof, whether a square can be dissected into finitely many (not necessarily congruent) triangles, each of which has interior angles $30^\circ, 75^\circ,$ and $75^\circ.$

MOAA Team Rounds, 2019.2
Tags: geometry , team , algebra
The lengths of the two legs of a right triangle are the two distinct roots of the quadratic $x^2 - 36x + 70$. What is the length of the triangle’s hypotenuse?

MOAA Team Rounds, 2018.2
Tags: algebra , team
If $x > 0$ and $x^2 +\frac{1}{x^2}= 14$, find $x^5 +\frac{1}{x^5}$.

2023 CMIMC Team, 11
Tags: team
A positive integer is [i]detestable[/i] if the sum of its digits is a multiple of $11$. How many positive integers below $10000$ are detestable? [i]Proposed by Giacomo Rizzo[/i]

MOAA Team Rounds, 2019.1
Tags: geometry , team
Jeffrey stands on a straight horizontal bridge that measures $20000$ meters across. He wishes to place a pole vertically at the center of the bridge so that the sum of the distances from the top of the pole to the two ends of the bridge is $20001$ meters. To the nearest meter, how long of a pole does Jeffrey need?

MOAA Team Rounds, TO4
Tags: algebra , team
Over all real numbers $x$, let $k$ be the minimum possible value of the expression $$\sqrt{x^2 + 9} +\sqrt{x^2 - 6x + 45}.$$ Determine $k^2$.

2025 CMIMC Team, 1
Tags: team
I define a "good day" as a day when both the day and the month evenly divide the concatenation of the two. For example, today (March $15$) is a good day since $3$ and $15$ both divide $315.$ However, March $9$ is not a good day since $9$ does not divide $39.$ How many good days are in March, April, and May combined?

2021 MOAA, 18
Tags: team
Let $\triangle ABC$ be a triangle with side length $BC= 4\sqrt{6}$. Denote $\omega$ as the circumcircle of $\triangle{ABC}$. Point $D$ lies on $\omega$ such that $AD$ is the diameter of $\omega$. Let $N$ be the midpoint of arc $BC$ that contains $A$. $H$ is the intersection of the altitudes in $\triangle{ABC}$ and it is given that $HN = HD= 6$. If the area of $\triangle{ABC}$ can be expressed as $\frac{a\sqrt{b}}{c}$, where $a,b,c$ are positive integers with $a$ and $c$ relatively prime and $b$ not divisible by the square of any prime, compute $a+b+c$. [i]Proposed by Andy Xu[/i]

2025 CMIMC Team, 5
Tags: team
Suppose we have a uniformly random function from $\{1, 2, 3, \ldots, 25\}$ to itself. Find the expected value of $$\sum_{x=1}^{25} (f(f(x))-x)^2.$$

2022 CMIMC, 5
Tags: team
For any integer $a$, let $f(a) = |a^4 - 36a^2 + 96a - 64|$. What is the sum of all values of $f(a)$ that are prime? [i]Proposed by Alexander Wang[/i]