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: 85335

2003 Manhattan Mathematical Olympiad, 2
Tags: pigeonhole principle
A tennis net is made of strings tied up together which make a grid consisting of small congruent squares as shown below. [asy] size(500); xaxis(-50,50); yaxis(-5,5); add(shift(-50,-5)*grid(100,10));[/asy] The size of the net is $100\times 10$ small squares. What is the maximal number of edges of small squares which can be cut without breaking the net into two pieces? (If an edge is cut, the cut is made in the middle, not at the ends.)

2018 BMT Spring, 1
Tags: combinatorics
Bob has $3$ different fountain pens and $11$ different ink colors. How many ways can he fill his fountain pens with ink if he can only put one ink in each pen?

2018 China Western Mathematical Olympiad, 7
Tags: number theory , least common multiple , inequalities
Let $p$ and $c$ be an prime and a composite, respectively. Prove that there exist two integers $m,n,$ such that $$0<m-n<\frac{\textup{lcm}(n+1,n+2,\cdots,m)}{\textup{lcm}(n,n+1,\cdots,m-1)}=p^c.$$

2024 CMIMC Team, 5
Tags: team
An ant is currently on a vertex of the top face on a 6-sided die. The ant wants to travel to the opposite vertex of the die (the vertex that is farthest from the start), and the ant can travel along edges of the die to other vertices that are on the top face of the die. Every second, the ant picks a valid edge to move along, and the die randomly flips to an adjacent face. If the ant is on any of the bottom vertices after the flip, it is crushed and dies. What is the probability that the ant makes it to its target? (If the ant makes it to the target and the die rolls to crush it, it achieved its dreams before dying, so this counts.) [i]Proposed by Lohith Tummala[/i]

2017 IMO Shortlist, G4
Tags: homothety , power of a point , excircle , geometry , tangent circle
In triangle $ABC$, let $\omega$ be the excircle opposite to $A$. Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.

1935 Moscow Mathematical Olympiad, 018
Tags: sum , sum of cubes , algebra
Evaluate the sum: $1^3 + 3^3 + 5^3 +... + (2n - 1)^3$.

I Soros Olympiad 1994-95 (Rus + Ukr), 9.1
Tags: number theory
The number $1995$ is divisible by both $19$ and $95$. How many four-digit numbers are there that are divisible by two-digit numbers formed by both its first two digits and its last two digits?

1997 AIME Problems, 4
Tags: number theory , relatively prime , pythagorean theorem , geometry
Circles of radii 5, 5, 8, and $m/n$ are mutually externally tangent, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

JBMO Geometry Collection, 2008
Tags: trigonometry , angle bisector , perpendicular bisector , power of a point , geometry
The vertices $ A$ and $ B$ of an equilateral triangle $ ABC$ lie on a circle $k$ of radius $1$, and the vertex $ C$ is in the interior of the circle $ k$. A point $ D$, different from $ B$, lies on $ k$ so that $ AD\equal{}AB$. The line $ DC$ intersects $ k$ for the second time at point $ E$. Find the length of the line segment $ CE$.

2022 Thailand TST, 3
Tags: number theory
Determine all integers $n\geqslant 2$ with the following property: every $n$ pairwise distinct integers whose sum is not divisible by $n$ can be arranged in some order $a_1,a_2,\ldots, a_n$ so that $n$ divides $1\cdot a_1+2\cdot a_2+\cdots+n\cdot a_n.$ [i]Arsenii Nikolaiev, Anton Trygub, Oleksii Masalitin, and Fedir Yudin[/i]