Found problems: 85335
2025 Harvard-MIT Mathematics Tournament, 8
Let $\triangle{ABC}$ be a triangle with incenter $I.$ The incircle of triangle $\triangle{ABC}$ touches $\overline{BC}$ at $D.$ Let $M$ be the midpoint of $\overline{BC},$ and let line $AI$ meet the circumcircle of triangle $\triangle{ABC}$ again at $L \neq A.$ Let $\omega$ be the circle centered at $L$ tangent to $AB$ and $AC.$ If $\omega$ intersects $\overline{AD}$ at point $P,$ prove that $\angle{IPM}=90^\circ.$
2020 Lusophon Mathematical Olympiad, 1
In certain country, the coins have the following values: $2^0, 2^1, 2^2,\dots 2^{10}$. A cash machine has $1000$ coins of each value and give the money using each coin(of each value) at most once. The customers order all the positive integers: $1,2,3,4,5,\dots$ (in this order) in coins.
a) Determine the first integer, such that the cash machine cannot provide.
b) In the moment that the first customer can not be attended, by the lack of coins, what are the coins which are not available in the cash machine?
2005 Austria Beginners' Competition, 4
We are given the triangle $ABC$ with an area of $2000$. Let $P,Q,R$ be the midpoints of the sides $BC$, $AC$, $AB$. Let $U,V,W$ be the midpoints of the sides $QR$, $PR$, $PQ$. The lengths of the line segments $AU$, $BV$, $CW$ are $x$, $y$, $z$. Show that there exists a triangle with side lengths $x$, $y$ and $z$ and caluclate it's area.
2020 Iran MO (3rd Round), 1
$1)$. Prove a graph with $2n$ vertices and $n+2$ edges has an independent set of size $n$ (there are $n$ vertices such that no two of them are adjacent ).
$2)$.Find the number of graphs with $2n$ vertices and $n+3$ edges , such that among any $n$ vertices there is an edge connecting two of them
1995 Portugal MO, 1
Joao Salta-Pocinhas jumps $1$ meter in the first jump, $2$ meters in the second, $4$ meters in the third, . . ., $2^{n-1}$ meters in jump number $n$. Is there any possibility for Joao to choose the directions of his jumps in order to get back to the starting point?
2023 Bulgaria National Olympiad, 1
Let $G$ be a graph on $n\geq 6$ vertices and every vertex is of degree at least 3. If $C_{1}, C_{2}, \dots, C_{k}$ are all the cycles in $G$, determine all possible values of $\gcd(|C_{1}|, |C_{2}|, \dots, |C_{k}|)$ where $|C|$ denotes the number of vertices in the cycle $C$.
2021 Malaysia IMONST 2, 3
Given a cube. On each edge of the cube, we write a number, either $1$ or $-1$. For each face of the cube, we multiply the four numbers on the edges of this face, and write the product on this face. Finally, we add all the eighteen numbers that we wrote down on the edges and face of the cube.
What is the smallest possible sum that we can get?
2013 AMC 10, 23
In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Distinct points $D$, $E$, and $F$ lie on segments $\overline{BC}$, $\overline{CA}$, and $\overline{DE}$, respectively, such that $\overline{AD}\perp\overline{BC}$, $\overline{DE}\perp\overline{AC}$, and $\overline{AF}\perp\overline{BF}$. The length of segment $\overline{DF}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
${ \textbf{(A)}\ 18\qquad\textbf{(B)}\ 21\qquad\textbf{(C)}\ 24\qquad\textbf{(D}}\ 27\qquad\textbf{(E)}\ 30 $
2016 Nordic, 1
Determine all sequences of non-negative integers $a_1, \ldots, a_{2016}$ all less than or equal to $2016$ satisfying $i+j\mid ia_i+ja_j$ for all $i, j\in \{ 1,2,\ldots, 2016\}$.
2013 Romania National Olympiad, 3
Find all real $x > 0$ and integer $n > 0$ so that $$ \lfloor x \rfloor+\left\{ \frac{1}{x}\right\}= 1.005 \cdot n.$$
2021 Sharygin Geometry Olympiad, 8.7
Let $ABCDE$ be a convex pentagon such that angles $CAB$, $BCA$, $ECD$, $DEC$ and $AEC$ are equal. Prove that $CE$ bisects $BD$.
2020 Purple Comet Problems, 6
Alex launches his boat into a river and heads upstream at a constant speed. At the same time at a point $8$ miles upstream from Alex, Alice launches her boat and heads downstream at a constant speed. Both boats move at $6$ miles per hour in still water, but the river is owing downstream at $2\frac{3}{10}$ miles per hour. Alex and Alice will meet at a point that is $\frac{m}{n}$ miles from Alex's starting point, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2004 China Team Selection Test, 3
Find all positive integer $ n$ satisfying the following condition: There exist positive integers $ m$, $ a_1$, $ a_2$, $ \cdots$, $ a_{m\minus{}1}$, such that $ \displaystyle n \equal{} \sum_{i\equal{}1}^{m\minus{}1} a_i(m\minus{}a_i)$, where $ a_1$, $ a_2$, $ \cdots$, $ a_{m\minus{}1}$ may not distinct and $ 1 \leq a_i \leq m\minus{}1$.
1999 Bulgaria National Olympiad, 1
The faces of a box with integer edge lengths are painted green. The box is partitioned into unit cubes. Find the dimensions of the box if the number of unit cubes with no green face is one third of the total number of cubes.
2025 USAMO, 1
Let $k$ and $d$ be positive integers. Prove that there exists a positive integer $N$ such that for every odd integer $n>N$, the digits in the base-$2n$ representation of $n^k$ are all greater than $d$.
1997 Irish Math Olympiad, 2
For a point $ M$ inside an equilateral triangle $ ABC$, let $ D,E,F$ be the feet of the perpendiculars from $ M$ onto $ BC,CA,AB$, respectively. Find the locus of all such points $ M$ for which $ \angle FDE$ is a right angle.
2008 Sharygin Geometry Olympiad, 1
(A.Zaslavsky) A convex polygon can be divided into 2008 congruent quadrilaterals. Is it true that this polygon has a center or an axis of symmetry?
2000 Harvard-MIT Mathematics Tournament, 45
Find all positive integers $x$ for which there exists a positive integer $y$ such that $\dbinom{x}{y}=1999000$
2006 Finnish National High School Mathematics Competition, 3
The numbers $p, 4p^2 + 1,$ and $6p^2 + 1$ are primes. Determine $p.$
2002 India IMO Training Camp, 4
Let $O$ be the circumcenter and $H$ the orthocenter of an acute triangle $ABC$. Show that there exist points $D$, $E$, and $F$ on sides $BC$, $CA$, and $AB$ respectively such that \[ OD + DH = OE + EH = OF + FH\] and the lines $AD$, $BE$, and $CF$ are concurrent.
2009 Purple Comet Problems, 5
Find $n$ so that $(4^{n+7})^3=(2^{n+23})^4.$
2017 AMC 8, 1
Which of the following values is largest?
$\textbf{(A) }2+0+1+7\qquad\textbf{(B) }2 \times 0 +1+7\qquad\textbf{(C) }2+0 \times 1 + 7\qquad\textbf{(D) }2+0+1 \times 7\qquad\textbf{(E) }2 \times 0 \times 1 \times 7$
1995 Vietnam Team Selection Test, 1
A graph has $ n$ vertices and $ \frac {1}{2}\left(n^2 \minus{} 3n \plus{} 4\right)$ edges. There is an edge such that, after removing it, the graph becomes unconnected. Find the greatest possible length $ k$ of a circuit in such a graph.
1966 IMO Longlists, 43
Given $5$ points in a plane, no three of them being collinear. Each two of these $5$ points are joined with a segment, and every of these segments is painted either red or blue; assume that there is no triangle whose sides are segments of equal color.
[b]a.)[/b] Show that:
[i](1)[/i] Among the four segments originating at any of the $5$ points, two are red and two are blue.
[i](2)[/i] The red segments form a closed way passing through all $5$ given points. (Similarly for the blue segments.)
[b]b.)[/b] Give a plan how to paint the segments either red or blue in order to have the condition (no triangle with equally colored sides) satisfied.
2024 Thailand TSTST, 12
We call polynomial $S(x)\in\mathbb{R}[x]$ sadeh whenever it's divisible by $x$ but not divisible by $x^2$.
For the polynomial $P(x)\in\mathbb{R}[x]$ we know that there exists a sadeh polynomial $Q(x)$ such that $P(Q(x))-Q(2x)$ is divisible by $x^2$. Prove that there exists sadeh polynomial $R(x)$ such that $P(R(x))-R(2x)$ is divisible by $x^{1401}$.