Found problems: 1001
2021 Sharygin Geometry Olympiad, 8.2
Three parallel lines $\ell_a, \ell_b, \ell_c$ pass through the vertices of triangle $ABC$. A line $a$ is the reflection of altitude $AH_a$ about $\ell_a$. Lines $b, c$ are defined similarly. Prove that $a, b, c$ are concurrent.
2011 Kazakhstan National Olympiad, 6
We call a square table of a binary, if at each cell is written a single number 0 or 1. The binary table is called regular if each row and each column exactly two units. Determine the number of regular size tables $n\times n$ ($n> 1$ - given a fixed positive integer). (We can assume that the rows and columns of the tables are numbered: the cases of coincidence in turn, reflect, and so considered different).
2013 ELMO Shortlist, 12
Let $ABC$ be a nondegenerate acute triangle with circumcircle $\omega$ and let its incircle $\gamma$ touch $AB, AC, BC$ at $X, Y, Z$ respectively. Let $XY$ hit arcs $AB, AC$ of $\omega$ at $M, N$ respectively, and let $P \neq X, Q \neq Y$ be the points on $\gamma$ such that $MP=MX, NQ=NY$. If $I$ is the center of $\gamma$, prove that $P, I, Q$ are collinear if and only if $\angle BAC=90^\circ$.
[i]Proposed by David Stoner[/i]
2010 India Regional Mathematical Olympiad, 5
Let $ABC$ be a triangle in which $\angle A = 60^\circ$. Let $BE$ and $CF$ be the bisectors of $\angle B$ and $\angle C$ with $E$ on $AC$ and $F$ on $AB$. Let $M$ be the reflection of $A$ in line $EF$. Prove that $M$ lies on $BC$.
2013 Sharygin Geometry Olympiad, 9
Let $T_1$ and $T_2$ be the points of tangency of the excircles of a triangle $ABC$ with its sides $BC$ and $AC$ respectively. It is known that the reflection of the incenter of $ABC$ across the midpoint of $AB$ lies on the circumcircle of triangle $CT_1T_2$. Find $\angle BCA$.
2014 Tuymaada Olympiad, 2
The points $K$ and $L$ on the side $BC$ of a triangle $\triangle{ABC}$ are such that $\widehat{BAK}=\widehat{CAL}=90^\circ$. Prove that the midpoint of the altitude drawn from $A$, the midpoint of $KL$ and the circumcentre of $\triangle{ABC}$ are collinear.
[i](A. Akopyan, S. Boev, P. Kozhevnikov)[/i]
2004 IberoAmerican, 2
In the plane are given a circle with center $ O$ and radius $ r$ and a point $ A$ outside the circle. For any point $ M$ on the circle, let $ N$ be the diametrically opposite point. Find the locus of the circumcenter of triangle $ AMN$ when $ M$ describes the circle.
2008 Poland - Second Round, 2
In the convex pentagon $ ABCDE$ following equalities holds: $ \angle ABD\equal{} \angle ACE, \angle ACB\equal{}\angle ACD, \angle ADC\equal{}\angle ADE$ and $ \angle ADB\equal{}\angle AEC$. The point $S$ is the intersection of the segments $BD$ and $CE$. Prove that lines $AS$ and $CD$ are perpendicular.
2015 SGMO, Q3
For all nonempty finite sets of point $S$ on the plane satisfying: $|S|$ is even and for all partitions of $S$ into two subsets $A,B$ of equal size, there is a reflection that maps $A$ to $B$.
1999 IMO Shortlist, 3
A set $ S$ of points from the space will be called [b]completely symmetric[/b] if it has at least three elements and fulfills the condition that for every two distinct points $ A$ and $ B$ from $ S$, the perpendicular bisector plane of the segment $ AB$ is a plane of symmetry for $ S$. Prove that if a completely symmetric set is finite, then it consists of the vertices of either a regular polygon, or a regular tetrahedron or a regular octahedron.
2023 Sharygin Geometry Olympiad, 16
Let $AH_A$ and $BH_B$ be the altitudes of a triangle $ABC$. The line $H_AH_B$ meets the circumcircle of $ABC$ at points $P$ and $Q$. Let $A'$ be the reflection of $A$ about $BC$, and $B'$ be the reflection of $B$ about $CA$. Prove that $A',B', P,Q$ are concyclic.
Geometry Mathley 2011-12, 6.4
Let $P$ be an arbitrary variable point in the plane of a triangle $ABC. A_1$ is the projection of $P$ onto $BC, A_2$ is the midpoint of line segment $PA_1, A_2P$ meets $BC$ at $A_3, A_4$ is the reflection of $P$ about $A_3$. Prove that $PA_4$ has a fixed point.
Trần Quang Hùng
2006 QEDMO 3rd, 6
The incircle of a triangle $ABC$ touches its sides $BC$, $CA$, $AB$ at the points $X$, $Y$, $Z$, respectively. Let $X^{\prime}$, $Y^{\prime}$, $Z^{\prime}$ be the reflections of these points $X$, $Y$, $Z$ in the external angle bisectors of the angles $CAB$, $ABC$, $BCA$, respectively. Show that $Y^{\prime}Z^{\prime}\parallel BC$, $Z^{\prime}X^{\prime}\parallel CA$ and $X^{\prime}Y^{\prime}\parallel AB$.
2022 Bosnia and Herzegovina IMO TST, 1
Let $ABC$ be a triangle such that $AB=AC$ and $\angle BAC$ is obtuse. Point $O$ is the circumcenter of triangle $ABC$, and $M$ is the reflection of $A$ in $BC$. Let $D$ be an arbitrary point on line $BC$, such that $B$ is in between $D$ and $C$. Line $DM$ cuts the circumcircle of $ABC$ in $E,F$. Circumcircles of triangles $ADE$ and $ADF$ cut $BC$ in $P,Q$ respectively. Prove that $DA$ is tangent to the circumcircle of triangle $OPQ$.
1992 Romania Team Selection Test, 3
Let $ABCD$ be a tetrahedron; $B', C', D'$ be the midpoints of the edges $AB, AC, AD$; $G_A, G_B, G_C, G_D$ be the barycentres of the triangles $BCD, ACD, ABD, ABC$, and $G$ be the barycentre of the tetrahedron. Show that $A, G, G_B, G_C, G_D$ are all on a sphere if and only if $A, G, B', C', D'$ are also on a sphere.
[i]Dan Brânzei[/i]
2020 Dutch IMO TST, 2
Given is a triangle $ABC$ with its circumscribed circle and $| AC | <| AB |$. On the short arc $AC$, there is a variable point $D\ne A$. Let $E$ be the reflection of $A$ wrt the inner bisector of $\angle BDC$. Prove that the line $DE$ passes through a fixed point, regardless of point $D$.
2024 Baltic Way, 14
Let $ABC$ be an acute triangle with circumcircle $\omega$. The altitudes $AD$, $BE$ and $CF$ of the triangle $ABC$ intersect at point $H$. A point $K$ is chosen on the line $EF$ such that $KH\parallel BC$. Prove that the reflection of $H$ in $KD$ lies on $\omega$.
2009 Sharygin Geometry Olympiad, 4
Let $ P$ and $ Q$ be the common points of two circles. The ray with origin $ Q$ reflects from the first circle in points $ A_1$, $ A_2$,$ \ldots$ according to the rule ''the angle of incidence is equal to the angle of reflection''. Another ray with origin $ Q$ reflects from the second circle in the points $ B_1$, $ B_2$,$ \ldots$ in the same manner. Points $ A_1$, $ B_1$ and $ P$ occurred to be collinear. Prove that all lines $ A_iB_i$ pass through P.
2010 AMC 12/AHSME, 18
A 16-step path is to go from $ ( \minus{} 4, \minus{}4)$ to $ (4,4)$ with each step increasing either the $x$-coordinate or the $y$-coordinate by 1. How many such paths stay outside or on the boundary of the square $ \minus{} 2 \le x \le 2$, $ \minus{} 2 \le y \le 2$ at each step?
$ \textbf{(A)}\ 92 \qquad \textbf{(B)}\ 144 \qquad \textbf{(C)}\ 1568 \qquad \textbf{(D)}\ 1698 \qquad \textbf{(E)}\ 12,\!800$
2007 Romania Team Selection Test, 4
Let $\mathcal O_{1}$ and $\mathcal O_{2}$ two exterior circles. Let $A$, $B$, $C$ be points on $\mathcal O_{1}$ and $D$, $E$, $F$ points on $\mathcal O_{1}$ such that $AD$ and $BE$ are the common exterior tangents to these two circles and $CF$ is one of the interior tangents to these two circles, and such that $C$, $F$ are in the interior of the quadrilateral $ABED$. If $CO_{1}\cap AB=\{M\}$ and $FO_{2}\cap DE=\{N\}$ then prove that $MN$ passes through the middle of $CF$.
2021 Caucasus Mathematical Olympiad, 4
In an acute triangle $ABC$ let $AH_a$ and $BH_b$ be altitudes. Let $H_aH_b$ intersect the circumcircle of $ABC$ at $P$ and $Q$. Let $A'$ be the reflection of $A$ in $BC$, and let $B'$ be the reflection of $B$ in $CA$. Prove that $A', B'$, $P$, $Q$ are concyclic.
1964 Miklós Schweitzer, 4
Let $ A_1,A_2,...,A_n$ be the vertices of a closed convex $ n$-gon $ K$ numbered consecutively. Show that at least $ n\minus{}3$
vertices $ A_i$ have the property that the reflection of $ A_i$ with respect to the midpoint of $ A_{i\minus{}1}A_{i\plus{}1}$ is contained in $ K$. (Indices are meant $ \textrm{mod} \;n\ .$)
1968 Spain Mathematical Olympiad, 8
We will assume that the sides of a square are reflective and we will designate them with the names of the four cardinal points. Marking a point on the side $N$ , determine in which direction a ray of light should exit (into the interior of the square) so that it returns to it after having undergone $n$ reflections on the side $E$ , another $n$ on the side $W$ , $m$ on the $S$ and $m - 1$ on the $N$, where $n$ and $m$ are known natural numbers. What happens if m and $n$ are not prime to each other? Calculate the length of the light ray considered as a function of $m$ and $n$, and of the length of the side of the square.
2011 IMO Shortlist, 6
Let $ABC$ be a triangle with $AB=AC$ and let $D$ be the midpoint of $AC$. The angle bisector of $\angle BAC$ intersects the circle through $D,B$ and $C$ at the point $E$ inside the triangle $ABC$. The line $BD$ intersects the circle through $A,E$ and $B$ in two points $B$ and $F$. The lines $AF$ and $BE$ meet at a point $I$, and the lines $CI$ and $BD$ meet at a point $K$. Show that $I$ is the incentre of triangle $KAB$.
[i]Proposed by Jan Vonk, Belgium and Hojoo Lee, South Korea[/i]
1968 Vietnam National Olympiad, 2
$L$ and $M$ are two parallel lines a distance $d$ apart. Given $r$ and $x$, construct a triangle $ABC$, with $A$ on $L$, and $B$ and $C$ on $M$, such that the inradius is $r$, and angle $A = x$. Calculate angles $B$ and $C$ in terms of $d$, $r$ and $x$. If the incircle touches the side $BC$ at $D$, find a relation between $BD$ and $DC$