Found problems: 3882
2014 China Girls Math Olympiad, 1
In the figure of [url]http://www.artofproblemsolving.com/Forum/download/file.php?id=50643&mode=view[/url]
$\odot O_1$ and $\odot O_2$ intersect at two points $A$, $B$.
The extension of $O_1A$ meets $\odot O_2$ at $C$, and the extension of $O_2A$ meets $\odot O_1$ at $D$,
and through $B$ draw $BE \parallel O_2A$ intersecting $\odot O_1$ again at $E$.
If $DE \parallel O_1A$, prove that $DC \perp CO_2$.
2010 Tournament Of Towns, 5
The quadrilateral $ABCD$ is inscribed in a circle with center $O$. The diagonals $AC$ and $BD$ do not pass through $O$. If the circumcentre of triangle $AOC$ lies on the line $BD$, prove that the circumcentre of triangle $BOD$ lies on the line $AC$.
2023 Serbia Team Selection Test, P2
A circle centered at $A$ intersects sides $AC$ and $AB$ of $\triangle ABC$ at $E$ and $F$, and the circumcircle of $\triangle ABC$ at $X$ and $Y$. Let $D$ be the point on $BC$ such that $AD$, $BE$, $CF$ concur. Let $P=XE\cap YF$ and $Q=XF\cap YE$.
Prove that the foot of the perpendicular from $D$ to $EF$ lies on $PQ$.
2000 India Regional Mathematical Olympiad, 5
The internal bisector of angle $A$ in a triangle $ABC$ with $AC > AB$ meets the circumcircle $\Gamma$ of the triangle in $D$. Join$D$ to the center $O$ of the circle $\Gamma$ and suppose that $DO$ meets $AC$ in $E$, possibly when extended. Given that $BE$ is perpendicular to $AD$, show that $AO$ is parallel to $BD$.
2013 All-Russian Olympiad, 2
Acute-angled triangle $ABC$ is inscribed into circle $\Omega$. Lines tangent to $\Omega$ at $B$ and $C$ intersect at $P$. Points $D$ and $E$ are on $AB$ and $AC$ such that $PD$ and $PE$ are perpendicular to $AB$ and $AC$ respectively. Prove that the orthocentre of triangle $ADE$ is the midpoint of $BC$.
2017 Iran MO (3rd round), 1
Let $ABC$ be a triangle. Suppose that $X,Y$ are points in the plane such that $BX,CY$ are tangent to the circumcircle of $ABC$, $AB=BX,AC=CY$ and $X,Y,A$ are in the same side of $BC$. If $I$ be the incenter of $ABC$ prove that $\angle BAC+\angle XIY=180$.
2013 North Korea Team Selection Test, 1
The incircle of a non-isosceles triangle $ABC$ with the center $I$ touches the sides $ BC, CA, AB$ at $ A_1 , B_1 , C_1 $ respectively. The line $AI$ meets the circumcircle of $ABC$ at $A_2 $. The line $B_1 C_1 $ meets the line $BC$ at $A_3 $ and the line $A_2 A_3 $ meets the circumcircle of $ABC$ at $A_4 (\ne A_2 ) $. Define $B_4 , C_4 $ similarly. Prove that the lines $ AA_4 , BB_4 , CC_4 $ are concurrent.
2025 China National Olympiad, 2
Let $ABC$ be a triangle with incenter $I$. Denote the midpoints of $AI$, $AC$ and $CI$ by $L$, $M$ and $N$ respectively. Point $D$ lies on segment $AM$ such that $BC= BD$. Let the incircle of triangle $ABD$ be tangent to $AD$ and $BD$ at $E$ and $F$ respectively. Denote the circumcenter of triangle $AIC$ by $J$, and the circumcircle of triangle $JMD$ by $\omega$. Lines $MN$ and $JL$ meet $\omega$ again at $P$ and $Q$ respectively. Prove that $PQ$, $LN$ and $EF$ are concurrent.
2009 Postal Coaching, 1
In a triangle $ABC$, let $D,E, F$ be interior points of sides $BC,CA,AB$ respectively. Let $AD,BE,CF$ meet the circumcircle of triangle $ABC$ in $K, L,M$ respectively. Prove that $\frac{AD}{DK} + \frac{BE}{EL} + \frac{CF}{FM} \ge 9$. When does the equality hold?
2019 Cono Sur Olympiad, 6
Let $ABC$ be an acute-angled triangle with $AB< AC$, and let $H$ be its orthocenter. The circumference with diameter $AH$ meets the circumscribed circumference of $ABC$ at $P\neq A$. The tangent to the circumscribed circumference of $ABC$ through $P$ intersects line $BC$ at $Q$. Show that $QP=QH$.
2019 Azerbaijan IMO TST, 2
Let $ABC$ ($BC > AC$) be an acute triangle with circumcircle $k$ centered at $O$. The tangent to $k$ at $C$ intersects the line $AB$ at the point $D$. The circumcircles of triangles $BCD, OCD$ and $AOB$ intersect the ray $CA$ (beyond $A$) at the points $Q, P$ and $K$, respectively, such that $P \in (AK)$ and $K \in (PQ)$. The line $PD$ intersects the circumcircle of triangle $BKQ$ at the point $T$, so that $P$ and $T$ are in different halfplanes with respect to $BQ$. Prove that $TB = TQ$.
2015 Belarus Team Selection Test, 2
In a cyclic quadrilateral $ABCD$, the extensions of sides $AB$ and $CD$ meet at point $P$, and the extensions of sides $AD$ and $BC$ meet at point $Q$. Prove that the distance between the orthocenters of triangles $APD$ and $AQB$ is equal to the distance between the orthocenters of triangles $CQD$ and $BPC$.
2021 Durer Math Competition (First Round), 3
Let $k_1$ and $k_2$ be two circles that are externally tangent at point $C$. We have a point $A$ on $k_1$ and a point $B$ on $k_2$ such that $C$ is an interior point of segment $AB$. Let $k_3$ be a circle that passes through points $A$ and $B$ and intersects circles $k_1$ and $k_2$ another time at points $M$ and $N$ respectively. Let $k_4$ be the circumscribed circle of triangle $CMN$. Prove that the centres of circles $k_1, k_2, k_3$ and $k_4$ all lie on the same circle.
2021 Peru Cono Sur TST., P3
Let $ABC$ be a triangle and $D$ is a point in $BC$. The line $DA$ cuts the circumcircle of $ABC$ in the point $E$. Let $M$ and $N$ be the midpoints of $AB$ and $CD$, respectively. Let $F=MN\cap AD$ and $G\neq F$ is the point of intersection of the circumcircles of $\triangle DNF$ and $\triangle ECF$. Prove that $B,F$ and $G$ are collinears.
2005 China Team Selection Test, 2
Let $\omega$ be the circumcircle of acute triangle $ABC$. Two tangents of $\omega$ from $B$ and $C$ intersect at $P$, $AP$ and $BC$ intersect at $D$. Point $E$, $F$ are on $AC$ and $AB$ such that $DE \parallel BA$ and $DF \parallel CA$.
(1) Prove that $F,B,C,E$ are concyclic.
(2) Denote $A_{1}$ the centre of the circle passing through $F,B,C,E$. $B_{1}$, $C_{1}$ are difined similarly. Prove that $AA_{1}$, $BB_{1}$, $CC_{1}$ are concurrent.
1995 Brazil National Olympiad, 4
A regular tetrahedron has side $L$. What is the smallest $x$ such that the tetrahedron can be passed through a loop of twine of length $x$?
Swiss NMO - geometry, 2008.1
Let $ABC$ be a triangle with $\angle BAC \ne 45^o$ and $\angle ABC \ne 135^o$. Let $P$ be the point on the line $AB$ with $\angle CPB = 45^o$. Let $O_1$ and $O_2$ be the centers of the circumcircles of the triangles $ACP$ and $BCP$ respectively. Show that the area of the square $CO_1P O_2$ is equal to the area of the triangle $ABC$.
2013 Dutch IMO TST, 2
Let $P$ be the point of intersection of the diagonals of a convex quadrilateral $ABCD$.Let $X,Y,Z$ be points on the interior of $AB,BC,CD$ respectively such that $\frac{AX}{XB}=\frac{BY}{YC}=\frac{CZ}{ZD}=2$. Suppose that $XY$ is tangent to the circumcircle of $\triangle CYZ$ and that $Y Z$ is tangent to the circumcircle of $\triangle BXY$.Show that $\angle APD=\angle XYZ$.
2017 Bulgaria EGMO TST, 3
Let $ ABC$ be a fixed triangle, and let $ A_1$, $ B_1$, $ C_1$ be the midpoints of sides $ BC$, $ CA$, $ AB$, respectively. Let $ P$ be a variable point on the circumcircle. Let lines $ PA_1$, $ PB_1$, $ PC_1$ meet the circumcircle again at $ A'$, $ B'$, $ C'$, respectively. Assume that the points $ A$, $ B$, $ C$, $ A'$, $ B'$, $ C'$ are distinct, and lines $ AA'$, $ BB'$, $ CC'$ form a triangle. Prove that the area of this triangle does not depend on $ P$.
[i]Author: Christopher Bradley, United Kingdom [/i]
2011 AIME Problems, 13
Point $P$ lies on the diagonal $AC$ of square $ABCD$ with $AP>CP$. Let $O_1$ and $O_2$ be the circumcenters of triangles $ABP$ and $CDP$ respectively. Given that $AB=12$ and $\angle O_1 P O_2 = 120^\circ$, then $AP=\sqrt{a}+\sqrt{b}$ where $a$ and $b$ are positive integers. Find $a+b$.
2007 CHKMO, 3
A convex quadrilateral $ABCD$ with $AC \neq BD$ is inscribed in a circle with center $O$. Let $E$ be the intersection of diagonals $AC$ and $BD$. If $P$ is a point inside $ABCD$ such that $\angle PAB+\angle PCB=\angle PBC+\angle PDC=90^\circ$, prove that $O$, $P$ and $E$ are collinear.
2013 Sharygin Geometry Olympiad, 15
(a) Triangles $A_1B_1C_1$ and $A_2B_2C_2$ are inscribed into triangle $ABC$ so that $C_1A_1 \perp BC$, $A_1B_1 \perp CA$, $B_1C_1 \perp AB$, $B_2A_2 \perp BC$, $C_2B_2 \perp CA$, $A_2C_2 \perp AB$. Prove that these triangles are equal.
(b) Points $A_1$, $B_1$, $C_1$, $A_2$, $B_2$, $C_2$ lie inside a triangle $ABC$ so that $A_1$ is on segment $AB_1$, $B_1$ is on segment $BC_1$, $C_1$ is on segment $CA_1$, $A_2$ is on segment $AC_2$, $B_2$ is on segment $BA_2$, $C_2$ is on segment $CB_2$, and the angles $BAA_1$, $CBB_2$, $ACC_1$, $CAA_2$, $ABB_2$, $BCC_2$ are equal. Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ are equal.
2001 239 Open Mathematical Olympiad, 2
In a convex quadrangle $ ABCD $, the rays $ DA $ and $ CB $ intersect at point $ Q $, and the rays $ BA $ and $ CD $ at the point $ P $. It turned out that $ \angle AQB = \angle APD $. The bisectors of the angles $ \angle AQB $ and $ \angle APD $ intersect the sides quadrangle at points $ X $, $ Y $ and $ Z $, $ T $ respectively. Circumscribed circles of triangles $ ZQT $ and $ XPY $ intersect at $ K $ inside quadrangle. Prove that $ K $ lies on the diagonal $ AC $.
2021 Brazil National Olympiad, 3
Let $ABC$ be a scalene triangle and $\omega$ is your incircle. The sides $BC,CA$ and $AB$ are tangents to $\omega$ in $X,Y,Z$ respectively. Let $M$ be the midpoint of $BC$ and $D$ is the intersection point of $BC$ with the angle bisector of $\angle BAC$. Prove that $\angle BAX=\angle MAC$ if and only if $YZ$ passes by the midpoint of $AD$.
2022 Yasinsky Geometry Olympiad, 6
Let $s$ be an arbitrary straight line passing through the incenter $I$ of the triangle $ABC$ . Line $s$ intersects lines $AB$ and $BC$ at points $D$ and $E$, respectively. Points $P$ and $Q$ are the centers of the circumscribed circles of triangles $DAI$ and $CEI$, respectively, and point $F$ is the second intersection point of these circles. Prove that the circumcircle of the triangle $PQF$ is always passes through a fixed point on the plane regardless of the position of the straight line $s$.
(Matvii Kurskyi)