Found problems: 1581
2000 India National Olympiad, 1
The incircle of $ABC$ touches $BC$, $CA$, $AB$ at $K$, $L$, $M$ respectively. The line through $A$ parallel to $LK$ meets $MK$ at $P$, and the line through $A$ parallel to $MK$ meets $LK$ at $Q$. Show that the line $PQ$ bisects $AB$ and bisects $AC$.
1997 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 10
The minimal value of $ f(x) \equal{} \sqrt{a^2 \plus{} x^2} \plus{} \sqrt{(x\minus{}b)^2 \plus{} c^2}$ is
A. $ a\plus{}b\plus{}c$
B. $ \sqrt{a^2 \plus{} (b \plus{} c)^2}$
C. $ \sqrt{b^2 \plus{} (a\plus{}c)^2}$
D. $ \sqrt{(a\plus{}b)^2 \plus{} c^2}$
E. None of these
2021 Thailand TST, 2
Let $ABCD$ be a cyclic quadrilateral. Points $K, L, M, N$ are chosen on $AB, BC, CD, DA$ such that $KLMN$ is a rhombus with $KL \parallel AC$ and $LM \parallel BD$. Let $\omega_A, \omega_B, \omega_C, \omega_D$ be the incircles of $\triangle ANK, \triangle BKL, \triangle CLM, \triangle DMN$.
Prove that the common internal tangents to $\omega_A$, and $\omega_C$ and the common internal tangents to $\omega_B$ and $\omega_D$ are concurrent.
2008 All-Russian Olympiad, 7
In convex quadrilateral $ ABCD$, the rays $ BA,CD$ meet at $ P$, and the rays $ BC,AD$ meet at $ Q$. $ H$ is the projection of $ D$ on $ PQ$. Prove that there is a circle inscribed in $ ABCD$ if and only if the incircles of triangles $ ADP,CDQ$ are visible from $ H$ under the same angle.
2002 India National Olympiad, 1
For a convex hexagon $ ABCDEF$ in which each pair of opposite sides is unequal, consider the following statements.
($ a_1$) $ AB$ is parallel to $ DE$. ($ a_2$)$ AE \equal{} BD$.
($ b_1$) $ BC$ is parallel to $ EF$. ($ b_2$)$ BF \equal{} CE$.
($ c_1$) $ CD$ is parallel to $ FA$. ($ c_2$) $ CA \equal{} DF$.
$ (a)$ Show that if all six of these statements are true then the hexagon is cyclic.
$ (b)$ Prove that, in fact, five of the six statements suffice.
2012 Today's Calculation Of Integral, 772
Given are three points $A(2,\ 0,\ 2),\ B(1,\ 1,\ 0),\ C(0,\ 0,\ 3)$ in the coordinate space. Find the volume of the solid of a triangle $ABC$ generated by a rotation about $z$-axis.
2011 Singapore MO Open, 1
In the acute-angled non-isosceles triangle $ABC$, $O$ is its circumcenter, $H$ is its orthocenter and $AB>AC$. Let $Q$ be a point on $AC$ such that the extension of $HQ$ meets the extension of $BC$ at the point $P$. Suppose $BD=DP$, where $D$ is the foot of the perpendicular from $A$ onto $BC$. Prove that $\angle ODQ=90^{\circ}$.
2020 MBMT, 20
Sam colors each tile in a 4 by 4 grid white or black. A coloring is called [i]rotationally symmetric[/i] if the grid can be rotated 90, 180, or 270 degrees to achieve the same pattern. Two colorings are called [i]rotationally distinct[/i] if neither can be rotated to match the other. How many rotationally distinct ways are there for Sam to color the grid such that the colorings are [i]not[/i] rotationally symmetric?
[i]Proposed by Gabriel Wu[/i]
Kvant 2019, M2564
Let $ABC$ be an acute-angled triangle with $AC<BC.$ A circle passes through $A$ and $B$ and crosses the segments $AC$ and $BC$ again at $A_1$ and $B_1$ respectively. The circumcircles of $A_1B_1C$ and $ABC$ meet each other at points $P$ and $C.$ The segments $AB_1$ and $A_1B$ intersect at $S.$ Let $Q$ and $R$ be the reflections of $S$ in the lines $CA$ and $CB$ respectively. Prove that the points $P,$ $Q,$ $R,$ and $C$ are concyclic.
2009 Tournament Of Towns, 4
A point is chosen on each side of a regular $2009$-gon. Let $S$ be the area of the $2009$-gon with vertices at these points. For each of the chosen points, reflect it across the midpoint of its side. Prove that the $2009$-gon with vertices at the images of these reflections also has area $S.$
[i](4 points)[/i]
2008 Sharygin Geometry Olympiad, 2
(A.Myakishev) Let triangle $ A_1B_1C_1$ be symmetric to $ ABC$ wrt the incenter of its medial triangle. Prove that the orthocenter of $ A_1B_1C_1$ coincides with the circumcenter of the triangle formed by the excenters of $ ABC$.
2002 AMC 10, 20
Let $ a$, $ b$, and $ c$ be real numbers such that $ a \minus{} 7b \plus{} 8c \equal{} 4$ and $ 8a \plus{} 4b \minus{} c \equal{} 7$. Then $ a^2 \minus{} b^2 \plus{} c^2$ is
$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8$
2013 Korea - Final Round, 4
For a triangle $ ABC $, let $ B_1 ,C_1 $ be the excenters of $ B, C $. Line $B_1 C_1 $ meets with the circumcircle of $ \triangle ABC $ at point $ D (\ne A) $. $ E $ is the point which satisfies $ B_1 E \bot CA $ and $ C_1 E \bot AB $. Let $ w $ be the circumcircle of $ \triangle ADE $. The tangent to the circle $ w $ at $ D $ meets $ AE $ at $ F $. $ G , H $ are the points on $ AE, w $ such that $ DGH \bot AE $. The circumcircle of $ \triangle HGF $ meets $ w $ at point $ I ( \ne H ) $, and $ J $ be the foot of perpendicular from $ D $ to $ AH $. Prove that $ AI $ passes the midpoint of $ DJ $.
2010 Germany Team Selection Test, 2
Determine all $n \in \mathbb{Z}^+$ such that a regular hexagon (i.e. all sides equal length, all interior angles same size) can be partitioned in finitely many $n-$gons such that they can be composed into $n$ congruent regular hexagons in a non-overlapping way upon certain rotations and translations.
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 $C_{n+2}-2C_{n+1}+C_n$
2022 Iran MO (3rd Round), 1
Triangle $ABC$ is assumed. The point $T$ is the second intersection of the symmedian of vertex $A$ with the circumcircle of the triangle $ABC$ and the point $D \neq A$ lies on the line $AC$ such that $BA=BD$. The line that at $D$ tangents to the circumcircle of the triangle $ADT$, intersects the circumcircle of the triangle $DCT$ for the second time at $K$. Prove that $\angle BKC = 90^{\circ}$(The symmedian of the vertex $A$, is the reflection of the median of the vertex $A$ through the angle bisector of this vertex).
2009 Baltic Way, 11
Let $M$ be the midpoint of the side $AC$ of a triangle $ABC$, and let $K$ be a point on the ray $BA$ beyond $A$. The line $KM$ intersects the side $BC$ at the point $L$. $P$is the point on the segment $BM$ such that $PM$ is the bisector of the angle $LPK$. The line $\ell$ passes through $A$ and is parallel to $BM$. Prove that the projection of the point $M$ onto the line $\ell$ belongs to the line $PK$.
2012 Albania National Olympiad, 5
Let $ABC$ be a triangle where $AC\neq BC$. Let $P$ be the foot of the altitude taken from $C$ to $AB$; and let $V$ be the orthocentre, $O$ the circumcentre of $ABC$, and $D$ the point of intersection between the radius $OC$ and the side $AB$. The midpoint of $CD$ is $E$.
a) Prove that the reflection $V'$ of $V$ in $AB$ is on the circumcircle of the triangle $ABC$.
b) In what ratio does the segment $EP$ divide the segment $OV$?
2000 Harvard-MIT Mathematics Tournament, 3
Using $3$ colors, red, blue and yellow, how many different ways can you color a cube (modulo rigid rotations)?
1996 Moldova Team Selection Test, 10
Given an equilateral triangle $ABC$ and a point $M$ in the plane ($ABC$). Let $A', B', C'$ be respectively the symmetric through $M$ of $A, B, C$.
[b]I.[/b] Prove that there exists a unique point $P$ equidistant from $A$ and $B'$, from $B$ and $C'$ and from $C$ and $A'$.
[b]II.[/b] Let $D$ be the midpoint of the side $AB$. When $M$ varies ($M$ does not coincide with $D$), prove that the circumcircle of triangle $MNP$ ($N$ is the intersection of the line $DM$ and $AP$) pass through a fixed point.
2015 India Regional MathematicaI Olympiad, 1
Let ABC be a triangle. Let B' and C' denote the reflection of B and C in the internal angle bisector of angle A. Show that the triangles ABC and AB'C' have the same incenter.
2014 Contests, 1
Let $ABC$ be an acute triangle, and let $X$ be a variable interior point on the minor arc $BC$ of its circumcircle. Let $P$ and $Q$ be the feet of the perpendiculars from $X$ to lines $CA$ and $CB$, respectively. Let $R$ be the intersection of line $PQ$ and the perpendicular from $B$ to $AC$. Let $\ell$ be the line through $P$ parallel to $XR$. Prove that as $X$ varies along minor arc $BC$, the line $\ell$ always passes through a fixed point. (Specifically: prove that there is a point $F$, determined by triangle $ABC$, such that no matter where $X$ is on arc $BC$, line $\ell$ passes through $F$.)
[i]Robert Simson et al.[/i]
2013 Baltic Way, 14
Circles $\alpha$ and $\beta$ of the same radius intersect in two points, one of which is $P$. Denote by $A$ and $B$, respectively, the points diametrically opposite to $P$ on each of $\alpha$ and $\beta$ . A third circle of the same radius passes through $P$ and intersects $\alpha$ and $\beta$ in the points $X$ and $Y$ , respectively. Show that the line $XY$ is parallel to the line $AB$.
2013 Uzbekistan National Olympiad, 4
Let circles $ \Gamma $ and $ \omega $ are circumcircle and incircle of the triangle $ABC$, the incircle touches sides $BC,CA,AB$ at the points $A_1,B_1,C_1$. Let $A_2$ and $B_2$ lies the lines $A_1I$ and $B_1I$ ($A_1$ and $A_2$ lies different sides from $I$, $B_1$ and $B_2$ lies different sides from $I$) such that $IA_2=IB_2=R$. Prove that :
(a) $AA_2=BB_2=IO$;
(b) The lines $AA_2$ and $BB_2$ intersect on the circle $ \Gamma ;$
2006 Iran MO (3rd Round), 3
In triangle $ABC$, if $L,M,N$ are midpoints of $AB,AC,BC$. And $H$ is orthogonal center of triangle $ABC$, then prove that \[LH^{2}+MH^{2}+NH^{2}\leq\frac14(AB^{2}+AC^{2}+BC^{2})\]