Found problems: 3882
2006 IMO Shortlist, 5
In triangle $ABC$, let $J$ be the center of the excircle tangent to side $BC$ at $A_{1}$ and to the extensions of the sides $AC$ and $AB$ at $B_{1}$ and $C_{1}$ respectively. Suppose that the lines $A_{1}B_{1}$ and $AB$ are perpendicular and intersect at $D$. Let $E$ be the foot of the perpendicular from $C_{1}$ to line $DJ$. Determine the angles $\angle{BEA_{1}}$ and $\angle{AEB_{1}}$.
[i]Proposed by Dimitris Kontogiannis, Greece[/i]
1983 IMO Longlists, 73
Let $ABC$ be a nonequilateral triangle. Prove that there exist two points $P$ and $Q$ in the plane of the triangle, one in the interior and one in the exterior of the circumcircle of $ABC$, such that the orthogonal projections of any of these two points on the sides of the triangle are vertices of an equilateral triangle.
2007 Ukraine Team Selection Test, 5
$ AA_{3}$ and $ BB_{3}$ are altitudes of acute-angled $ \triangle ABC$. Points $ A_{1}$ and $ B_{1}$ are second points of intersection lines $ AA_{3}$ and $ BB_{3}$ with circumcircle of $ \triangle ABC$ respectively. $ A_{2}$ and $ B_{2}$ are points on $ BC$ and $ AC$ respectively. $ A_{1}A_{2}\parallel AC$, $ B_{1}B_{2}\parallel BC$. Point $ M$ is midpoint of $ A_{2}B_{2}$. $ \angle BCA \equal{} x$. Find $ \angle A_{3}MB_{3}$.
2009 Germany Team Selection Test, 3
In an acute triangle $ ABC$ segments $ BE$ and $ CF$ are altitudes. Two circles passing through the point $ A$ and $ F$ and tangent to the line $ BC$ at the points $ P$ and $ Q$ so that $ B$ lies between $ C$ and $ Q$. Prove that lines $ PE$ and $ QF$ intersect on the circumcircle of triangle $ AEF$.
[i]Proposed by Davood Vakili, Iran[/i]
2021 Saudi Arabia Training Tests, 14
Let $BC$ be a fixed chord of a circle $\omega$. Let $A$ be a variable point on the major arc $BC$ of $\omega$. Let $H$ be the orthocenter of $ABC$. The points $D, E$ lie on $AB, AC$ such that $H$ is the midpoint of $DE$. $O_A$ is the circumcenter of $ADE$. Prove that as $A$ varies, $O_A$ lies on a fixed circle.
2014 Indonesia MO Shortlist, G6
Given an $ABC$ acute triangle with $O$ the center of the circumscribed circle. Suppose that $\omega$ is a circle that is tangent to the line $AO$ at point $A$ and also tangent to the line $BC$. Prove that $\omega$ is also tangent to the circumcircle of the triangle $BOC$.
2009 Singapore Team Selection Test, 2
Let $H$ be the orthocentre of $\triangle ABC$ and let $P$ be a point on the circumcircle of $\triangle ABC$, distinct from $A,B,C$. Let $E$ and $F$ be the feet of altitudes from $H$ onto $AC$ and $AB$ respectively. Let $PAQB$ and $PARC$ be parallelograms. Suppose $QA$ meets $RH$ at $X$ and $RA$ meets $QH$ at $Y$. Prove that $XE$ is parallel to $YF$.
2003 Chile National Olympiad, 6
Consider a triangle $ ABC $. On the line $ AC $ take a point $ B_1 $ such that $ AB = AB_1 $ and in addition, $ B_1 $ and $ C $ are located on the same side of the line with respect to the point $ A $. The bisector of the angle $ A $ intersects the side $ BC $ at a point that we will denote as $ A_1 $. Let $ P $ and $ R $ be the circumscribed circles of the triangles $ ABC $ and $ A_1B_1C $ respectively. They intersect at points $ C $ and $ Q $. Prove that the tangent to the circle $ R $ at the point $ Q $ is parallel to the line $ AC $.
2009 India IMO Training Camp, 1
Let $ ABC$ be a triangle with $ \angle A = 60^{\circ}$.Prove that if $ T$ is point of contact of Incircle And Nine-Point Circle, Then $ AT = r$,
$ r$ being inradius.
2021 Latvia Baltic Way TST, P12
Five points $A,B,C,P,Q$ are chosen so that $A,B,C$ aren't collinear. The following length conditions hold: $\frac{AP}{BP}=\frac{AQ}{BQ}=\frac{21}{20}$ and $\frac{BP}{CP}=\frac{BQ}{CQ}=\frac{20}{19}$. Prove that line $PQ$ goes through the circumcentre of $\triangle ABC$.
2010 Sharygin Geometry Olympiad, 20
The incircle of an acute-angled triangle $ABC$ touches $AB, BC, CA$ at points $C_1, A_1, B_1$ respectively. Points $A_2, B_2$ are the midpoints of the segments $B_1C_1, A_1C_1$ respectively. Let $P$ be a common point of the incircle and the line $CO$, where $O$ is the circumcenter of triangle $ABC.$ Let also $A'$ and $B'$ be the second common points of $PA_2$ and $PB_2$ with the incircle. Prove that a common point of $AA'$ and $BB'$ lies on the altitude of the triangle dropped from the vertex $C.$
2023 Iran MO (3rd Round), 3
In triangle $\triangle ABC$ points $M,N$ lie on $BC$ st : $\angle BAM= \angle MAN= \angle NAC$ . Points $P,Q$ are on the angle bisector of $BAC$, on the same side of $BC$ as A , st :
$$\frac{1}{3} \angle BAC = \frac{1}{2} \angle BPC = \angle BQC$$
Let $E = AM \cap CQ$ and $F = AN \cap BQ$ . Prove that the common tangents to $(EPF), (EQF)$ and the circumcircle of $\triangle ABC$ , are concurrent.
2004 Bulgaria Team Selection Test, 1
The points $P$ and $Q$ lie on the diagonals $AC$ and $BD$, respectively, of a quadrilateral $ABCD$ such that $\frac{AP}{AC} + \frac{BQ}{BD} =1$. The line $PQ$ meets the sides $AD$ and $BC$ at points $M$ and $N$. Prove that the circumcircles of the triangles $AMP$, $BNQ$, $DMQ$, and $CNP$ are concurrent.
2018 Moscow Mathematical Olympiad, 4
$ABCD$ is convex and $AB\not \parallel CD,BC \not \parallel DA$. $P$ is variable point on $AD$. Circumcircles of $\triangle ABP$ and $\triangle CDP$ intersects at $Q$. Prove, that all lines $PQ$ goes through fixed point.
2013 Dutch BxMO/EGMO TST, 5
Let $ABCD$ be a cyclic quadrilateral for which $|AD| =|BD|$. Let $M$ be the intersection of $AC$ and $BD$. Let $I$ be the incentre of $\triangle BCM$. Let $N$ be the second intersection pointof $AC$ and the circumscribed circle of $\triangle BMI$. Prove that $|AN| \cdot |NC| = |CD | \cdot |BN|$.
2018 Junior Balkan Team Selection Tests - Romania, 3
Let $ABCD$ be a cyclic quadrilateral. The line parallel to $BD$ passing through $A$ meets the line parallel to $AC$ passing through $B$ at $E$. The circumcircle of triangle $ABE$ meets the lines $EC$ and $ED$, again, at $F$ and $G$, respectively. Prove that the lines $AB, CD$ and $FG$ are either parallel or concurrent.
2017 Iran Team Selection Test, 5
In triangle $ABC$, arbitrary points $P,Q$ lie on side $BC$ such that $BP=CQ$ and $P$ lies between $B,Q$.The circumcircle of triangle $APQ$ intersects sides $AB,AC$ at $E,F$ respectively.The point $T$ is the intersection of $EP,FQ$.Two lines passing through the midpoint of $BC$ and parallel to $AB$ and $AC$, intersect $EP$ and $FQ$ at points $X,Y$ respectively.
Prove that the circumcircle of triangle $TXY$ and triangle $APQ$ are tangent to each other.
[i]Proposed by Iman Maghsoudi[/i]
2021 Thailand TSTST, 3
A triangle $ABC$ with $AB<AC<BC$ is given. The point $P$ is the center of an excircle touching the line segment $AB$ at $D$. The point $Q$ is the center of an excircle touching the line segment $AC$ at $E$. The circumcircle of the triangle $ADE$ intersects $\overline{PE}$ and $\overline{QD}$ again at $G$ and $H$ respectively. The line perpendicular to $\overline{AG}$ at $G$ intersects the side $AB$ at $R$. The line perpendicular to $\overline{AH}$ at $H$ intersects the side $AC$ at $S$. Prove that $\overline{DE}$ and $\overline{RS}$ are parallel.
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.
2021 Taiwan TST Round 3, G
Let $ABC$ be a triangle with $AB<AC$, and let $I_a$ be its $A$-excenter. Let $D$ be the projection of $I_a$ to $BC$. Let $X$ be the intersection of $AI_a$ and $BC$, and let $Y,Z$ be the points on $AC,AB$, respectively, such that $X,Y,Z$ are on a line perpendicular to $AI_a$. Let the circumcircle of $AYZ$ intersect $AI_a$ again at $U$. Suppose that the tangent of the circumcircle of $ABC$ at $A$ intersects $BC$ at $T$, and the segment $TU$ intersects the circumcircle of $ABC$ at $V$. Show that $\angle BAV=\angle DAC$.
[i]Proposed by usjl.[/i]
2014 Vietnam National Olympiad, 1
Given a circle $(O)$ and two fixed points $B,C$ on $(O),$ and an arbitrary point $A$ on $(O)$ such that the triangle $ABC$ is acute. $M$ lies on ray $AB,$ $N$ lies on ray $AC$ such that $MA=MC$ and $NA=NB.$ Let $P$ be the intersection of $(AMN)$ and $(ABC),$ $P\ne A.$ $MN$ intersects $BC$ at $Q.$
a) Prove that $A,P,Q$ are collinear.
b) $D$ is the midpoint of $BC.$ Let $K$ be the intersection of $(M,MA)$ and $(N,NA),$ $K\ne A.$ $d$ is the line passing through $A$ and perpendicular to $AK.$ $E$ is the intersection of $d$ and $BC.$ $(ADE)$ intersects $(O)$ at $F,$ $F\ne A.$ Prove that $AF$ passes through a fixed point.
2017 Harvard-MIT Mathematics Tournament, 6
In convex quadrilateral $ABCD$ we have $AB=15$, $BC=16$, $CD=12$, $DA=25$, and $BD=20$. Let $M$ and $\gamma$ denote the circumcenter and circumcircle of $\triangle ABD$. Line $CB$ meets $\gamma$ again at $F$, line $AF$ meets $MC$ at $G$, and line $GD$ meets $\gamma$ again at $E$. Determine the area of pentagon $ABCDE$.
2014 Contests, 4
$ABC$ is an acute triangle with orthocenter $H$. Points $D$ and $E$ lie on segment $BC$. Circumcircle of $\triangle BHC$ instersects with segments $AD$,$AE$ at $P$ and $Q$, respectively. Prove that if $BD^2+CD^2=2DP\cdot DA$ and $BE^2+CE^2=2EQ\cdot EA$, then $BP=CQ$.
2014 France Team Selection Test, 5
Let $\omega$ be the circumcircle of a triangle $ABC$. Denote by $M$ and $N$ the midpoints of the sides $AB$ and $AC$, respectively, and denote by $T$ the midpoint of the arc $BC$ of $\omega$ not containing $A$. The circumcircles of the triangles $AMT$ and $ANT$ intersect the perpendicular bisectors of $AC$ and $AB$ at points $X$ and $Y$, respectively; assume that $X$ and $Y$ lie inside the triangle $ABC$. The lines $MN$ and $XY$ intersect at $K$. Prove that $KA=KT$.
2001 AMC 12/AHSME, 24
In $ \triangle ABC$, $ \angle ABC \equal{} 45^\circ$. Point $ D$ is on $ \overline{BC}$ so that $ 2 \cdot BD \equal{} CD$ and $ \angle DAB \equal{} 15^\circ$. Find $ \angle ACB$.
[asy]
pair A, B, C, D;
A = origin;
real Bcoord = 3*sqrt(2) + sqrt(6);
B = Bcoord/2*dir(180);
C = sqrt(6)*dir(120);
draw(A--B--C--cycle);
D = (C-B)/2.4 + B;
draw(A--D);
label("$A$", A, dir(0));
label("$B$", B, dir(180));
label("$C$", C, dir(110));
label("$D$", D, dir(130));
[/asy]
$ \textbf{(A)} \ 54^\circ \qquad \textbf{(B)} \ 60^\circ \qquad \textbf{(C)} \ 72^\circ \qquad \textbf{(D)} \ 75^\circ \qquad \textbf{(E)} \ 90^\circ$