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

2020-2021 OMMC, 5
Tags: ommc
Let $N$ be an 3 digit integer in base $10$ such that the sum of its digits in base $4$ is half the sum of its digits in base $8$. In base $10$, find the largest possible value of $N$.

OMMC POTM, 2023 7
Tags: number theory , ommc , Digits
Let $N$ be a positive integer. Prove that at least one of the numbers $N$ of $3N$ contains at least one of the digits $1,2,9$. [i]Proposed by Evan Chang (squareman), USA[/i]

2021-2022 OMMC, 14
Tags: ommc
The corners of a $2$-dimensional room in the shape of an isosceles right triangle are labeled $A$, $B$, $C$ where $AB = BC$. Walls $BC$ and $CA$ are mirrors. A laser is shot from $A$, hits off of each of the mirrors once and lands at a point $X$ on $AB$. Let $Y$ be the point where the laser hits off $AC$. If $\tfrac{AB}{AX} = 64$, $\tfrac{CA}{AY} = \tfrac pq$ for coprime positive integers $p$, $q$. Find $p + q$. [i]Proposed by Sid Doppalapudi[/i]

2020-2021 OMMC, 9
Tags: ommc
The infinite sequence of integers $a_1, a_2, \cdots $ is defined recursively as follows: $a_1 = 3$, $a_2 = 7$, and $a_n$ equals the alternating sum $$a_1 - 2a_2 + 3a_3 - 4a_4 + \cdots (-1)^n \cdot (n-1)a_{n-1}$$ for all $n > 2$. Let $a_x$ be the smallest positive multiple of $1090$ appearing in this sequence. Find the remainder of $a_x$ when divided by $113$.

2020-2021 OMMC, 5
Tags: ommc
Two points $A, B$ are randomly chosen on a circle with radius $100.$ For a positive integer $x$, denote $P(x)$ as the probability that the length of $AB$ is less than $x$. Find the minimum possible integer value of $x$ such that $\text{P}(x) > \frac{2}{3}$.

2021-2022 OMMC, 18
Tags: ommc
Define mutually externally tangent circles $\omega_1$, $\omega_2$, and $\omega_3$. Let $\omega_1$ and $\omega_2$ be tangent at $P$. The common external tangents of $\omega_1$ and $\omega_2$ meet at $Q$. Let $O$ be the center of $\omega_3$. If $QP = 420$ and $QO = 427$, find the radius of $\omega_3$. [i]Proposed by Tanishq Pauskar and Mahith Gottipati[/i]

2020-2021 OMMC, 14
Tags: ommc
There exist positive integers $N, M$ such that $N$'s remainders modulo the four integers $6, 36,$ $216,$ and $M$ form an increasing nonzero geometric sequence in that order. Find the smallest possible value of $M$.

OMMC POTM, 2023 1
Tags: Coloring , combinatorics , ommc
Define a $100 \times 100$ square grid $G$. Initially color all cells of $G$ white. A move consists of selecting a $1 \times 7$ or $7 \times 1$ subgrid of $G$ and flipping the colors of all cells in this subgrid from white to black or vice versa. Is it possible that after a series of moves, all cells are colored black? [i]Proposed by Evan Chang (squareman), USA[/i]

2020-2021 OMMC, 1
Tags: ommc , algebra
A man rows at a speed of $2$ mph in still water. He set out on a trip towards a spot $2$ miles downstream. He rowed with the current until he was halfway there, then turned back and rowed against the current for $15$ minutes. Then, he turned around again and rowed with the current until he reached his destination. The entire trip took him $70$ minutes. The speed of the current can be represented as $\frac{p}{q}$ mph where $p,q$ are relatively prime positive integers. Find $10p+q$.

2021-2022 OMMC, 19
Tags: ommc
$N$ people have a series of calls. Each call is between two people, and is started by exactly one of them. Each person starts at most $10$ calls. Two people can call at most once. In any group of $3$ people, there are at least two people who have a call. Find the maximum possible value of $N$. [i]Proposed by Serena Xu[/i]