Found problems: 3882
1990 IMO Longlists, 96
Suppose that points $X, Y,Z$ are located on sides $BC, CA$, and $AB$, respectively, of triangle $ABC$ in such a way that triangle $XY Z$ is similar to triangle $ABC$. Prove that the orthocenter of triangle $XY Z$ is the circumcenter of triangle $ABC.$
2014 Turkey EGMO TST, 5
Let $ABC$ be a triangle with circumcircle $\omega$ and let $\omega_A$ be a circle drawn outside $ABC$ and tangent to side $BC$ at $A_1$ and tangent to $\omega$ at $A_2$. Let the circles $\omega_B$ and $\omega_C$ and the points $B_1, B_2, C_1, C_2$ are defined similarly. Prove that if the lines $AA_1, BB_1, CC_1$ are concurrent, then the lines $AA_2, BB_2, CC_2$ are also concurrent.
2003 Balkan MO, 2
Let $ABC$ be a triangle, and let the tangent to the circumcircle of the triangle $ABC$ at $A$ meet the line $BC$ at $D$. The perpendicular to $BC$ at $B$ meets the perpendicular bisector of $AB$ at $E$. The perpendicular to $BC$ at $C$ meets the perpendicular bisector of $AC$ at $F$. Prove that the points $D$, $E$ and $F$ are collinear.
[i]Valentin Vornicu[/i]
2011 Croatia Team Selection Test, 3
Let $K$ and $L$ be the points on the semicircle with diameter $AB$. Denote intersection of $AK$ and $AL$ as $T$ and let $N$ be the point such that $N$ is on segment $AB$ and line $TN$ is perpendicular to $AB$. If $U$ is the intersection of perpendicular bisector of $AB$ an $KL$ and $V$ is a point on $KL$ such that angles $UAV$ and $UBV$ are equal. Prove that $NV$ is perpendicular to $KL$.
2001 Canada National Olympiad, 3
Let $ABC$ be a triangle with $AC > AB$. Let $P$ be the intersection point of the perpendicular bisector of $BC$ and the internal angle bisector of $\angle{A}$. Construct points $X$ on $AB$ (extended) and $Y$ on $AC$ such that $PX$ is perpendicular to $AB$ and $PY$ is perpendicular to $AC$. Let $Z$ be the intersection point of $XY$ and $BC$.
Determine the value of $\frac{BZ}{ZC}$.
2012 Greece JBMO TST, 3
Let $ABC$ be an acute triangle with $AB<AC<BC$, inscribed in circle $c(O,R)$ (with center $O$ and radius $R$). Let $O_1$ be the symmetric point of $O$ wrt $AC$. Circle $c_1(O_1,R)$ intersects $BC$ at $Z$. If the extension of the altitude $AD$ intersects the cicrumscribed circle $c(O,R)$ at point $E$, prove that $EC$ is perpendicular on $AZ$.
2014 Sharygin Geometry Olympiad, 12
Circles $\omega_1$ and $\omega_2$ meet at points $A$ and $B$. Let points $K_1$ and $K_2 $ of $\omega_1$ and $\omega_2$ respectively be such that $K_1A$ touches $\omega_2$, and $K_2A$ touches $\omega_1$. The circumcircle of triangle $K_1BK_2$ meets lines $AK_1$ and $AK_2$ for the second time at points $L_1$ and $L_2$ respectively. Prove that $L_1$ and $L_2$ are equidistant from line $AB$.
2015 AoPS Mathematical Olympiad, 5
Let $ABC$ be a triangle with orthocenter $h$. Let $AH$, $BH$, and $CH$ intersect the circumcircle of $\triangle ABC$ at points $D$, $E$, and $F$. Find the maximum value of $\frac{[DEF]}{[ABC]}$. (Here $[X]$ denotes the area of $X$.)
[i]Proposed by tkhalid.[/i]
1998 Slovenia National Olympiad, Problem 3
A point $A$ is outside a circle $\mathcal K$ with center $O$. Line $AO$ intersects the circle at $B$ and $C$, and a tangent through $A$ touches the circle in $D$. Let $E$ be an arbitrary point on the line $BD$ such that $D$ lies between $B$ and $E$. The circumcircle of the triangle $DCE$ meets line $AO$ at $C$ and $F$ and line $AD$ at $D$ and $G$. Prove that the lines $BD$ and $FG$ are parallel.
2005 AIME Problems, 14
In triangle $ABC$, $AB=13$, $BC=15$, and $CA=14$. Point $D$ is on $\overline{BC}$ with $CD=6.$ Point $E$ is on $\overline{BC}$ such that $\angle BAE\cong \angle CAD.$ Given that $BE=\frac pq$ where $p$ and $q$ are relatively prime positive integers, find $q.$
2018 Oral Moscow Geometry Olympiad, 3
On the extensions of sides $CA$ and $AB$ of triangle $ABC$ beyond points $A$ and $B$, respectively, the segments $AE = BC$ and $BF = AC$ are drawn. A circle is tangent to segment $BF$ at point $N$, side $BC$ and the extension of side $AC$ beyond point $C$. Point $M$ is the midpoint of segment $EF$. Prove that the line $MN$ is parallel to the bisector of angle $A$.
2009 Serbia National Math Olympiad, 6
Triangle ABC has incircle w centered as S that touches the sides BC,CA and AB at P,Q and R respectively. AB isn't equal AC, the lines QR and BC intersects at point M, the circle that passes through points B and C touches the circle w at point N, circumcircle of triangle MNP intersects with line AP at L (L isn't equal to P). Then prove that S,L and M lie on the same line
2013 ELMO Shortlist, 5
Let $\omega_1$ and $\omega_2$ be two orthogonal circles, and let the center of $\omega_1$ be $O$. Diameter $AB$ of $\omega_1$ is selected so that $B$ lies strictly inside $\omega_2$. The two circles tangent to $\omega_2$, passing through $O$ and $A$, touch $\omega_2$ at $F$ and $G$. Prove that $FGOB$ is cyclic.
[i]Proposed by Eric Chen[/i]
1999 Brazil National Olympiad, 6
Given any triangle $ABC$, show how to construct $A'$ on the side $AB$, $B'$ on the side $BC$ and $C'$ on the side $CA$, such that $ABC$ and $A'B'C'$ are similar (with $\angle A = \angle A', \angle B = \angle B', \angle C = \angle C'$) and $A'B'C'$ has the least possible area.
2002 China Second Round Olympiad, 1
In $\triangle ABC$, $\angle A = 60$, $AB>AC$, point $O$ is the circumcenter and $H$ is the intersection point of two altitudes $BE$ and $CF$. Points $M$ and $N$ are on the line segments $BH$ and $HF$ respectively, and satisfy $BM=CN$. Determine the value of $\frac{MH+NH}{OH}$.
2002 Moldova National Olympiad, 4
Let the triangle $ ADB_1$ s.t. $ m(\angle DAB_1)\ne 90^\circ$.On the sides of this triangle externally are constructed the squares $ ABCD$
and $ AB_1C_1D_1$ with centers $ O_1$ and $ O_2$, respectively.Prove that the circumcircles of the triangles $ BAB_1$, $ DAD_1$ and
$ O_1AO_2$ share a common point, that differs from $ A$.
2017 Bosnia and Herzegovina Team Selection Test, Problem 6
Given is an acute triangle $ABC$. $M$ is an arbitrary point at the side $AB$ and $N$ is midpoint of $AC$. The foots of the perpendiculars from $A$ to $MC$ and $MN$ are points $P$ and $Q$. Prove that center of the circumcircle of triangle $PQN$ lies on the fixed line for all points $M$ from the side $AB$.
2007 Germany Team Selection Test, 3
In triangle $ ABC$ we have $ a \geq b$ and $ a \geq c.$ Prove that the ratio of circumcircle radius to incircle diameter is at least as big as the length of the centroidal axis $ s_a$ to the altitude $ a_a.$ When do we have equality?
2016 Stars of Mathematics, 3
Let $ ABC $ be a triangle, $ M_A $ be the midpoint of the side $ BC, $ and $ P_A $ be the orthogonal projection of $ A $ on $ BC. $ Similarly, define $ M_B,M_C,P_B,P_C. M_BM_C $ intersects $ P_BP_C $ at $ S_A, $ and the tangent of the circumcircle of $ ABC $ at $ A $ meets $ BC $ at $ T_A. $ Similarly, define $ S_B,S_C,T_B,T_C. $
Show that the perpendiculars through $ A,B,C, $ to $ S_AT_A,S_BT_B, $ respectively, $ S_CT_C, $ are concurent.
[i]Flavian Georgescu[/i]
2014 Indonesia MO, 2
Let $ABC$ be a triangle. Suppose $D$ is on $BC$ such that $AD$ bisects $\angle BAC$. Suppose $M$ is on $AB$ such that $\angle MDA = \angle ABC$, and $N$ is on $AC$ such that $\angle NDA = \angle ACB$. If $AD$ and $MN$ intersect on $P$, prove that $AD^3 = AB \cdot AC \cdot AP$.
Croatia MO (HMO) - geometry, 2018.3
Let $k$ be a circle centered at $O$. Let $\overline{AB}$ be a chord of that circle and $M$ its midpoint. Tangent on $k$ at points $A$ and $B$ intersect at $T$. The line $\ell$ goes through $T$, intersect the shorter arc $AB$ at the point $C$ and the longer arc $AB$ at the point $D$, so that $|BC| = |BM|$. Prove that the circumcenter of the triangle $ADM$ is the reflection of $O$ across the line $AD$
2022-2023 OMMC, 19
Let $\triangle ABC$ be a triangle with $AB = 7$, $AC = 8$, and $BC = 3$. Let $P_1$ and $P_2$ be two distinct points on line $AC$ ($A, P_1, C, P_2$ appear in that order on the line) and $Q_1$ and $Q_2$ be two distinct points on line $AB$ ($A, Q_1, B, Q_2$ appear in that order on the line) such that $BQ_1 = P_1Q_1 = P_1C$ and $BQ_2 = P_2Q_2 = P_2C$. Find the distance between the circumcenters of $BP_1P_2$ and $CQ_1Q_2$.
2023 Turkey MO (2nd round), 6
On a triangle $ABC$, points $D$, $E$, $F$ are given on the segments $BC$, $AC$, $AB$ respectively such that $DE \parallel AB$, $DF \parallel AC$ and $\frac{BD}{DC}=\frac{AB^2}{AC^2}$ holds. Let the circumcircle of $AEF$ meet $AD$ at $R$ and the line that is tangent to the circumcircle of $ABC$ at $A$ at $S$ again. Let the line $EF$ intersect $BC$ at $L$ and $SR$ at $T$. Prove that $SR$ bisects $AB$ if and only if $BS$ bisects $TL$.
2017 Oral Moscow Geometry Olympiad, 3
On the plane, a non-isosceles triangle is given, a circle circumscribed around it and the center of its inscribed circle are marked. Using only a ruler without tick marks and drawing no more than seven lines, construct the diameter of the circumcircle.
2013 Sharygin Geometry Olympiad, 19
a) The incircle of a triangle $ABC$ touches $AC$ and $AB$ at points $B_0$ and $C_0$ respectively. The bisectors of angles $B$ and $C$ meet the perpendicular bisector to the bisector $AL$ in points $Q$ and $P$ respectively. Prove that the lines $PC_0, QB_0$ and $BC$ concur.
b) Let $AL$ be the bisector of a triangle $ABC$. Points $O_1$ and $O_2$ are the circumcenters of triangles $ABL$ and $ACL$ respectively. Points $B_1$ and $C_1$ are the projections of $C$ and $B$ to the bisectors of angles $B$ and $C$ respectively. Prove that the lines $O_1C_1, O_2B_1,$ and $BC$ concur.
c) Prove that the two points obtained in pp. a) and b) coincide.