Found problems: 343
2007 Bulgaria Team Selection Test, 1
In isosceles triangle $ABC(AC=BC)$ the point $M$ is in the segment $AB$ such that $AM=2MB,$ $F$ is the midpoint of $BC$ and $H$ is the orthogonal projection of $M$ in $AF.$ Prove that $\angle BHF=\angle ABC.$
2022 Taiwan TST Round 1, 1
In the triangle $ABC$ let $B'$ and $C'$ be the midpoints of the sides $AC$ and $AB$ respectively and $H$ the foot of the altitude passing through the vertex $A$. Prove that the circumcircles of the triangles $AB'C'$,$BC'H$, and $B'CH$ have a common point $I$ and that the line $HI$ passes through the midpoint of the segment $B'C'.$
2001 India IMO Training Camp, 1
If on $ \triangle ABC$, trinagles $ AEB$ and $ AFC$ are constructed externally such that $ \angle AEB\equal{}2 \alpha$, $ \angle AFB\equal{} 2 \beta$.
$ AE\equal{}EB$, $ AF\equal{}FC$.
COnstructed externally on $ BC$ is triangle $ BDC$ with $ \angle DBC\equal{} \beta$ , $ \angle BCD\equal{} \alpha$.
Prove that 1. $ DA$ is perpendicular to $ EF$.
2. If $ T$ is the projection of $ D$ on $ BC$, then prove that $ \frac{DA}{EF}\equal{} 2 \frac{DT}{BC}$.
2007 Junior Balkan Team Selection Tests - Romania, 3
Let $ABC$ be a right triangle with $A = 90^{\circ}$ and $D \in (AC)$. Denote by $E$ the reflection of $A$ in the line $BD$ and $F$ the intersection point of $CE$ with the perpendicular in $D$ to $BC$. Prove that $AF, DE$ and $BC$ are concurrent.
2001 All-Russian Olympiad, 2
Let the circle $ {\omega}_{1}$ be internally tangent to another circle $ {\omega}_{2}$ at $ N$.Take a point $ K$ on $ {\omega}_{1}$ and draw a tangent $ AB$ which intersects $ {\omega}_{2}$ at $ A$ and $ B$. Let $M$ be the midpoint of the arc $ AB$ which is on the opposite side of $ N$. Prove that, the circumradius of the $ \triangle KBM$ doesnt depend on the choice of $ K$.
2005 JBMO Shortlist, 1
Let $ABC$ be an acute-angled triangle inscribed in a circle $k$. It is given that the tangent from $A$ to the circle meets the line $BC$ at point $P$. Let $M$ be the midpoint of the line segment $AP$ and $R$ be the second intersection point of the circle $k$ with the line $BM$. The line $PR$ meets again the circle $k$ at point $S$ different from $R$.
Prove that the lines $AP$ and $CS$ are parallel.
2010 Germany Team Selection Test, 2
Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ meet at $E$ and the lines $AD$ and $BC$ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E$, $G$ and $H$.
[i]Proposed by David Monk, United Kingdom[/i]
2005 All-Russian Olympiad Regional Round, 11.4
11.4 Let $AA_1$ and $BB_1$ are altitudes of an acute non-isosceles triangle $ABC$, $A'$ is a midpoint of $BC$ and $B'$ is a midpoint of $AC$. A segement $A_1B_1$ intersects $A'B'$ at point $C'$. Prove that $CC'\perp HO$, where $H$ is a orthocenter and $O$ is a circumcenter of $ABC$.
([i]L. Emel'yanov[/i])
1970 IMO Shortlist, 6
In the triangle $ABC$ let $B'$ and $C'$ be the midpoints of the sides $AC$ and $AB$ respectively and $H$ the foot of the altitude passing through the vertex $A$. Prove that the circumcircles of the triangles $AB'C'$,$BC'H$, and $B'CH$ have a common point $I$ and that the line $HI$ passes through the midpoint of the segment $B'C'.$
2006 All-Russian Olympiad, 6
Consider a tetrahedron $SABC$. The incircle of the triangle $ABC$ has the center $I$ and touches its sides $BC$, $CA$, $AB$ at the points $E$, $F$, $D$, respectively. Let $A^{\prime}$, $B^{\prime}$, $C^{\prime}$ be the points on the segments $SA$, $SB$, $SC$ such that $AA^{\prime}=AD$, $BB^{\prime}=BE$, $CC^{\prime}=CF$, and let $S^{\prime}$ be the point diametrically opposite to the point $S$ on the circumsphere of the tetrahedron $SABC$. Assume that the line $SI$ is an altitude of the tetrahedron $SABC$. Show that $S^{\prime}A^{\prime}=S^{\prime}B^{\prime}=S^{\prime}C^{\prime}$.
1992 India Regional Mathematical Olympiad, 4
$ABCD$ is a cyclic quadrilateral with $AC \perp BD$; $AC$ meets $BD$ at $E$. Prove that \[ EA^2 + EB^2 + EC^2 + ED^2 = 4 R^2 \]
where $R$ is the radius of the circumscribing circle.
2010 Contests, 3
A triangle $ ABC$ is inscribed in a circle $ C(O,R)$ and has incenter $ I$. Lines $ AI,BI,CI$ meet the circumcircle $ (O)$ of triangle $ ABC$ at points $ D,E,F$ respectively. The circles with diameter $ ID,IE,IF$ meet the sides $ BC,CA, AB$ at pairs of points $ (A_1,A_2), (B_1, B_2), (C_1, C_2)$ respectively.
Prove that the six points $ A_1,A_2, B_1, B_2, C_1, C_2$ are concyclic.
Babis
2003 China Team Selection Test, 2
Denote by $\left(ABC\right)$ the circumcircle of a triangle $ABC$.
Let $ABC$ be an isosceles right-angled triangle with $AB=AC=1$ and $\measuredangle CAB=90^{\circ}$. Let $D$ be the midpoint of the side $BC$, and let $E$ and $F$ be two points on the side $BC$.
Let $M$ be the point of intersection of the circles $\left(ADE\right)$ and $\left(ABF\right)$ (apart from $A$).
Let $N$ be the point of intersection of the line $AF$ and the circle $\left(ACE\right)$ (apart from $A$).
Let $P$ be the point of intersection of the line $AD$ and the circle $\left(AMN\right)$.
Find the length of $AP$.
2005 Italy TST, 2
The circle $\Gamma$ and the line $\ell$ have no common points. Let $AB$ be the diameter of $\Gamma$ perpendicular to $\ell$, with $B$ closer to $\ell$ than $A$. An arbitrary point $C\not= A$, $B$ is chosen on $\Gamma$. The line $AC$ intersects $\ell$ at $D$. The line $DE$ is tangent to $\Gamma$ at $E$, with $B$ and $E$ on the same side of $AC$. Let $BE$ intersect $\ell$ at $F$, and let $AF$ intersect $\Gamma$ at $G\not= A$. Let $H$ be the reflection of $G$ in $AB$. Show that $F,C$, and $H$ are collinear.
2014 Contests, 1
Let $ABC$ be a triangle, let ${A}'$, ${B}'$, ${C}'$ be the orthogonal projections of the vertices $A$ ,$B$ ,$C$ on the lines $BC$, $CA$ and $AB$, respectively, and let $X$ be a point on the line $A{A}'$.Let $\gamma_{B}$ be the circle through $B$ and $X$, centred on the line $BC$, and let $\gamma_{C}$ be the circle through $C$ and $X$, centred on the line $BC$.The circle $\gamma_{B}$ meets the lines $AB$ and $B{B}'$ again at $M$ and ${M}'$, respectively, and the circle $\gamma_{C}$ meets the lines $AC$ and $C{C}'$ again at $N$ and ${N}'$, respectively.Show that the points $M$, ${M}'$, $N$ and ${N}'$ are collinear.
Swiss NMO - geometry, 2010.9
Let $ k$ and $ k'$ two concentric circles centered at $ O$, with $ k'$ being larger than $ k$. A line through $ O$ intersects $ k$ at $ A$ and $ k'$ at $ B$ such that $ O$ seperates $ A$ and $ B$. Another line through $ O$ intersects $ k$ at $ E$ and $ k'$ at $ F$ such that $ E$ separates $ O$ and $ F$.
Show that the circumcircle of $ \triangle{OAE}$ and the circles with diametres $ AB$ and $ EF$ have a common point.
2018 Romania Team Selection Tests, 1
In triangle $ABC$, let $\omega$ be the excircle opposite to $A$. Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.
JBMO Geometry Collection, 2005
Let $ABC$ be an acute-angled triangle inscribed in a circle $k$. It is given that the tangent from $A$ to the circle meets the line $BC$ at point $P$. Let $M$ be the midpoint of the line segment $AP$ and $R$ be the second intersection point of the circle $k$ with the line $BM$. The line $PR$ meets again the circle $k$ at point $S$ different from $R$.
Prove that the lines $AP$ and $CS$ are parallel.
2006 Turkey MO (2nd round), 1
Points $P$ and $Q$ on side $AB$ of a convex quadrilateral $ABCD$ are given such that $AP = BQ.$ The circumcircles of triangles $APD$ and $BQD$ meet again at $K$ and those of $APC$ and $BQC$ meet again at $L$. Show that the points $D,C,K,L$ lie on a circle.
2013 ELMO Shortlist, 13
In $\triangle ABC$, $AB<AC$. $D$ and $P$ are the feet of the internal and external angle bisectors of $\angle BAC$, respectively. $M$ is the midpoint of segment $BC$, and $\omega$ is the circumcircle of $\triangle APD$. Suppose $Q$ is on the minor arc $AD$ of $\omega$ such that $MQ$ is tangent to $\omega$. $QB$ meets $\omega$ again at $R$, and the line through $R$ perpendicular to $BC$ meets $PQ$ at $S$. Prove $SD$ is tangent to the circumcircle of $\triangle QDM$.
[i]Proposed by Ray Li[/i]
2007 Bundeswettbewerb Mathematik, 3
In triangle $ ABC$ points $ E$ and $ F$ lie on sides $ AC$ and $ BC$ such that segments $ AE$ and $ BF$ have equal length, and circles formed by $ A,C,F$ and by $ B,C,E,$ respectively, intersect at point $ C$ and another point $ D.$ Prove that that the line $ CD$ bisects $ \angle ACB.$
2016 Indonesia TST, 3
Circles $\Omega $ and $\omega $ are tangent at a point $P$ ($\omega $ lies inside $\Omega $). A chord $AB$ of $\Omega $ is tangent to $\omega $ at $C;$ the line $PC$ meets $\Omega $ again at $Q.$ Chords $QR$ and $QS$ of $ \Omega $ are tangent to $\omega .$ Let $I,X,$ and $Y$ be the incenters of the triangles $APB,$ $ARB,$ and $ASB,$ respectively. Prove that $\angle PXI+\angle PYI=90^{\circ }.$
2011 IMO Shortlist, 3
Let $ABCD$ be a convex quadrilateral whose sides $AD$ and $BC$ are not parallel. Suppose that the circles with diameters $AB$ and $CD$ meet at points $E$ and $F$ inside the quadrilateral. Let $\omega_E$ be the circle through the feet of the perpendiculars from $E$ to the lines $AB,BC$ and $CD$. Let $\omega_F$ be the circle through the feet of the perpendiculars from $F$ to the lines $CD,DA$ and $AB$. Prove that the midpoint of the segment $EF$ lies on the line through the two intersections of $\omega_E$ and $\omega_F$.
[i]Proposed by Carlos Yuzo Shine, Brazil[/i]
2009 Hungary-Israel Binational, 1
Given is the convex quadrilateral $ ABCD$. Assume that there exists a point $ P$ inside the quadrilateral for which the triangles $ ABP$ and $ CDP$ are both isosceles right triangles with the right angle at the common vertex $ P$. Prove that there exists a point $ Q$ for which the triangles $ BCQ$ and $ ADQ$ are also isosceles right triangles with the right angle at the common vertex $ Q$.
2009 Indonesia TST, 3
Let $ ABC$ be an acute triangle with $ \angle BAC\equal{}60^{\circ}$. Let $ P$ be a point in triangle $ ABC$ with $ \angle APB\equal{}\angle BPC\equal{}\angle CPA\equal{}120^{\circ}$. The foots of perpendicular from $ P$ to $ BC,CA,AB$ are $ X,Y,Z$, respectively. Let $ M$ be the midpoint of $ YZ$.
a) Prove that $ \angle YXZ\equal{}60^{\circ}$
b) Prove that $ X,P,M$ are collinear.