Found problems: 85335
1991 Greece National Olympiad, 2
Let $O$ be the circumcenter of triangle $ABC$ and let $A_1,B_1,C_1$ be the midpoints of arcs $BC, CA,AB$ respectively. If $I$ is the incenter of triangle $ABC$, prove that $$\overrightarrow{OI}= \overrightarrow{OA_1}+ \overrightarrow{OB_1}+ \overrightarrow{OC_1}.$$
2009 IMO Shortlist, 3
Let $ABC$ be a triangle. The incircle of $ABC$ touches the sides $AB$ and $AC$ at the points $Z$ and $Y$, respectively. Let $G$ be the point where the lines $BY$ and $CZ$ meet, and let $R$ and $S$ be points such that the two quadrilaterals $BCYR$ and $BCSZ$ are parallelogram.
Prove that $GR=GS$.
[i]Proposed by Hossein Karke Abadi, Iran[/i]
1996 Estonia National Olympiad, 3
The vertices of the quadrilateral $ABCD$ lie on a single circle. The diagonals of this rectangle divide the angles of the rectangle at vertices $A$ and $B$ and divides the angles at vertices $C$ and $D$ in a $1: 2$ ratio. Find angles of the quadrilateral $ABCD$.
Kvant 2020, M2625
A connected checkered figure is drawn on a checkered paper. It is known that the figure can be cut both into $2\times 2$ squares and into (possibly rotated) [url=https://upload.wikimedia.org/wikipedia/commons/thumb/3/30/Tetromino-skew2.svg/1200px-Tetromino-skew2.svg.png]skew-tetrominoes[/url]. Prove that there is a hole in the figure.
[i]Proposed by Y. Markelov and A. Sairanov[/i]
1983 Swedish Mathematical Competition, 5
Show that a unit square can be covered with three equal disks with radius less than $\frac{1}{\sqrt{2}}$.
What is the smallest possible radius?
1999 Tuymaada Olympiad, 1
50 knights of King Arthur sat at the Round Table. A glass of white or red wine stood before each of them. It is known that at least one glass of red wine and at least one glass of white wine stood on the table. The king clapped his hands twice. After the first clap every knight with a glass of red wine before him took a glass from his left neighbour. After the second clap every knight with a glass of white wine (and possibly something more) before him gave this glass to the left neughbour of his left neighbour. Prove that some knight was left without wine.
[i]Proposed by A. Khrabrov, incorrect translation from Hungarian[/i]
2004 China Team Selection Test, 3
$ S$ is a non-empty subset of the set $ \{ 1, 2, \cdots, 108 \}$, satisfying:
(1) For any two numbers $ a,b \in S$ ( may not distinct), there exists $ c \in S$, such that $ \gcd(a,c)\equal{}\gcd(b,c)\equal{}1$.
(2) For any two numbers $ a,b \in S$ ( may not distinct), there exists $ c' \in S$, $ c' \neq a$, $ c' \neq b$, such that $ \gcd(a, c') > 1$, $ \gcd(b,c') >1$.
Find the largest possible value of $ |S|$.
2016 Ecuador Juniors, 3
Let $P_1P_2 . . . P_{2016 }$ be a cyclic polygon of $2016$ sides. Let $K$ be a point inside the polygon and let $M$ be the midpoint of the segment $P_{1000}P_{2000}$. Knowing that $KP_1 = KP_{2011} = 2016$ and $KM$ is perpendicular to $P_{1000}P_{2000}$, find the length of segment $KP_{2016}$.
2009 Romania National Olympiad, 3
Find all functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that verify the relation
$$ f\left( x^3+y^3 \right) =xf\left( y^2 \right) + yf\left( x^2 \right) , $$
for all real numbers $ x,y. $
2014 Purple Comet Problems, 4
Find the least positive integer $n$ such that the prime factorizations of $n$, $n + 1$, and $n + 2$ each have exactly two factors (as $4$ and $6$ do, but $12$ does not).
2000 AIME Problems, 5
Given eight distinguishable rings, let $n$ be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand. The order of rings on each finger is significant, but it is not required that each finger have a ring. Find the leftmost three nonzero digits of $n.$
2010 Iran MO (3rd Round), 6
Suppose that $X$ is a set with $n$ elements and $\mathcal F\subseteq X^{(k)}$ and $X_1,X_2,...,X_s$ is a partition of $X$. We know that for every $A,B\in \mathcal F$ and every $1\le j\le s$, $E=B\cap (\bigcup_{i=1}^{j}X_i)\neq A\cap (\bigcup_{i=1}^{j} X_i)=F$ shows that none of $E,F$ contains the other one. Prove that
\[|\mathcal F|\le \max_{\sum\limits_{i=1}^{S}w_i=k}\prod_{j=1}^{s}\binom{|X_j|}{w_j}\]
(15 points)
Exam time was 5 hours and 20 minutes.
2020 Moldova Team Selection Test, 1
All members of geometrical progression $(b_n)_{n\geq1}$ are members of some arithmetical progression. It is known that $b_1$ is an integer. Prove that all members of this geometrical progression are integers. (progression is infinite)
2021 Durer Math Competition Finals, 13
The trapezoid $ABCD$ satisfies $AB \parallel CD$, $AB = 70$, $AD = 32$ and $BC = 49$. We also know that $\angle ABC = 3 \angle ADC$. How long is the base $CD$?
2009 South africa National Olympiad, 1
Determine the smallest integer $n > 1$ with the property that $n^2(n - 1)$ is divisible by 2009.
2021 USAJMO, 4
Carina has three pins, labeled $A, B$, and $C$, respectively, located at the origin of the coordinate plane. In a [i]move[/i], Carina may move a pin to an adjacent lattice point at distance $1$ away. What is the least number of moves that Carina can make in order for triangle $ABC$ to have area 2021?
(A lattice point is a point $(x, y)$ in the coordinate plane where $x$ and $y$ are both integers, not necessarily positive.)
2013 China Girls Math Olympiad, 3
In a group of $m$ girls and $n$ boys, any two persons either know each other or do not know each other. For any two boys and any two girls, there are at least one boy and one girl among them,who do not know each other. Prove that the number of unordered pairs of (boy, girl) who know each other does not exceed $m+\frac{n(n-1)}{2}$.
2006 AMC 12/AHSME, 20
A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vertex exactly once?
$ \textbf{(A) } \frac {1}{2187} \qquad \textbf{(B) } \frac {1}{729} \qquad \textbf{(C) } \frac {2}{243} \qquad \textbf{(D) } \frac {1}{81} \qquad \textbf{(E) } \frac {5}{243}$
Kvant 2024, M2804
There are two equal circles of radius $1$ placed inside the triangle $ABC$ with side $BC = 6$. The circles are tangent to each other, one is inscribed in angle $B$, the other one is inscribed in angle $C$.
(a) Prove that the centroid $M$ of the triangle $ABC$ does not lie inside any of the given circles.
(b) Prove that if $M$ lies on one of the circles, then the triangle $ABC$ is isosceles.
2022 Brazil National Olympiad, 3
Let $\{a_n\}_{n=0}^{\infty}$ be a sequence of integers numbers. Let $\Delta^1a_n=a_{n+1}-a_n$ for a non-negative integer $n$. Define $\Delta^Ma_n= \Delta^{M-1}a_{n+1}- \Delta^{M-1}a_n$. A sequence is [i]patriota[/i] if there are positive integers $k,l$ such that $a_{n+k}=\Delta^Ma_{n+l}$ for all non-negative integers $n$. Determine, with proof, whether exists a sequence that the last value of $M$ for which the sequence is [i]patriota[/i] is $2022$.
2011 Korea - Final Round, 2
$ABC$ is a triangle such that $AC<AB<BC$ and $D$ is a point on side $AB$ satisfying $AC=AD$.
The circumcircle of $ABC$ meets with the bisector of angle $A$ again at $E$ and meets with $CD$ again at $F$.
$K$ is an intersection point of $BC$ and $DE$. Prove that $CK=AC$ is a necessary and sufficient condition for $DK \cdot EF = AC \cdot DF$.
KoMaL A Problems 2023/2024, A. 876
Find all non negative integers $a{}$ and $b{}$ satisfying $5^a+6=31^b$
[i]Proposed by Erik Füredi, Budapest[/i]
2014 Contests, 3
The triangle $ABC$ is inscribed in a circle $w_1$. Inscribed in a triangle circle touchs the sides $BC$ in a point $N$. $w_2$ — the circle inscribed in a segment $BAC$ circle of $w_1$, and passing through a point $N$. Let points $O$ and $J$ — the centers of circles $w_2$ and an extra inscribed circle (touching side $BC$) respectively. Prove, that lines $AO$ and $JN$ are parallel.
2010 Contests, 1
The picture below shows the way Juan wants to divide a square field in three regions, so that all three of them share a well at vertex $B$. If the side length of the field is $60$ meters, and each one of the three regions has the same area, how far must the points $M$ and $N$ be from $D$?
Note: the area of each region includes the area the well occupies.
[asy]
pair A=(0,0),B=(60,0),C=(60,-60),D=(0,-60),M=(0,-40),N=(20,-60);
pathpen=black;
D(MP("A",A,W)--MP("B",B,NE)--MP("C",C,SE)--MP("D",D,SW)--cycle);
D(B--MP("M",M,W));
D(B--MP("N",N,S));
D(CR(B,3));[/asy]
2010 Romania National Olympiad, 4
Let $a,b,c,d$ be positive integers, and let $p=a+b+c+d$. Prove that if $p$ is a prime, then $p$ is not a divisor of $ab-cd$.
[i]Marian Andronache[/i]