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

2020 Moldova EGMO TST, 4
Tags: geometry , incenter , angle bisector
The incircle of triangle $ABC$ touches $AC$ and $BC$ respectively $P$ and $Q$. Let $N$ and $M$ be the midpoints of the sides $AC$ and $BC$ respectively.$AM$ and $BP$,$BN$ and $AQ$ intersects at the points $X$ and $Y$ respectively. If the points $C,X$ and $Y$ are collinear , then prove that $CX$ is the angle bisector of $\angle ACB$.

2019 Hanoi Open Mathematics Competitions, 13
Tags: triangle inequality , geometry , equilateral , distance
Find all points inside a given equilateral triangle such that the distances from it to three sides of the given triangle are the side lengths of a triangle.

2025 Sharygin Geometry Olympiad, 24
Tags: geometry , 3d geometry , geometric inequality
The insphere of a tetrahedron $ABCD$ touches the faces $ABC$, $BCD$, $CDA$, $DAB$ at $D^{\prime}$, $A^{\prime}$, $B^{\prime}$, $C^{\prime}$ respectively. Denote by $S_{AB}$ the area of the triangle $AC^{\prime}B^{\prime}$. Define similarly $S_{AC}$, $S_{BC},$ $S_{AD}$, $S_{BD}$, $S_{CD}$. Prove that there exists a triangle with sidelengths $\sqrt{S_{AB}S_{CD}}$, $\sqrt{S_{AC}S_{BD}}$ , $\sqrt{S_{AD}S_{BC}}$. Proposed by: S.Arutyunyan

2012 Cuba MO, 6
Tags: geometry
Let $ABC$ be a right triangle at $A$, and let $AD$ be the relative height to the hypotenuse. Let $N$ be the intersection of the bisector of the angle of vertex $C$ with $AD$. Prove that $$AD \cdot BC = AB \cdot DC + BD \cdot AN.$$

2009 Switzerland - Final Round, 4
Tags: combinatorics
Let $n$ be a natural number. Each cell of a $n \times n$ square contains one of $n$ different symbols, such that each of the symbols is in exactly $n$ cells. Show that a row or a column exists that contains at least \sqrt{n} different symbols.

2019 Harvard-MIT Mathematics Tournament, 8
Tags: hmmt , geometry
In triangle $ABC$ with $AB < AC$, let $H$ be the orthocenter and $O$ be the circumcenter. Given that the midpoint of $OH$ lies on $BC$, $BC = 1$, and the perimeter of $ABC$ is 6, find the area of $ABC$.

1997 Akdeniz University MO, 3
Tags: equation , number theory , sequence
$(x_n)$ be a sequence with $x_1=0$, $$x_{n+1}=5x_n + \sqrt{24x_n^2+1}$$. Prove that for $k \geq 2$ $x_k$ is a natural number.

2006 AMC 12/AHSME, 22
Tags: floor function , function , inequalities , algebra
Suppose $ a, b,$ and $ c$ are positive integers with $ a \plus{} b \plus{} c \equal{} 2006$, and $ a!b!c! \equal{} m\cdot10^n$, where $ m$ and $ n$ are integers and $ m$ is not divisible by 10. What is the smallest possible value of $ n$? $ \textbf{(A) } 489 \qquad \textbf{(B) } 492 \qquad \textbf{(C) } 495 \qquad \textbf{(D) } 498 \qquad \textbf{(E) } 501$

2021 IMO Shortlist, C8
Tags: combinatorics
Determine the largest integer $N$ for which there exists a table $T$ of integers with $N$ rows and $100$ columns that has the following properties: $\text{(i)}$ Every row contains the numbers $1$, $2$, $\ldots$, $100$ in some order. $\text{(ii)}$ For any two distinct rows $r$ and $s$, there is a column $c$ such that $|T(r,c) - T(s, c)|\geq 2$. (Here $T(r,c)$ is the entry in row $r$ and column $c$.)

2009 Argentina Iberoamerican TST, 3
Tags: search , symmetry , combinatorics
Within a group of $ 2009$ people, every two people has exactly one common friend. Find the least value of the difference between the person with maximum number of friends and the person with minimum number of friends.

1962 AMC 12/AHSME, 31
Tags: ratio
The ratio of the interior angles of two regular polygons with sides of unit length is $ 3: 2$. How many such pairs are there? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ \text{infinitely many}$

1992 Miklós Schweitzer, 5
Tags: number theory , real analysis
Prove that if the $a_i$'s are different natural numbers, then $\sum_ {j = 1}^n a_j ^ 2 \prod_{k \neq j} \frac{a_j + a_k}{a_j-a_k}$ is a square number.

Estonia Open Junior - geometry, 2012.1.3
Tags: ratio , geometry , rectangle , right triangle
A rectangle $ABEF$ is drawn on the leg $AB$ of a right triangle $ABC$, whose apex $F$ is on the leg $AC$. Let $X$ be the intersection of the diagonal of the rectangle $AE$ and the hypotenuse $BC$ of the triangle. In what ratio does point $X$ divide the hypotenuse $BC$ if it is known that $| AC | = 3 | AB |$ and $| AF | = 2 | AB |$?

1991 Vietnam National Olympiad, 1
Tags: combinatorics
$1991$ students sit around a circle and play the following game. Starting from some student $A$ and counting clockwise, each student on turn says a number. The numbers are $1,2,3,1,2,3,...$ A student who says $2$ or $3$ must leave the circle. The game is over when there is only one student left. What position was the remaining student sitting at the beginning of the game?

2014 Danube Mathematical Competition, 3
Tags: midpoint , geometry , right angle , orthocenter
Let $ABC$ be a triangle with $\angle A<90^o, AB \ne AC$. Denote $H$ the orthocenter of triangle $ABC$, $N$ the midpoint of segment $[AH]$, $M$ the midpoint of segment $[BC]$ and $D$ the intersection point of the angle bisector of $\angle BAC$ with the segment $[MN]$. Prove that $<ADH=90^o$

2021/2022 Tournament of Towns, P7
Tags: combinatorics , combinatorial geometry
A checkered square of size $2\times2$ is covered by two triangles. Is it necessarily true that: [list=a] [*]at least one of its four cells is fully covered by one of the triangles; [*]some square of size $1\times1$ can be placed into one of these triangles? [/list] [i]Alexandr Shapovalov[/i]

2002 Chile National Olympiad, 5
Tags: analytic geometry , geometry
Given a right triangle $T$, where the coordinates of its vertices are integers, let $E$ be the number of points of integer coordinates that belong to the edge of the triangle $T$, $I$ the number of points of integer coordinates that belong to the interior of the triangle $T$. Show that the area $A(T)$ of triangle $T$ is given by: $A(T) = \frac{E}{2}+I -1$.

2017 APMO, 5
Tags: combinatorics
Let $n$ be a positive integer. A pair of $n$-tuples $(a_1,\cdots{}, a_n)$ and $(b_1,\cdots{}, b_n)$ with integer entries is called an [i]exquisite pair[/i] if $$|a_1b_1+\cdots{}+a_nb_n|\le 1.$$ Determine the maximum number of distinct $n$-tuples with integer entries such that any two of them form an exquisite pair. [i]Pakawut Jiradilok and Warut Suksompong, Thailand[/i]

2020 Princeton University Math Competition, A1/B3
Tags: algebra
Let $f(x) =\frac{x+a}{x+b}$ satisfy $f(f(f(x))) = x$ for real numbers $a, b$. If the maximum value of a is $p/q$, where $p, q$ are relatively prime integers, what is $|p| + |q|$?

1988 Mexico National Olympiad, 5
Tags: number theory , coprime , greatest common divisor
If $a$ and $b$ are coprime positive integers and $n$ an integer, prove that the greatest common divisor of $a^2+b^2-nab$ and $a+b$ divides $n+2$.

V Soros Olympiad 1998 - 99 (Russia), 11.9
Tags: algebra , number theory
It is known that unequal numbers $a$,$b$ and $c$ are successive members of an arithmetic progression, all of them are greater than $1000$ and all are squares of natural numbers. Find the smallest possible value of $b$.

2010 Pan African, 1
Tags: prime number , number theory
a) Show that it is possible to pair off the numbers $1,2,3,\ldots ,10$ so that the sums of each of the five pairs are five different prime numbers. b) Is it possible to pair off the numbers $1,2,3,\ldots ,20$ so that the sums of each of the ten pairs are ten different prime numbers?

1992 National High School Mathematics League, 11
Tags:
For real numbers $a_1,a_2,\cdots,a_{100}$, $a_1=a_2=1,a_3=2$. For any positive integer $n$, $a_na_{n+1}a_{n+2}\neq1,a_na_{n+1}a_{n+2}a_{n+3}=a_n+a_{n+1}+a_{n+2}+a_{n+3}$, then $a_1+a_2+\cdots+a_{100}=$________.

2010 Purple Comet Problems, 5
Tags:
If $a$ and $b$ are positive integers such that $a \cdot b = 2400,$ find the least possible value of $a + b.$

2009 AMC 12/AHSME, 3
Tags: percent
Twenty percent less than $ 60$ is one-third more than what number? $ \textbf{(A)}\ 16\qquad \textbf{(B)}\ 30\qquad \textbf{(C)}\ 32\qquad \textbf{(D)}\ 36\qquad \textbf{(E)}\ 48$