Found problems: 3882
2021 China Girls Math Olympiad, 7
In an acute triangle $ABC$, $AB \neq AC$, $O$ is its circumcenter. $K$ is the reflection of $B$ over $AC$ and $L$ is the reflection of $C$ over $AB$. $X$ is a point within $ABC$ such that $AX \perp BC, XK=XL$. Points $Y, Z$ are on $\overline{BK}, \overline{CL}$ respectively, satisfying $XY \perp CK, XZ \perp BL$.
Proof that $B, C, Y, O, Z$ lie on a circle.
2018 Bosnia And Herzegovina - Regional Olympiad, 4
Let $P$ be a point on circumcircle of triangle $ABC$ on arc $\stackrel{\frown}{BC}$ which does not contain point $A$. Let lines $AB$ and $CP$ intersect at point $E$, and lines $AC$ and $BP$ intersect at $F$. If perpendicular bisector of side $AB$ intersects $AC$ in point $K$, and perpendicular bisector of side $AC$ intersects side $AB$ in point $J$, prove that:
${\left(\frac{CE}{BF}\right)}^2=\frac{AJ\cdot JE}{AK \cdot KF}$
2017 Bosnia and Herzegovina Team Selection Test, Problem 1
Incircle of triangle $ ABC$ touches $ AB,AC$ at $ P,Q$. $ BI, CI$ intersect with $ PQ$ at $ K,L$. Prove that circumcircle of $ ILK$ is tangent to incircle of $ ABC$ if and only if $ AB\plus{}AC\equal{}3BC$.
1966 IMO Longlists, 56
In a tetrahedron, all three pairs of opposite (skew) edges are mutually perpendicular. Prove that the midpoints of the six edges of the tetrahedron lie on one sphere.
2020 Balkan MO Shortlist, G5
Let $ABC$ be an isosceles triangle with $AB = AC$ and $\angle A = 45^o$. Its circumcircle $(c)$ has center $O, M$ is the midpoint of $BC$ and $D$ is the foot of the perpendicular from $C$ to $AB$. With center $C$ and radius $CD$ we draw a circle which internally intersects $AC$ at the point $F$ and the circle $(c)$ at the points $Z$ and $E$, such that $Z$ lies on the small arc $BC$ and $E$ on the small arc $AC$. Prove that the lines $ZE$, $CO$, $FM$ are concurrent.
[i]Brazitikos Silouanos, Greece[/i]
2012 Danube Mathematical Competition, 2
Let $ABC$ be an acute triangle and let $A_1$, $B_1$, $C_1$ be points on the sides $BC, CA$ and $AB$, respectively. Show that the triangles $ABC$ and $A_1B_1C_1$ are similar ($\angle A = \angle A_1, \angle B = \angle B_1,\angle C = \angle C_1$) if and only if the orthocentre of the triangle $A_1B_1C_1$ and the circumcentre of the triangle $ABC$ coincide.
2010 Contests, 3
In an acute-angled triangle $ABC$, $CF$ is an altitude, with $F$ on $AB$, and $BM$ is a median, with $M$ on $CA$. Given that $BM=CF$ and $\angle MBC=\angle FCA$, prove that triangle $ABC$ is equilateral.
2012 ELMO Shortlist, 10
Let $A_1A_2A_3A_4A_5A_6A_7A_8$ be a cyclic octagon. Let $B_i$ by the intersection of $A_iA_{i+1}$ and $A_{i+3}A_{i+4}$. (Take $A_9 = A_1$, $A_{10} = A_2$, etc.) Prove that $B_1, B_2, \ldots , B_8$ lie on a conic.
[i]David Yang.[/i]
2018 OMMock - Mexico National Olympiad Mock Exam, 5
Let $ABC$ be a triangle with circumcirle $\Gamma$, and let $M$ and $N$ be the respective midpoints of the minor arcs $AB$ and $AC$ of $\Gamma$. Let $P$ and $Q$ be points such that $AB=BP$, $AC=CQ$, and $P$, $B$, $C$, $Q$ lie on $BC$ in that order. Prove that $PM$ and $QN$ meet at a point on $\Gamma$.
[i]Proposed by Victor DomÃnguez[/i]
2013 Lusophon Mathematical Olympiad, 2
Let $ABC$ be an acute triangle. The circumference with diameter $AB$ intersects sides $AC$ and $BC$ at $E$ and $F$ respectively. The tangent lines to the circumference at the points $E$ and $F$ meet at $P$. Show that $P$ belongs to the altitude from $C$ of triangle $ABC$.
2017 Junior Balkan Team Selection Tests - Romania, 3
Let $I$ be the incenter of the scalene $\Delta ABC$, such, $AB<AC$, and let $I'$ be the reflection of point $I$ in line $BC$. The angle bisector $AI$ meets $BC$ at $D$ and circumcircle of $\Delta ABC$ at $E$. The line $EI'$ meets the circumcircle at $F$. Prove, that,
$\text{(i) } \frac{AI}{IE}=\frac{ID}{DE}$
$\text{(ii) } IA=IF$
2011 Indonesia MO, 8
Given a triangle $ABC$. Its incircle is tangent to $BC, CA, AB$ at $D, E, F$ respectively. Let $K, L$ be points on $CA, AB$ respectively such that $K \neq A \neq L, \angle EDK = \angle ADE, \angle FDL = \angle ADF$. Prove that the circumcircle of $AKL$ is tangent to the incircle of $ABC$.
2010 Sharygin Geometry Olympiad, 3
Points $A', B', C'$ lie on sides $BC, CA, AB$ of triangle $ABC.$ for a point $X$ one has $\angle AXB =\angle A'C'B' + \angle ACB$ and $\angle BXC = \angle B'A'C' +\angle BAC.$ Prove that the quadrilateral $XA'BC'$ is cyclic.
2013 China Team Selection Test, 2
Let $P$ be a given point inside the triangle $ABC$. Suppose $L,M,N$ are the midpoints of $BC, CA, AB$ respectively and \[PL: PM: PN= BC: CA: AB.\] The extensions of $AP, BP, CP$ meet the circumcircle of $ABC$ at $D,E,F$ respectively. Prove that the circumcentres of $APF, APE, BPF, BPD, CPD, CPE$ are concyclic.
2007 Canada National Olympiad, 5
Let the incircle of triangle $ ABC$ touch sides $ BC,\, CA$ and $ AB$ at $ D,\, E$ and $ F,$ respectively. Let $ \omega,\,\omega_{1},\,\omega_{2}$ and $ \omega_{3}$ denote the circumcircles of triangle $ ABC,\, AEF,\, BDF$ and $ CDE$ respectively.
Let $ \omega$ and $ \omega_{1}$ intersect at $ A$ and $ P,\,\omega$ and $ \omega_{2}$ intersect at $ B$ and $ Q,\,\omega$ and $ \omega_{3}$ intersect at $ C$ and $ R.$
$ a.$ Prove that $ \omega_{1},\,\omega_{2}$ and $ \omega_{3}$ intersect in a common point.
$ b.$ Show that $ PD,\, QE$ and $ RF$ are concurrent.
1967 Bulgaria National Olympiad, Problem 3
It is given a right-angled triangle $ABC$ and its circumcircle $k$.
(a) prove that the radii of the circle $k_1$ tangent to the cathets of the triangle and to the circle $k$ is equal to the diameter of the incircle of the triangle ABC.
(b) on the circle $k$ there may be found a point $M$ for which the sum $MA+MB+MC$ is as large as possible.
1986 IMO Longlists, 41
Let $M,N,P$ be the midpoints of the sides $BC, CA, AB$ of a triangle $ABC$. The lines $AM, BN, CP$ intersect the circumcircle of $ABC$ at points $A',B', C'$, respectively. Show that if $A'B'C'$ is an equilateral triangle, then so is $ABC.$
2014 ELMO Shortlist, 8
In triangle $ABC$ with incenter $I$ and circumcenter $O$, let $A',B',C'$ be the points of tangency of its circumcircle with its $A,B,C$-mixtilinear circles, respectively. Let $\omega_A$ be the circle through $A'$ that is tangent to $AI$ at $I$, and define $\omega_B, \omega_C$ similarly. Prove that $\omega_A,\omega_B,\omega_C$ have a common point $X$ other than $I$, and that $\angle AXO = \angle OXA'$.
[i]Proposed by Sammy Luo[/i]
EGMO 2017, 6
Let $ABC$ be an acute-angled triangle in which no two sides have the same length. The reflections of the centroid $G$ and the circumcentre $O$ of $ABC$ in its sides $BC,CA,AB$ are denoted by $G_1,G_2,G_3$ and $O_1,O_2,O_3$, respectively. Show that the circumcircles of triangles $G_1G_2C$, $G_1G_3B$, $G_2G_3A$, $O_1O_2C$, $O_1O_3B$, $O_2O_3A$ and $ABC$ have a common point.
[i]The centroid of a triangle is the intersection point of the three medians. A median is a line connecting a vertex of the triangle to the midpoint of the opposite side.[/i]
2007 Korea - Final Round, 1
Let $ O$ be the circumcenter of an acute triangle $ ABC$ and let $ k$ be the circle with center $ P$ that is tangent to $ O$ at $ A$ and tangent to side $ BC$ at $ D$. Circle $ k$ meets $ AB$ and $ AC$ again at $ E$ and $ F$ respectively. The lines $ OP$ and $ EP$ meet $ k$ again at $ I$ and $ G$. Lines $ BO$ and $ IG$ intersect at $ H$. Prove that $ \frac{{DF}^2}{AF}\equal{}GH$.
2018 Moscow Mathematical Olympiad, 3
$O$ is circumcircle and $AH$ is the altitude of $\triangle ABC$. $P$ is the point on line $OC$ such that $AP \perp OC$. Prove, that midpoint of $AB$ lies on the line $HP$.
2016 India Regional Mathematical Olympiad, 1
Given are two circles $\omega_1,\omega_2$ which intersect at points $X,Y$. Let $P$ be an arbitrary point on $\omega_1$. Suppose that the lines $PX,PY$ meet $\omega_2$ again at points $A,B$ respectively. Prove that the circumcircles of all triangles $PAB$ have the same radius.
2012 Oral Moscow Geometry Olympiad, 5
Given a circle and a chord $AB$, different from the diameter. Point $C$ moves along the large arc $AB$. The circle passing through passing through points $A, C$ and point $H$ of intersection of altitudes of of the triangle $ABC$, re-intersects the line $BC$ at point $P$. Prove that line $PH$ passes through a fixed point independent of the position of point $C$.
2021 Pan-African, 2
Let $\Gamma$ be a circle, $P$ be a point outside it, and $A$ and $B$ the intersection points between $\Gamma$ and the tangents from $P$ to $\Gamma$. Let $K$ be a point on the line $AB$, distinct from $A$ and $B$ and let $T$ be the second intersection point of $\Gamma$ and the circumcircle of the triangle $PBK$.Also, let $P'$ be the reflection of $P$ in point $A$.
Show that $\angle PBT=\angle P'KA$
2008 Romania Team Selection Test, 1
Let $ ABC$ be a triangle with $ \measuredangle{BAC} < \measuredangle{ACB}$. Let $ D$, $ E$ be points on the sides $ AC$ and $ AB$, such that the angles $ ACB$ and $ BED$ are congruent. If $ F$ lies in the interior of the quadrilateral $ BCDE$ such that the circumcircle of triangle $ BCF$ is tangent to the circumcircle of $ DEF$ and the circumcircle of $ BEF$ is tangent to the circumcircle of $ CDF$, prove that the points $ A$, $ C$, $ E$, $ F$ are concyclic.
[i]Author: Cosmin Pohoata[/i]