Found problems: 3882
2014 ELMO Shortlist, 2
$ABCD$ is a cyclic quadrilateral inscribed in the circle $\omega$. Let $AB \cap CD = E$, $AD \cap BC = F$. Let $\omega_1, \omega_2$ be the circumcircles of $AEF, CEF$, respectively. Let $\omega \cap \omega_1 = G$, $\omega \cap \omega_2 = H$. Show that $AC, BD, GH$ are concurrent.
[i]Proposed by Yang Liu[/i]
2006 IberoAmerican, 1
In a scalene triangle $ABC$ with $\angle A = 90^\circ,$ the tangent line at $A$ to its circumcircle meets line $BC$ at $M$ and the incircle touches $AC$ at $S$ and $AB$ at $R.$
The lines $RS$ and $BC$ intersect at $N,$ while the lines $AM$ and $SR$ intersect at $U.$
Prove that the triangle $UMN$ is isosceles.
2010 Malaysia National Olympiad, 7
Let $ABC$ be a triangle in which $AB=AC$ and let $I$ be its incenter. It is known that $BC=AB+AI$. Let $D$ be a point on line $BA$ extended beyond $A$ such that $AD=AI$. Prove that $DAIC$ is a cyclic quadrilateral.
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 $KL$.
2005 Uzbekistan National Olympiad, 4
Let $ABCD$ is a cyclic. $K,L,M,N$ are midpoints of segments $AB$, $BC$ $CD$ and $DA$. $H_{1},H_{2},H_{3},H_{4}$ are orthocenters of $AKN$ $KBL$ $LCM$ and $MND$. Prove that $H_{1}H_{2}H_{3}H_{4}$ is a paralelogram.
2005 Korea Junior Math Olympiad, 5
In $\triangle ABC$, let the bisector of $\angle BAC$ hit the circumcircle at $M$. Let $P$ be the intersection of $CM$ and $AB$. Denote by $(V,WX,YZ)$ the intersection of the line passing $V$ perpendicular to $WX$ with the line $YZ$. Prove that the points $(P,AM,AC), (P,AC,AM), (P,BC,MB)$ are collinear.
[hide=Restatement]In isosceles triangle $APX$ with $AP=AX$, select a point $M$ on the altitude. $PM$ intersects $AX$ at $C$. The circumcircle of $ACM$ intersects $AP$ at $B$. A line passing through $P$ perpendicular to $BC$ intersects $MB$ at $Z$. Show that $XZ$ is perpendicular to $AP$.[/hide]
2009 German National Olympiad, 5
Let a triangle $ ABC$. $ E,F$ in segment $ AB$ so that $ E$ lie between $ AF$ and half of circle with diameter $ EF$ is tangent with $ BC,CA$ at $ G,H$. $ HF$ cut $ GE$ at $ S$, $ HE$ cut $ FG$ at $ T$. Prove that $ C$ is midpoint of $ ST$.
2021-IMOC, G11
The incircle of $\triangle ABC$ tangents $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. The projections of $B$, $C$ to $AD$ are $U$, $V$, respectively; the projections of $C$, $A$ to $BE$ are $W$, $X$, respectively; and the projections of $A$, $B$ to $CF$ are $Y$, $Z$, respectively. Show that the circumcircle of the triangle formed by $UX$, $VY$, $WZ$ is tangent to the incircle of $\triangle ABC$.
2017 CCA Math Bonanza, I15
Let $ABC$, $AB<AC$ be an acute triangle inscribed in circle $\Gamma$ with center $O$. The altitude from $A$ to $BC$ intersects $\Gamma$ again at $A_1$. $OA_1$ intersects $BC$ at $A_2$ Similarly define $B_1$, $B_2$, $C_1$, and $C_2$. Then $B_2C_2=2\sqrt{2}$. If $B_2C_2$ intersects $AA_2$ at $X$ and $BC$ at $Y$, then $XB_2=2$ and $YB_2=k$. Find $k^2$.
[i]2017 CCA Math Bonanza Individual Round #15[/i]
Kyiv City MO Juniors 2003+ geometry, 2020.8.5
Given a triangle $ABC, O$ is the center of the circumcircle, $M$ is the midpoint of $BC, W$ is the second intersection of the bisector of the angle $C$ with this circle. A line parallel to $BC$ passing through $W$, intersects$ AB$ at the point $K$ so that $BK = BO$. Find the measure of angle $WMB$.
(Anton Trygub)
2000 Moldova National Olympiad, Problem 8
In an isosceles triangle $ABC$ with $BC=AC$ and $\angle B<60^\circ$, $I$ is the incenter and $O$ the circumcenter. The circle with center $E$ that passes through $A,O$ and $I$ intersects the circumcircle of $\triangle ABC$ again at point $D$. Prove that the lines $DE$ and $CO$ intersect on the circumcircle of $ABC$.
2006 Iran Team Selection Test, 5
Let $ABC$ be an acute angle triangle.
Suppose that $D,E,F$ are the feet of perpendicluar lines from $A,B,C$ to $BC,CA,AB$.
Let $P,Q,R$ be the feet of perpendicular lines from $A,B,C$ to $EF,FD,DE$.
Prove that
\[ 2(PQ+QR+RP)\geq DE+EF+FD \]
2008 Balkan MO Shortlist, G2
Given a scalene acute triangle $ ABC$ with $ AC>BC$ let $ F$ be the foot of the altitude from $ C$. Let $ P$ be a point on $ AB$, different from $ A$ so that $ AF\equal{}PF$. Let $ H,O,M$ be the orthocenter, circumcenter and midpoint of $ [AC]$. Let $ X$ be the intersection point of $ BC$ and $ HP$. Let $ Y$ be the intersection point of $ OM$ and $ FX$ and let $ OF$ intersect $ AC$ at $ Z$. Prove that $ F,M,Y,Z$ are concyclic.
2014 Canada National Olympiad, 4
The quadrilateral $ABCD$ is inscribed in a circle. The point $P$ lies in the interior of $ABCD$, and $\angle P AB = \angle P BC = \angle P CD = \angle P DA$. The lines $AD$ and $BC$ meet at $Q$, and the lines $AB$ and $CD$ meet at $R$. Prove that the lines $P Q$ and $P R$ form the same angle as the diagonals of $ABCD$.
2006 Vietnam National Olympiad, 2
Let $ABCD$ be a convex quadrilateral. Take an arbitrary point $M$ on the line $AB$, and let $N$ be the point of intersection of the circumcircles of triangles $MAC$ and $MBC$ (different from $M$). Prove that:
a) The point $N$ lies on a fixed circle;
b) The line $MN$ passes though a fixed point.
2016 India IMO Training Camp, 1
Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let $A_1,B_1$ and $C_1$ be respectively the midpoints of the arcs $BAC,CBA$ and $ACB$ of $\Gamma$. Show that the inradius of triangle $A_1B_1C_1$ is not less than the inradius of triangle $ABC$.
2005 IMAR Test, 1
The incircle of triangle $ABC$ touches the sides $BC,CA,AB$ at the points $D,E,F$, respectively. Let $K$ be a point on the side $BC$ and let $M$ be the point on the line segment $AK$ such that $AM=AE=AF$. Denote by $L,N$ the incenters of triangles $ABK,ACK$, respectively.
Prove that $K$ is the foot of the altitude from $A$ if and only if $DLMN$ is a square.
[hide="Remark"]This problem is slightly connected to [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=344774#p344774]GMB-IMAR 2005, Juniors, Problem 2[/url]
[/hide]
[i]Bogdan Enescu[/i]
2012 Sharygin Geometry Olympiad, 6
Let $ABC$ be an isosceles triangle with $BC = a$ and $AB = AC = b$. Segment $AC$ is the base of an isosceles triangle $ADC$ with $AD = DC = a$ such that points $D$ and $B$ share the opposite sides of AC. Let $CM$ and $CN$ be the bisectors in triangles $ABC$ and $ADC$ respectively. Determine the circumradius of triangle $CMN$.
(M.Rozhkova)
2016 ELMO Problems, 6
Elmo is now learning olympiad geometry. In triangle $ABC$ with $AB\neq AC$, let its incircle be tangent to sides $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. The internal angle bisector of $\angle BAC$ intersects lines $DE$ and $DF$ at $X$ and $Y$, respectively. Let $S$ and $T$ be distinct points on side $BC$ such that $\angle XSY=\angle XTY=90^\circ$. Finally, let $\gamma$ be the circumcircle of $\triangle AST$.
(a) Help Elmo show that $\gamma$ is tangent to the circumcircle of $\triangle ABC$.
(b) Help Elmo show that $\gamma$ is tangent to the incircle of $\triangle ABC$.
[i]James Lin[/i]
2021 Canadian Mathematical Olympiad Qualification, 4
Let $O$ be the centre of the circumcircle of triangle $ABC$ and let $I$ be the centre of the incircle of triangle $ABC$. A line passing through the point $I$ is perpendicular to the line $IO$ and passes through the incircle at points $P$ and $Q$. Prove that the diameter of the circumcircle is equal to the perimeter of triangle $OPQ$.
2008 Moldova National Olympiad, 9.3
From the vertex $ A$ of the equilateral triangle $ ABC$ a line is drown that intercepts the segment $ [BC]$ in the point $ E$. The point $ M \in (AE$ is such that $ M$ external to $ ABC$, $ \angle AMB \equal{} 20 ^\circ$ and $ \angle AMC \equal{} 30 ^ \circ$. What is the measure of the angle $ \angle MAB$?
2012 Argentina National Olympiad, 3
In the triangle $ABC$ the incircle is tangent to the sides $AB$ and $AC$ at $D$ and $E$ respectively. The line $DE$ intersects the circumcircle at $P$ and $Q$, with $P$ in the small arc $AB$ and $Q$ in the small arc $AC$. If $P$ is the midpoint of the arc $AB$, find the angle A and the ratio $\frac{PQ}{BC}$.
2006 Poland - Second Round, 2
Point $C$ is a midpoint of $AB$. Circle $o_1$ which passes through $A$ and $C$ intersect circle $o_2$ which passes through $B$ and $C$ in two different points $C$ and $D$. Point $P$ is a midpoint of arc $AD$ of circle $o_1$ which doesn't contain $C$. Point $Q$ is a midpoint of arc $BD$ of circle $o_2$ which doesn't contain $C$. Prove that $PQ \perp CD$.
Geometry Mathley 2011-12, 15.4
Let $ABC$ be a fixed triangle. Point $D$ is an arbitrary point on the side $BC$. Point $P$ is fixed on $AD$. The circumcircle of triangle $BPD$ meets $AB$ at $E$ distinct from $B$. Point $Q$ varies on $AP$. Let $BQ$ and $CQ$ meet the circumcircles of triangles $BPD, CPD$ respectively at $F,Z$ distinct from $B,C$. Prove that the circumcircle $EFZ$ is through a fixed point distinct from $E$ and this fixed point is on the circumcircle of triangle $CPD$.
Kostas Vittas
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.