Found problems: 343
2010 Indonesia TST, 2
Circles $ \Gamma_1$ and $ \Gamma_2$ are internally tangent to circle $ \Gamma$ at $ P$ and $ Q$, respectively. Let $ P_1$ and $ Q_1$ are on $ \Gamma_1$ and $ \Gamma_2$ respectively such that $ P_1Q_1$ is the common tangent of $ P_1$ and $ Q_1$. Assume that $ \Gamma_1$ and $ \Gamma_2$ intersect at $ R$ and $ R_1$. Define $ O_1,O_2,O_3$ as the intersection of $ PQ$ and $ P_1Q_1$, the intersection of $ PR$ and $ P_1R_1$, and the intersection $ QR$ and $ Q_1R_1$. Prove that the points $ O_1,O_2,O_3$ are collinear.
[i]Rudi Adha Prihandoko, Bandung[/i]
2012 ELMO Problems, 1
In acute triangle $ABC$, let $D,E,F$ denote the feet of the altitudes from $A,B,C$, respectively, and let $\omega$ be the circumcircle of $\triangle AEF$. Let $\omega_1$ and $\omega_2$ be the circles through $D$ tangent to $\omega$ at $E$ and $F$, respectively. Show that $\omega_1$ and $\omega_2$ meet at a point $P$ on $BC$ other than $D$.
[i]Ray Li.[/i]
2011 Balkan MO Shortlist, G1
Let $ABCD$ be a convex quadrangle such that $AB=AC=BD$ (vertices are labelled in circular order). The lines $AC$ and $BD$ meet at point $O$, the circles $ABC$ and $ADO$ meet again at point $P$, and the lines $AP$ and $BC$ meet at the point $Q$. Show that the angles $COQ$ and $DOQ$ are equal.
1983 AMC 12/AHSME, 27
A large sphere is on a horizontal field on a sunny day. At a certain time the shadow of the sphere reaches out a distance of $10$ m from the point where the sphere touches the ground. At the same instant a meter stick (held vertically with one end on the ground) casts a shadow of length $2$ m. What is the radius of the sphere in meters? (Assume the sun's rays are parallel and the meter stick is a line segment.)
$ \textbf{(A)}\ \frac{5}{2}\qquad\textbf{(B)}\ 9 - 4\sqrt{5}\qquad\textbf{(C)}\ 8\sqrt{10} - 23\qquad\textbf{(D)}\ 6 - \sqrt{15}\qquad\textbf{(E)}\ 10\sqrt{5} - 20 $
2006 Tuymaada Olympiad, 3
A line $d$ is given in the plane. Let $B\in d$ and $A$ another point, not on $d$, and such that $AB$ is not perpendicular on $d$. Let $\omega$ be a variable circle touching $d$ at $B$ and letting $A$ outside, and $X$ and $Y$ the points on $\omega$ such that $AX$ and $AY$ are tangent to the circle. Prove that the line $XY$ passes through a fixed point.
[i]Proposed by F. Bakharev [/i]
Russian TST 2018, P2
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 $\omega$.
2014 Sharygin Geometry Olympiad, 21
Let $ABCD$ be a circumscribed quadrilateral. Its incircle $\omega$ touches the sides $BC$ and $DA$ at points $E$ and $F$ respectively. It is known that lines $AB,FE$ and $CD$ concur. The circumcircles of triangles $AED$ and $BFC$ meet $\omega$ for the second time at points $E_1$ and $F_1$. Prove that $EF$ is parallel to $E_1 F_1$.
2012 Macedonia National Olympiad, 4
A fixed circle $k$ and collinear points $E,F$ and $G$ are given such that the points $E$ and $G$ lie outside the circle $k$ and $F$ lies inside the circle $k$. Prove that, if $ABCD$ is an arbitrary quadrilateral inscribed in the circle $k$ such that the points $E,F$ and $G$ lie on lines $AB,AD$ and $DC$ respectively, then the side $BC$ passes through a fixed point collinear with $E,F$ and $G$, independent of the quadrilateral $ABCD$.
2004 IMO Shortlist, 8
Given a cyclic quadrilateral $ABCD$, let $M$ be the midpoint of the side $CD$, and let $N$ be a point on the circumcircle of triangle $ABM$. Assume that the point $N$ is different from the point $M$ and satisfies $\frac{AN}{BN}=\frac{AM}{BM}$. Prove that the points $E$, $F$, $N$ are collinear, where $E=AC\cap BD$ and $F=BC\cap DA$.
[i]Proposed by Dusan Dukic, Serbia and Montenegro[/i]
2013 NIMO Problems, 12
In $\triangle ABC$, $AB = 40$, $BC = 60$, and $CA = 50$. The angle bisector of $\angle A$ intersects the circumcircle of $\triangle ABC$ at $A$ and $P$. Find $BP$.
[i]Proposed by Eugene Chen[/i]
2010 China Team Selection Test, 1
Let $\omega$ be a semicircle and $AB$ its diameter. $\omega_1$ and $\omega_2$ are two different circles, both tangent to $\omega$ and to $AB$, and $\omega_1$ is also tangent to $\omega_2$. Let $P,Q$ be the tangent points of $\omega_1$ and $\omega_2$ to $AB$ respectively, and $P$ is between $A$ and $Q$. Let $C$ be the tangent point of $\omega_1$ and $\omega$. Find $\tan\angle ACQ$.
Oliforum Contest IV 2013, 2
Given an acute angled triangle $ABC$ with $M$ being the mid-point of $AB$ and $P$ and $Q$ are the feet of heights from $A$ to $BC$ and $B$ to $AC$ respectively. Show that if the line $AC$ is tangent to the circumcircle of $BMP$ then the line $BC$ is tangent to the circumcircle of $AMQ$.
2008 AMC 10, 10
Points $ A$ and $ B$ are on a circle of radius $ 5$ and $ AB\equal{}6$. Point $ C$ is the midpoint of the minor arc $ AB$. What is the length of the line segment $ AC$?
$ \textbf{(A)}\ \sqrt{10} \qquad
\textbf{(B)}\ \frac{7}{2} \qquad
\textbf{(C)}\ \sqrt{14} \qquad
\textbf{(D)}\ \sqrt{15} \qquad
\textbf{(E)}\ 4$
2005 Harvard-MIT Mathematics Tournament, 10
Let $AB$ be a diameter of a semicircle $\Gamma$. Two circles, $\omega_1$ and $\omega_2$, externally tangent to each other and internally tangent to $\Gamma$, are tangent to the line $AB$ at $P$ and $Q$, respectively, and to semicircular arc $AB$ at $C$ and $D$, respectively, with $AP<AQ$. Suppose $F$ lies on $\Gamma$ such that $ \angle FQB = \angle CQA $ and that $ \angle ABF = 80^\circ $. Find $ \angle PDQ $ in degrees.
2006 Switzerland Team Selection Test, 2
Let $D$ be inside $\triangle ABC$ and $E$ on $AD$ different to $D$. Let $\omega_1$ and $\omega_2$ be the circumscribed circles of $\triangle BDE$ and $\triangle CDE$ respectively. $\omega_1$ and $\omega_2$ intersect $BC$ in the interior points $F$ and $G$ respectively. Let $X$ be the intersection between $DG$ and $AB$ and $Y$ the intersection between $DF$ and $AC$. Show that $XY$ is $\|$ to $BC$.
2010 Indonesia TST, 2
Circles $ \Gamma_1$ and $ \Gamma_2$ are internally tangent to circle $ \Gamma$ at $ P$ and $ Q$, respectively. Let $ P_1$ and $ Q_1$ are on $ \Gamma_1$ and $ \Gamma_2$ respectively such that $ P_1Q_1$ is the common tangent of $ P_1$ and $ Q_1$. Assume that $ \Gamma_1$ and $ \Gamma_2$ intersect at $ R$ and $ R_1$. Define $ O_1,O_2,O_3$ as the intersection of $ PQ$ and $ P_1Q_1$, the intersection of $ PR$ and $ P_1R_1$, and the intersection $ QR$ and $ Q_1R_1$. Prove that the points $ O_1,O_2,O_3$ are collinear.
[i]Rudi Adha Prihandoko, Bandung[/i]
2006 Romania Team Selection Test, 3
Let $\gamma$ be the incircle in the triangle $A_0A_1A_2$. For all $i\in\{0,1,2\}$ we make the following constructions (all indices are considered modulo 3): $\gamma_i$ is the circle tangent to $\gamma$ which passes through the points $A_{i+1}$ and $A_{i+2}$; $T_i$ is the point of tangency between $\gamma_i$ and $\gamma$; finally, the common tangent in $T_i$ of $\gamma_i$ and $\gamma$ intersects the line $A_{i+1}A_{i+2}$ in the point $P_i$. Prove that
a) the points $P_0$, $P_1$ and $P_2$ are collinear;
b) the lines $A_0T_0$, $A_1T_1$ and $A_2T_2$ are concurrent.
2012 ELMO Shortlist, 2
In triangle $ABC$, $P$ is a point on altitude $AD$. $Q,R$ are the feet of the perpendiculars from $P$ to $AB,AC$, and $QP,RP$ meet $BC$ at $S$ and $T$ respectively. the circumcircles of $BQS$ and $CRT$ meet $QR$ at $X,Y$.
a) Prove $SX,TY, AD$ are concurrent at a point $Z$.
b) Prove $Z$ is on $QR$ iff $Z=H$, where $H$ is the orthocenter of $ABC$.
[i]Ray Li.[/i]
1957 AMC 12/AHSME, 18
Circle $ O$ has diameters $ AB$ and $ CD$ perpendicular to each other. $ AM$ is any chord intersecting $ CD$ at $ P$. Then $ AP\cdot AM$ is equal to:
[asy]defaultpen(linewidth(.8pt));
unitsize(2cm);
pair O = origin;
pair A = (-1,0);
pair B = (1,0);
pair C = (0,1);
pair D = (0,-1);
pair M = dir(45);
pair P = intersectionpoint(O--C,A--M);
draw(Circle(O,1));
draw(A--B);
draw(C--D);
draw(A--M);
label("$A$",A,W);
label("$B$",B,E);
label("$C$",C,N);
label("$D$",D,S);
label("$M$",M,NE);
label("$O$",O,NE);
label("$P$",P,NW);[/asy]$ \textbf{(A)}\ AO\cdot OB \qquad \textbf{(B)}\ AO\cdot AB\qquad \textbf{(C)}\ CP\cdot CD \qquad \textbf{(D)}\ CP\cdot PD\qquad$
$ \textbf{(E)}\ CO\cdot OP$
2019 EGMO, 4
Let $ABC$ be a triangle with incentre $I$. The circle through $B$ tangent to $AI$ at $I$ meets side $AB$ again at $P$. The circle through $C$ tangent to $AI$ at $I$ meets side $AC$ again at $Q$. Prove that $PQ$ is tangent to the incircle of $ABC.$
2018 Morocco TST., 3
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 $\omega$.
2025 Bangladesh Mathematical Olympiad, P6
Suppose $X$ and $Y$ are the common points of two circles $\omega_1$ and $\omega_2$. The third circle $\omega$ is internally tangent to $\omega_1$ and $\omega_2$ in $P$ and $Q$, respectively. Segment $XY$ intersects $\omega$ in points $M$ and $N$. Rays $PM$ and $PN$ intersect $\omega_1$ in points $A$ and $D$; rays $QM$ and $QN$ intersect $\omega_2$ in points $B$ and $C$, respectively. Prove that $AB = CD$.
2010 ITAMO, 3
Let $ABCD$ be a convex quadrilateral. such that $\angle CAB = \angle CDA$ and $\angle BCA = \angle ACD$. If $M$ be the midpoint of $AB$, prove that $\angle BCM = \angle DBA$.
1996 Canadian Open Math Challenge, 7
Triangle $ABC$ is right angled at $A$. The circle with center $A$ and radius $AB$ cuts $BC$ and $AC$ internally at $D$ and $E$ respectively. If $BD = 20$ and $DC = 16$, determine $AC^2$.
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]