Found problems: 1581
2011 Brazil Team Selection Test, 2
Five points $A_1,A_2,A_3,A_4,A_5$ lie on a plane in such a way that no three among them lie on a same straight line. Determine the maximum possible value that the minimum value for the angles $\angle A_iA_jA_k$ can take where $i, j, k$ are distinct integers between $1$ and $5$.
2008 Federal Competition For Advanced Students, Part 2, 3
We are given a line $ g$ with four successive points $ P$, $ Q$, $ R$, $ S$, reading from left to right. Describe a straightedge and compass construction yielding a square $ ABCD$ such that $ P$ lies on the line $ AD$, $ Q$ on the line $ BC$, $ R$ on the line $ AB$ and $ S$ on the line $ CD$.
2002 India IMO Training Camp, 18
Consider the square grid with $A=(0,0)$ and $C=(n,n)$ at its diagonal ends. Paths from $A$ to $C$ are composed of moves one unit to the right or one unit up. Let $C_n$ (n-th catalan number) be the number of paths from $A$ to $C$ which stay on or below the diagonal $AC$. Show that the number of paths from $A$ to $C$ which cross $AC$ from below at most twice is equal to $C_{n+2}-2C_{n+1}+C_n$
2010 All-Russian Olympiad, 3
Lines tangent to circle $O$ in points $A$ and $B$, intersect in point $P$. Point $Z$ is the center of $O$. On the minor arc $AB$, point $C$ is chosen not on the midpoint of the arc. Lines $AC$ and $PB$ intersect at point $D$. Lines $BC$ and $AP$ intersect at point $E$. Prove that the circumcentres of triangles $ACE$, $BCD$, and $PCZ$ are collinear.
1997 Hungary-Israel Binational, 3
Can a closed disk can be decomposed into a union of two congruent parts having no common point?
2022 China Second Round A2, 2
$A,B,C,D,E$ are points on a circle $\omega$, satisfying $AB=BD$, $BC=CE$. $AC$ meets $BE$ at $P$. $Q$ is on $DE$ such that $BE//AQ$. Suppose $\odot(APQ)$ intersects $\omega$ again at $T$. $A'$ is the reflection of $A$ wrt $BC$. Prove that $A'BPT$ lies on the same circle.
1991 IberoAmerican, 2
A square is divided in four parts by two perpendicular lines, in such a way that three of these parts have areas equal to 1. Show that the square has area equal to 4.
2008 Balkan MO Shortlist, C3
Let $ n$ be a positive integer. Consider a rectangle $ (90n\plus{}1)\times(90n\plus{}5)$ consisting of unit squares. Let $ S$ be the set of the vertices of these squares. Prove that the number of distinct lines passing through at least two points of $ S$ is divisible by $ 4$.
2013 Harvard-MIT Mathematics Tournament, 31
Let $ABCD$ be a quadrilateral inscribed in a unit circle with center $O$. Suppose that $\angle AOB = \angle COD = 135^\circ$, $BC=1$. Let $B^\prime$ and $C^\prime$ be the reflections of $A$ across $BO$ and $CO$ respectively. Let $H_1$ and $H_2$ be the orthocenters of $AB^\prime C^\prime$ and $BCD$, respectively. If $M$ is the midpoint of $OH_1$, and $O^\prime$ is the reflection of $O$ about the midpoint of $MH_2$, compute $OO^\prime$.
2013 Gulf Math Olympiad, 3
There are $n$ people standing on a circular track. We want to perform a number of [i]moves[/i] so that we end up with a situation where the distance between every two neighbours is the same. The [i]move[/i] that is allowed consists in selecting two people and asking one of them to walk a distance $d$ on the circular track clockwise, and asking the other to walk the same distance on the track anticlockwise. The two people selected and the quantity $d$ can vary from move to move.
Prove that it is possible to reach the desired situation (where the distance between every two neighbours is the same) after at most $n-1$ moves.
2007 AIME Problems, 15
Four circles $\omega,$ $\omega_{A},$ $\omega_{B},$ and $\omega_{C}$ with the same radius are drawn in the interior of triangle $ABC$ such that $\omega_{A}$ is tangent to sides $AB$ and $AC$, $\omega_{B}$ to $BC$ and $BA$, $\omega_{C}$ to $CA$ and $CB$, and $\omega$ is externally tangent to $\omega_{A},$ $\omega_{B},$ and $\omega_{C}$. If the sides of triangle $ABC$ are $13,$ $14,$ and $15,$ the radius of $\omega$ can be represented in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
2007 Kyiv Mathematical Festival, 4
The point $D$ at the side $AB$ of triangle $ABC$ is given. Construct points $E,F$ at sides $BC, AC$ respectively such that the midpoints of $DE$ and $DF$ are collinear with $B$ and the midpoints of $DE$ and $EF$ are collinear with $C.$
1993 Greece National Olympiad, 7
Three numbers, $a_1$, $a_2$, $a_3$, are drawn randomly and without replacement from the set $\{1, 2, 3, \dots, 1000\}$. Three other numbers, $b_1$, $b_2$, $b_3$, are then drawn randomly and without replacement from the remaining set of 997 numbers. Let $p$ be the probability that, after a suitable rotation, a brick of dimensions $a_1 \times a_2 \times a_3$ can be enclosed in a box of dimensions $b_1 \times b_2 \times b_3$, with the sides of the brick parallel to the sides of the box. If $p$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?
2011 Poland - Second Round, 1
Points $A,B,C,D,E,F$ lie in that order on semicircle centered at $O$, we assume that $AD=BE=CF$. $G$ is a common point of $BE$ and $AD$, $H$ is a common point of $BE$ and $CD$. Prove that:
\[\angle AOC=2\angle GOH.\]
2016 India PRMO, 6
Suppose a circle $C$ of radius $\sqrt2$ touches the $Y$ -axis at the origin $(0, 0)$. A ray of light $L$, parallel to the $X$-axis, reflects on a point $P$ on the circumference of $C$, and after reflection, the reflected ray $L'$ becomes parallel to the $Y$ -axis. Find the distance between the ray $L$ and the $X$-axis.
2013 Online Math Open Problems, 26
Let $ABC$ be a triangle with $AB=13$, $AC=25$, and $\tan A = \frac{3}{4}$. Denote the reflections of $B,C$ across $\overline{AC},\overline{AB}$ by $D,E$, respectively, and let $O$ be the circumcenter of triangle $ABC$. Let $P$ be a point such that $\triangle DPO\sim\triangle PEO$, and let $X$ and $Y$ be the midpoints of the major and minor arcs $\widehat{BC}$ of the circumcircle of triangle $ABC$. Find $PX \cdot PY$.
[i]Proposed by Michael Kural[/i]
1997 Flanders Math Olympiad, 4
Thirteen birds arrive and sit down in a plane. It's known that from each 5-tuple of birds, at least four birds sit on a circle. Determine the greatest $M \in \{1, 2, ..., 13\}$ such that from these 13 birds, at least $M$ birds sit on a circle, but not necessarily $M + 1$ birds sit on a circle. (prove that your $M$ is optimal)
1996 Bundeswettbewerb Mathematik, 3
Let $ABC$ be a triangle, and erect three rectangles $ABB_1A_2$, $BCC_1B_2$, $CAA_1C_2$ externally on its sides $AB$, $BC$, $CA$, respectively. Prove that the perpendicular bisectors of the segments $A_1A_2$, $B_1B_2$, $C_1C_2$ are concurrent.
1995 Turkey MO (2nd round), 2
Let $ABC$ be an acute triangle and let $k_{1},k_{2},k_{3}$ be the circles with diameters $BC,CA,AB$, respectively. Let $K$ be the radical center of these circles. Segments $AK,CK,BK$ meet $k_{1},k_{2},k_{3}$ again at $D,E,F$, respectively. If the areas of triangles $ABC,DBC,ECA,FAB$ are $u,x,y,z$, respectively, prove that \[u^{2}=x^{2}+y^{2}+z^{2}.\]
1986 Iran MO (2nd round), 1
$O$ is a point in the plane. Let $O'$ be an arbitrary point on the axis $Ox$ of the plane and let $M$ be an arbitrary point. Rotate $M$, $90^\circ$ clockwise around $O$ to get the point $M'$ and rotate $M$, $90^\circ$ anticlockwise around $O'$ to get the point $M''.$ Prove that the midpoint of the segment $MM''$ is a fixed point.
2004 All-Russian Olympiad, 2
Let $ I(A)$ and $ I(B)$ be the centers of the excircles of a triangle $ ABC,$ which touches the sides $ BC$ and $ CA$ in its interior. Furthermore let $ P$ a point on the circumcircle $ \omega$ of the triangle $ ABC.$ Show that the center of the segment which connects the circumcenters of the triangles $ I(A)CP$ and $ I(B)CP$ coincides with the center of the circle $ \omega.$
2011 Belarus Team Selection Test, 3
Let $ABC$ be an acute triangle with $D, E, F$ the feet of the altitudes lying on $BC, CA, AB$ respectively. One of the intersection points of the line $EF$ and the circumcircle is $P.$ The lines $BP$ and $DF$ meet at point $Q.$ Prove that $AP = AQ.$
[i]Proposed by Christopher Bradley, United Kingdom[/i]
2010 China Team Selection Test, 1
Given acute triangle $ABC$ with $AB>AC$, let $M$ be the midpoint of $BC$. $P$ is a point in triangle $AMC$ such that $\angle MAB=\angle PAC$. Let $O,O_1,O_2$ be the circumcenters of $\triangle ABC,\triangle ABP,\triangle ACP$ respectively. Prove that line $AO$ passes through the midpoint of $O_1 O_2$.
2009 Greece National Olympiad, 2
Consider a triangle $ABC$ with circumcenter $O$ and let $A_1,B_1,C_1$ be the midpoints of the sides $BC,AC,AB,$ respectively.
Points $A_2,B_2,C_2$ are defined as $\overrightarrow{OA_2}=\lambda \cdot \overrightarrow{OA_1}, \ \overrightarrow{OB_2}=\lambda \cdot \overrightarrow{OB_1}, \ \overrightarrow{OC_2}=\lambda \cdot \overrightarrow{OC_1},$ where $\lambda >0.$
Prove that lines $AA_2,BB_2,CC_2$ are concurrent.
2024 Israel TST, P2
Triangle $ABC$ is inscribed in circle $\Omega$ with center $O$. The incircle of $ABC$ is tangent to $BC$, $AC$, $AB$ at $D$, $E$, $F$ respectively, and its center is $I$. The reflection of the tangent line to $\Omega$ at $A$ with respect to $EF$ will be denoted $\ell_A$. We similarly define $\ell_B$, $\ell_C$.
Show that the orthocenter of the triangle with sides $\ell_A$, $\ell_B$, $\ell_C$ lies on $OI$.