Found problems: 393
2009 Tuymaada Olympiad, 2
$ M$ is the midpoint of base $ BC$ in a trapezoid $ ABCD$. A point $ P$ is chosen on the base $ AD$. The line $ PM$ meets the line $ CD$ at a point $ Q$ such that $ C$ lies between $ Q$ and $ D$. The perpendicular to the bases drawn through $ P$ meets the line $ BQ$ at $ K$. Prove that $ \angle QBC \equal{} \angle KDA$.
[i]Proposed by S. Berlov[/i]
2008 IMO Shortlist, 3
Let $ ABCD$ be a convex quadrilateral and let $ P$ and $ Q$ be points in $ ABCD$ such that $ PQDA$ and $ QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $ E$ on the line segment $ PQ$ such that $ \angle PAE \equal{} \angle QDE$ and $ \angle PBE \equal{} \angle QCE$. Show that the quadrilateral $ ABCD$ is cyclic.
[i]Proposed by John Cuya, Peru[/i]
2007 China Team Selection Test, 1
Let $ ABC$ be a triangle. Circle $ \omega$ passes through points $ B$ and $ C.$ Circle $ \omega_{1}$ is tangent internally to $ \omega$ and also to sides $ AB$ and $ AC$ at $ T,\, P,$ and $ Q,$ respectively. Let $ M$ be midpoint of arc $ BC\, ($containing $ T)$ of $ \omega.$ Prove that lines $ PQ,\,BC,$ and $ MT$ are concurrent.
2007 IMO Shortlist, 3
The diagonals of a trapezoid $ ABCD$ intersect at point $ P$. Point $ Q$ lies between the parallel lines $ BC$ and $ AD$ such that $ \angle AQD \equal{} \angle CQB$, and line $ CD$ separates points $ P$ and $ Q$. Prove that $ \angle BQP \equal{} \angle DAQ$.
[i]Author: Vyacheslav Yasinskiy, Ukraine[/i]
2014 Tuymaada Olympiad, 2
The points $K$ and $L$ on the side $BC$ of a triangle $\triangle{ABC}$ are such that $\widehat{BAK}=\widehat{CAL}=90^\circ$. Prove that the midpoint of the altitude drawn from $A$, the midpoint of $KL$ and the circumcentre of $\triangle{ABC}$ are collinear.
[i](A. Akopyan, S. Boev, P. Kozhevnikov)[/i]
2000 All-Russian Olympiad, 7
Two circles are internally tangent at $N$. The chords $BA$ and $BC$ of the larger circle are tangent to the smaller circle at $K$ and $M$ respectively. $Q$ and $P$ are midpoint of arcs $AB$ and $BC$ respectively. Circumcircles of triangles $BQK$ and $BPM$ are intersect at $L$. Show that $BPLQ$ is a parallelogram.
2006 QEDMO 3rd, 6
The incircle of a triangle $ABC$ touches its sides $BC$, $CA$, $AB$ at the points $X$, $Y$, $Z$, respectively. Let $X^{\prime}$, $Y^{\prime}$, $Z^{\prime}$ be the reflections of these points $X$, $Y$, $Z$ in the external angle bisectors of the angles $CAB$, $ABC$, $BCA$, respectively. Show that $Y^{\prime}Z^{\prime}\parallel BC$, $Z^{\prime}X^{\prime}\parallel CA$ and $X^{\prime}Y^{\prime}\parallel AB$.
2011 IMO Shortlist, 5
Let $ABC$ be a triangle with incentre $I$ and circumcircle $\omega$. Let $D$ and $E$ be the second intersection points of $\omega$ with $AI$ and $BI$, respectively. The chord $DE$ meets $AC$ at a point $F$, and $BC$ at a point $G$. Let $P$ be the intersection point of the line through $F$ parallel to $AD$ and the line through $G$ parallel to $BE$. Suppose that the tangents to $\omega$ at $A$ and $B$ meet at a point $K$. Prove that the three lines $AE,BD$ and $KP$ are either parallel or concurrent.
[i]Proposed by Irena Majcen and Kris Stopar, Slovenia[/i]
1992 Romania Team Selection Test, 3
Let $ABCD$ be a tetrahedron; $B', C', D'$ be the midpoints of the edges $AB, AC, AD$; $G_A, G_B, G_C, G_D$ be the barycentres of the triangles $BCD, ACD, ABD, ABC$, and $G$ be the barycentre of the tetrahedron. Show that $A, G, G_B, G_C, G_D$ are all on a sphere if and only if $A, G, B', C', D'$ are also on a sphere.
[i]Dan Brânzei[/i]
1989 Polish MO Finals, 2
$k_1, k_2, k_3$ are three circles. $k_2$ and $k_3$ touch externally at $P$, $k_3$ and $k_1$ touch externally at $Q$, and $k_1$ and $k_2$ touch externally at $R$. The line $PQ$ meets $k_1$ again at $S$, the line $PR$ meets $k_1$ again at $T$. The line $RS$ meets $k_2$ again at $U$, and the line $QT$ meets $k_3$ again at $V$. Show that $P, U, V$ are collinear.
2000 National Olympiad First Round, 29
One of the external common tangent lines of the two externally tangent circles with center $O_1$ and $O_2$ touches the circles at $B$ and $C$, respectively. Let $A$ be the common point of the circles. The line $BA$ meets the circle with center $O_2$ at $A$ and $D$. If $|BA|=5$ and $|AD|=4$, then what is $|CD|$?
$ \textbf{(A)}\ \sqrt{20}
\qquad\textbf{(B)}\ \sqrt{27}
\qquad\textbf{(C)}\ 6
\qquad\textbf{(D)}\ \frac{15}2
\qquad\textbf{(E)}\ 4\sqrt5
$
2003 ITAMO, 3
Let a semicircle is given with diameter $AB$ and centre $O$ and let $C$ be a arbitrary point on the segment $OB$. Point $D$ on the semicircle is such that $CD$ is perpendicular to $AB$. A circle with centre $P$ is tangent to the arc $BD$ at $F$ and to the segment $CD$ and $AB$ at $E$ and $G$ respectively. Prove that the triangle $ADG$ is isosceles.
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]
2012 Morocco TST, 4
Let $ABC$ be an acute triangle with circumcircle $\Omega$. Let $B_0$ be the midpoint of $AC$ and let $C_0$ be the midpoint of $AB$. Let $D$ be the foot of the altitude from $A$ and let $G$ be the centroid of the triangle $ABC$. Let $\omega$ be a circle through $B_0$ and $C_0$ that is tangent to the circle $\Omega$ at a point $X\not= A$. Prove that the points $D,G$ and $X$ are collinear.
[i]Proposed by Ismail Isaev and Mikhail Isaev, Russia[/i]
2007 Vietnam Team Selection Test, 2
Let $ABC$ be an acute triangle with incricle $(I)$. $(K_{A})$ is the cricle such that $A\in (K_{A})$ and $AK_{A}\perp BC$ and it in-tangent for $(I)$ at $A_{1}$, similary we have $B_{1},C_{1}$.
a) Prove that $AA_{1},BB_{1},CC_{1}$ are concurrent, called point-concurrent is $P$.
b) Assume circles $(J_{A}),(J_{B}),(J_{C})$ are symmetry for excircles $(I_{A}),(I_{B}),(I_{C})$ across midpoints of $BC,CA,AB$ ,resp. Prove that $P_{P/(J_{A})}=P_{P/(J_{B})}=P_{P/(J_{C})}$.
Note. If $(O;R)$ is a circle and $M$ is a point then $P_{M/(O)}=OM^{2}-R^{2}$.
2005 Postal Coaching, 6
Let $ABCD$ be a trapezoid such that $AB$ is parallel to $CD$, and let $E$ be the midpoint of its side $BC$. Suppose we can inscribe a circle into the quadrilateral $ABED$, and that we can inscribe a circle into the quadrilateral $AECD$. Denote $|AB|=a$, $|BC|=b$, $|CD|=c$, $|DA|=d$. Prove that \[a+c=\frac{b}{3}+d;\] \[\frac{1}{a}+\frac{1}{c}=\frac{3}{b}\]
MIPT Undergraduate Contest 2019, 1.1 & 2.1
In $\mathbb{R}^3$, let there be a cube $Q$ and a sequence of other cubes, all of which are homothetic to $Q$ with coefficients of homothety that are each smaller than $1$. Prove that if this sequence of homothetic cubes completely fills $Q$, the sum of their coefficients of homothety is not less than $4$.
2006 China Team Selection Test, 1
$ABCD$ is a trapezoid with $AB || CD$. There are two circles $\omega_1$ and $\omega_2$ is the trapezoid such that $\omega_1$ is tangent to $DA$, $AB$, $BC$ and $\omega_2$ is tangent to $BC$, $CD$, $DA$. Let $l_1$ be a line passing through $A$ and tangent to $\omega_2$(other than $AD$), Let $l_2$ be a line passing through $C$ and tangent to $\omega_1$ (other than $CB$).
Prove that $l_1 || l_2$.
1998 IMO Shortlist, 5
Let $ABC$ be a triangle, $H$ its orthocenter, $O$ its circumcenter, and $R$ its circumradius. Let $D$ be the reflection of the point $A$ across the line $BC$, let $E$ be the reflection of the point $B$ across the line $CA$, and let $F$ be the reflection of the point $C$ across the line $AB$. Prove that the points $D$, $E$ and $F$ are collinear if and only if $OH=2R$.
2013 Taiwan TST Round 1, 6
Let $ABCD$ be a convex quadrilateral with non-parallel sides $BC$ and $AD$. Assume that there is a point $E$ on the side $BC$ such that the quadrilaterals $ABED$ and $AECD$ are circumscribed. Prove that there is a point $F$ on the side $AD$ such that the quadrilaterals $ABCF$ and $BCDF$ are circumscribed if and only if $AB$ is parallel to $CD$.
2009 Singapore Team Selection Test, 1
Two circles are tangent to each other internally at a point $\ T $. Let the chord $\ AB $ of the larger circle be tangent to the smaller circle at a point $\ P $. Prove that the line $\ TP $ bisects $\ \angle ATB $.
2013 Czech-Polish-Slovak Match, 1
Suppose $ABCD$ is a cyclic quadrilateral with $BC = CD$. Let $\omega$ be the circle with center $C$ tangential to the side $BD$. Let $I$ be the centre of the incircle of triangle $ABD$. Prove that the straight line passing through $I$, which is parallel to $AB$, touches the circle $\omega$.
2008 Hong Kong TST, 4
Two circles $ C_1,C_2$ with different radii are given in the plane, they touch each other externally at $ T$. Consider any points $ A\in C_1$ and $ B\in C_2$, both different from $ T$, such that $ \angle ATB \equal{} 90^{\circ}$.
(a) Show that all such lines $ AB$ are concurrent.
(b) Find the locus of midpoints of all such segments $ AB$.
2005 Taiwan TST Round 2, 2
Let $O$ be the circumcenter of an acute-angled triangle $ABC$ with ${\angle B<\angle C}$. The line $AO$ meets the side $BC$ at $D$. The circumcenters of the triangles $ABD$ and $ACD$ are $E$ and $F$, respectively. Extend the sides $BA$ and $CA$ beyond $A$, and choose on the respective extensions points $G$ and $H$ such that ${AG=AC}$ and ${AH=AB}$. Prove that the quadrilateral $EFGH$ is a rectangle if and only if ${\angle ACB-\angle ABC=60^{\circ }}$.
[i]Proposed by Hojoo Lee, Korea[/i]
2010 Contests, 2
Bisectors $AA_1$ and $BB_1$ of a right triangle $ABC \ (\angle C=90^\circ )$ meet at a point $I.$ Let $O$ be the circumcenter of triangle $CA_1B_1.$ Prove that $OI \perp AB.$