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: 85

OMMC POTM, 2024 2
Tags: algebra , inequalities , ommc
Let $a,b,c$, and $d$ be real numbers such that $$a+b = c +d+ 12$$ and $$ab + cd - 28 = bc + ad.$$ Find the minimum possible value of $a^4+b^4+c^4+d^4$.

2021-2022 OMMC, 11
Tags: ommc
Let $ABC$ be a triangle such that $AB = 7$, $BC = 8$, and $CA = 9$. There exists a unique point $X$ such that $XB = XC$ and $XA$ is tangent to the circumcircle of $ABC$. If $XA = \tfrac ab$, where $a$ and $b$ are coprime positive integers, find $a + b$. [i]Proposed by Alexander Wang[/i]

2020-2021 OMMC, 10
Tags: ommc
How many ways are there to arrange the numbers $1$ through $8$ into a $2$ by $4$ grid such that the sum of the numbers in each of the two rows are all multiples of $6,$ and the sum of the numbers in each of the four columns are all multiples of $3$?

2021-2022 OMMC, 3
Tags: ommc
Evan has $10$ cards numbered $1$ through $10$. He chooses some of the cards and takes the product of the numbers on them. When the product is divided by $3$, the remainder is $1$. Find the maximum number of cards he could have chose. [i]Proposed by Evan Chang [/i]

OMMC POTM, 2021 12
Tags: algebra , polynomial , ommc
Let $r,s,t$ be the roots of $x^3+6x^2+7x+8$. Find $$(r^2+s+t)(s^2+t+r)(t^2+r+s).$$ [i]Proposed by Evan Chang (squareman), USA[/i]

2021-2022 OMMC, 7
Tags: ommc
How many ordered triples of integers $(x,y,z)$ satisfy \[36x^2+100y^2+225z^2=12600?\] [i]Proposed by Bill Fei and Mahith Gottipati [/i]

2020-2021 OMMC, 8
Tags: ommc , geometry
Triangle $ABC$ has circumcircle $\omega$. The angle bisectors of $\angle A$ and $\angle B$ intersect $\omega$ at points $D$ and $E$ respectively. $DE$ intersects $BC$ and $AC$ at $X$ and $Y$ respectively. Given $DX = 7,$ $XY = 8$ and $YE = 9,$ the area of $\triangle ABC$ can be written as $\frac{a\sqrt{b}}{c}$ where $a, b, c$ are positive integers, $\gcd(a,c) = 1,$ and $b$ is square free. Find $a+b+c.$

2020-2021 OMMC, 4
Tags: ommc , Sequences
The sum $$\frac{1^2-2}{1!} + \frac{2^2-2}{2!} + \frac{3^2-2}{3!} + \cdots + \frac{2021^2 - 2}{2021!}$$ $ $ \\ can be expressed as a rational number $N$. Find the last 3 digits of $2021! \cdot N$.

OMMC POTM, 2023 3
Tags: number theory , ommc
Three natural numbers are such that the product of any two of them is divisible by the sum of those two numbers. Prove that these numbers have a common divisor greater than $1$. [i]Proposed by Evan Chang (squareman), USA[/i]

2021-2022 OMMC, 22
Tags: ommc
A positive integer $N$ is [i]apt[/i] if for each integer $0 < k < 1009$, there exists exactly one divisor of $N$ with a remainder of $k$ when divided by $1009$. For a prime $p$, suppose there exists an [i]apt[/i] positive integer $N$ where $\tfrac Np$ is an integer but $\tfrac N{p^2}$ is not. Find the number of possible remainders when $p$ is divided by $1009$. [i]Proposed by Evan Chang[/i]

2020-2021 OMMC, 6
Tags: ommc
Jason and Jared take turns placing blocks within a game board with dimensions $3 \times 300$, with Jason going first, such that no two blocks can overlap. The player who cannot place a block within the boundaries loses. Jason can only place $2 \times 100$ blocks, and Jared can only place $2 \times n$ blocks where $n$ is some positive integer greater than 3. Find the smallest integer value of $n$ that still guarantees Jason a win (given both players are playing optimally).

2021-2022 OMMC, 5
Tags: ommc
$12$ distinct points are equally spaced around a circle. How many ways can Bryan choose $3$ points (not in any order) out of these $12$ points such that they form an acute triangle (Rotations of a set of points are considered distinct). [i]Proposed by Bryan Guo [/i]

2020-2021 OMMC, 3
Tags: ommc
The intersection of two squares with perimeter $8$ is a rectangle with diagonal length $1$. Given that the distance between the centers of the two squares is $2$, the perimeter of the rectangle can be expressed as $P$. Find $10P$.

2021-2022 OMMC, 24
Tags: ommc
In $\triangle ABC$, angle $B$ is obtuse, $AB = 42$ and $BC = 69$. Let $M$ and $N$ be the midpoints of $AB$ and $BC$, respectively. The angle bisectors of $\angle CAB$ and $\angle ABC$ meet $BC$ and $CA$ at $D$ and $E$ respectively. Let $X$ and $Y$ be the midpoints of $AD$ and $AN$ respectively. Let $CY$ and $BX$ meet $AB$ and $CA$ at $P$ and $Q$. If $EM$ and $PQ$ meet on $BC$, find $CA$. [i]Proposed by Sid Doppalapudi[/i]

2021-2022 OMMC, 2
Tags: ommc , combinatorics
Alex writes down some distinct integers on a blackboard. For each pair of integers, he writes the positive difference of those on a piece of paper. Find the sum of all $n\leq2022$ such that it is possible for the numbers on the paper to contain only the positive integers between $1$ and $n$, inclusive exactly once. [i]Proposed by Alexander Wang[/i]

OMMC POTM, 2023 5
Tags: geometry , combinatorics , combinatorial geometry , ommc
$10$ rectangles have their vertices lie on a circle. The vertices divide the circle into $40$ equal arcs. Prove that two of the rectangles are congruent. [i]Proposed by Evan Chang (squareman), USA[/i]

2021-2022 OMMC, 16
Tags: ommc
In $\triangle ABC$ with $AB = 10$, $BC = 12$, and $AC = 14$, let $E$ and $F$ be the midpoints of $AB$ and $AC$. If a circle passing through $B$ and $C$ is tangent to the circumcircle of $AEF$ at point $X \ne A$, find $AX$. [i]Proposed by Vivian Loh [/i]

2020-2021 OMMC, 6
Tags: ommc
Find the minimum possible value of $$\left(\sqrt{x^2+4} + \sqrt{x^2+7\sqrt{3}x + 49}\right)^2$$ over all real numbers.

2021-2022 OMMC, 9
Tags: ommc
$12$ people stand in a row. Each person is given a red shirt or a blue shirt. Every minute, exactly one pair of two people with the same color currently standing next to each other in the row leave. After $6$ minutes, everyone has left. How many ways could the shirts have been assigned initially? [i]Proposed by Evan Chang[/i]

OMMC POTM, 2022 11
Tags: combinatorics , ommc
Let $S$ be the set of colorings of a $100 \times 100$ grid where each square is colored black or white and no $2\times2$ subgrid is colored like a chessboard. A random such coloring is chosen: what is the probability there is a path of black squares going from the top row to the bottom row where any two consecutive squares in the path are adjacent? [i]Proposed by Evan Chang (squareman), USA [/i]

2020-2021 OMMC, 8
Tags: ommc , geometry
Let triangle $MAD$ be inscribed in circle $O$ with diameter $85$ such that $MA = 68$ and $DA = 40$. The altitudes from $M, D$ to sides $AD$ and $MA$, respectively, intersect the tangent to circle $O$ at $A$ at $X$ and $Y$ respectively. $XA \times YA$ can be expressed as $\frac{a}{b}$, where $a$ and $ b$ are relatively prime positive integers. Find $a + b$.

2020-2021 OMMC, 2
Tags: ommc , algebra , functions
The function $f(x)$ is defined on the reals such that $$f\left(\frac{1-4x}{4-x}\right) = 4-xf(x)$$ for all $x \ne 4$. There exists two distinct real numbers $a, b \ne 4$ such that $f(a) = f(b) = \frac{5}{2}$. $a+b$ can be represented as $\frac{p}{q}$ where $p, q$ are relatively prime positive integers. Find $10p + q$.

OMMC POTM, 2023 4
Tags: geometry , ommc , cyclic quadrilateral
Let $ABCD$ be a quadrilateral inscribed in a circle with center $O$. Points $X$ and $Y$ lie on sides $AB$ and $CD$, respectively. Suppose the circumcircles of $CDX$ and $ABY$ meet line $XY$ again at $P$ and $Q$ respectively. Show that $OP=OQ$. [i]Proposed by Evan Chang (squareman), USA[/i]

OMMC POTM, 2022 6
Tags: geometry , geometric transformation , rotation , ommc
Let $G$ be the centroid of $\triangle ABC.$ A rotation $120^\circ$ clockwise about $G$ takes $B$ and $C$ to $B_1$ and $C_1$ respectively. A rotation $120^\circ$ counterclockwise about $G$ takes $B$ and $C$ to $B_2$ and $C_2$ respectively. Prove $\triangle AB_1C_2$ and $\triangle AB_2C_1$ are equilateral. [i]Proposed by Evan Chang (squareman), USA [/i] [img]https://cdn.artofproblemsolving.com/attachments/3/b/46b4f09edcf17755df2dea3546881475db6eff.png[/img]

2020-2021 OMMC, 5
Tags: ommc
How many nonempty subsets of ${1, 2, \dots, 15}$ are there such that the sum of the squares of each subset is a multiple of $5$?