Found problems: 670
2018 Junior Balkan Team Selection Tests - Romania, 3
Let $ABCD$ be a cyclic quadrilateral. The line parallel to $BD$ passing through $A$ meets the line parallel to $AC$ passing through $B$ at $E$. The circumcircle of triangle $ABE$ meets the lines $EC$ and $ED$, again, at $F$ and $G$, respectively. Prove that the lines $AB, CD$ and $FG$ are either parallel or concurrent.
2023 Bangladesh Mathematical Olympiad, P6
Let $\triangle ABC$ be an acute angle triangle and $\omega$ be its circumcircle. Let $N$ be a point on arc $AC$ not containing $B$ and $S$ be a point on line $AB$. The line tangent to $\omega$ at $N$ intersects $BC$ at $T$, $NS$ intersects $\omega$ at $K$. Assume that $\angle NTC = \angle KSB$. Prove that $CK\parallel AN \parallel TS$.
2006 Romania Team Selection Test, 2
Let $A$ be point in the exterior of the circle $\mathcal C$. Two lines passing through $A$ intersect the circle $\mathcal C$ in points $B$ and $C$ (with $B$ between $A$ and $C$) respectively in $D$ and $E$ (with $D$ between $A$ and $E$). The parallel from $D$ to $BC$ intersects the second time the circle $\mathcal C$ in $F$. Let $G$ be the second point of intersection between the circle $\mathcal C$ and the line $AF$ and $M$ the point in which the lines $AB$ and $EG$ intersect. Prove that
\[ \frac 1{AM} = \frac 1{AB} + \frac 1{AC}. \]
2005 Taiwan TST Round 1, 2
$P$ is a point in the interior of $\triangle ABC$, and $\angle ABP = \angle PCB = 10^\circ$.
(a) If $\angle PBC = 10^\circ$ and $\angle ACP = 20^\circ$, what is the value of $\angle BAP$?
(b) If $\angle PBC = 20^\circ$ and $\angle ACP = 10^\circ$, what is the value of $\angle BAP$?
2007 Harvard-MIT Mathematics Tournament, 26
$ABCD$ is a cyclic quadrilateral in which $AB=4$, $BC=3$, $CD=2$, and $AD=5$. Diagonals $AC$ and $BD$ intersect at $X$. A circle $\omega$ passes through $A$ and is tangent to $BD$ at $X$. $\omega$ intersects $AB$ and $AD$ at $Y$ and $Z$ respectively. Compute $YZ/BD$.
2007 Serbia National Math Olympiad, 2
In a scalene triangle $ABC , AD, BE , CF$ are the angle bisectors $(D \in BC , E \in AC , F \in AB)$. Points $K_{a}, K_{b}, K_{c}$ on the incircle of triangle $ABC$ are such that $DK_{a}, EK_{b}, FK_{c}$ are tangent to the incircle and $K_{a}\not\in BC , K_{b}\not\in AC , K_{c}\not\in AB$. Let $A_{1}, B_{1}, C_{1}$ be the midpoints of sides $BC , CA, AB$ , respectively. Prove that the lines $A_{1}K_{a}, B_{1}K_{b}, C_{1}K_{c}$ intersect on the incircle of triangle $ABC$.
IV Soros Olympiad 1997 - 98 (Russia), 10.1
Two sides of the cyclic quadrilateral $ABCD$ are known: $AB = a$, $BC = b$. A point $K$ is taken on the side $CD$ so that $CK = m$. A circle passing through $B$, $K$ and $D$ intersects line $DA$ at a point $M$, different from $D$. Find $AM$.
2007 All-Russian Olympiad, 3
$BB_{1}$ is a bisector of an acute triangle $ABC$. A perpendicular from $B_{1}$ to $BC$ meets a smaller arc $BC$ of a circumcircle of $ABC$ in a point $K$. A perpendicular from $B$ to $AK$ meets $AC$ in a point $L$. $BB_{1}$ meets arc $AC$ in $T$. Prove that $K$, $L$, $T$ are collinear.
[i]V. Astakhov[/i]
2005 Postal Coaching, 16
The diagonals AC and BD of a cyclic ABCD intersect at E. Let O be circumcentre of ABCD. If midpoints of AB, CD, OE are collinear prove that AD=BC.
Bomb
[color=red][Moderator edit: The problem is wrong. See also http://www.mathlinks.ro/Forum/viewtopic.php?t=53090 .][/color]
2010 Ukraine Team Selection Test, 2
Let $ABCD$ be a quadrilateral inscribled in a circle with the center $O, P$ be the point of intersection of the diagonals $AC$ and $BD$, $BC\nparallel AD$. Rays $AB$ and $DC$ intersect at the point $E$. The circle with center $I$ inscribed in the triangle $EBC$ touches $BC$ at point $T_1$. The $E$-excircle with center $J$ in the triangle $EAD$ touches the side $AD$ at the point T$_2$. Line $IT_1$ and $JT_2$ intersect at $Q$. Prove that the points $O, P$, and $Q$ lie on a straight line.
2021 Yasinsky Geometry Olympiad, 6
Given a quadrilateral $ABCD$, around which you can circumscribe a circle. The perpendicular bisectors of sides $AD$ and $CD$ intersect at point $Q$ and intersect sides $BC$ and $AB$ at points $P$ and $K$ resepctively. It turned out that the points $K, B, P, Q$ lie on the same circle. Prove that the points $A, Q, C$ lie on one line.
(Olena Artemchuk)
2003 Bundeswettbewerb Mathematik, 3
Consider a cyclic quadrilateral $ABCD$, and let $S$ be the intersection of $AC$ and $BD$.
Let $E$ and $F$ the orthogonal projections of $S$ on $AB$ and $CD$ respectively.
Prove that the perpendicular bisector of segment $EF$ meets the segments $AD$ and $BC$ at their midpoints.
2011 Saudi Arabia Pre-TST, 3.2
Prove that for each $n \ge 4$ a parallelogram can be dissected in $n$ cyclic quadrilaterals.
1986 IMO Longlists, 69
Let $AX,BY,CZ$ be three cevians concurrent at an interior point $D$ of a triangle $ABC$. Prove that if two of the quadrangles $DY AZ,DZBX,DXCY$ are circumscribable, so is the third.
2008 Indonesia TST, 3
Let $\Gamma_1$ and $\Gamma_2$ be two circles that tangents each other at point $N$, with $\Gamma_2$ located inside $\Gamma_1$. Let $A, B, C$ be distinct points on $\Gamma_1$ such that $AB$ and $AC$ tangents $\Gamma_2$ at $D$ and $E$, respectively. Line $ND$ cuts $\Gamma_1$ again at $K$, and line $CK$ intersects line $DE$ at $I$.
(i) Prove that $CK$ is the angle bisector of $\angle ACB$.
(ii) Prove that $IECN$ and $IBDN$ are cyclic quadrilaterals.
1995 APMO, 4
Let $C$ be a circle with radius $R$ and centre $O$, and $S$ a fixed point in the interior of $C$. Let $AA'$ and $BB'$ be perpendicular chords through $S$. Consider the rectangles $SAMB$, $SBN'A'$, $SA'M'B'$, and $SB'NA$. Find the set of all points $M$, $N'$, $M'$, and $N$ when $A$ moves around the whole circle.
2009 Jozsef Wildt International Math Competition, W. 25
Let $ABCD$ be a quadrilateral in which $\widehat{A}=\widehat{C}=90^{\circ}$. Prove that $$\frac{1}{BD}(AB+BC+CD+DA)+BD^2\left (\frac{1}{AB\cdot AD}+\frac{1}{CB\cdot CD}\right )\geq 2\left (2+\sqrt{2}\right )$$
1994 Turkey MO (2nd round), 2
Let $ABCD$ be a cyclic quadrilateral $\angle{BAD}< 90^\circ$ and $\angle BCA = \angle DCA$. Point $E$ is taken on segment $DA$ such that $BD=2DE$. The line through $E$ parallel to $CD$ intersects the diagonal $AC$ at $F$. Prove that \[ \frac{AC\cdot BD}{AB\cdot FC}=2.\]
2014 Germany Team Selection Test, 2
Let $ABCD$ be a convex cyclic quadrilateral with $AD=BD$. The diagonals $AC$ and $BD$ intersect in $E$. Let the incenter of triangle $\triangle BCE$ be $I$. The circumcircle of triangle $\triangle BIE$ intersects side $AE$ in $N$.
Prove
\[ AN \cdot NC = CD \cdot BN. \]
2022 Bulgaria JBMO TST, 2
Let $ABC$ ($AB < AC$) be a triangle with circumcircle $k$. The tangent to $k$ at $A$ intersects the line $BC$ at $D$ and the point $E\neq A$ on $k$ is such that $DE$ is tangent to $k$. The point $X$ on line $BE$ is such that $B$ is between $E$ and $X$ and $DX = DA$ and the point $Y$ on the line $CX$ is such that $Y$ is between $C$ and $X$ and $DY = DA$. Prove that the lines $BC$ and $YE$ are perpendicular.
1979 Czech And Slovak Olympiad IIIA, 3
If in a quadrilateral $ABCD$ whose vertices lie on a circle of radius $1$, holds $$|AB| \cdot |BC| \cdot |CD|\cdot |DA| \ge 4$$, then $ABCD$ is a square. Prove it.
[hide=Hint given in contest] You can use Ptolemy's formula $|AB| \cdot |CD| + |BC|\cdot |AD|= |AC| \cdot|BD|$[/hide]
2021 Cono Sur Olympiad, 2
Let $ABC$ be a triangle and $I$ its incenter. The lines $BI$ and $CI$ intersect the circumcircle of $ABC$ again at $M$ and $N$, respectively. Let $C_1$ and $C_2$ be the circumferences of diameters $NI$ and $MI$, respectively. The circle $C_1$ intersects $AB$ at $P$ and $Q$, and the circle $C_2$ intersects $AC$ at $R$ and $S$. Show that $P$, $Q$, $R$ and $S$ are concyclic.
Geometry Mathley 2011-12, 9.3
Let $ABCD$ be a quadrilateral inscribed in a circle $(O)$. Let $(O_1), (O_2), (O_3), (O_4)$ be the circles going through $(A,B), (B,C),(C,D),(D,A)$. Let $X, Y,Z, T$ be the second intersection of the pairs of the circles: $(O_1)$ and $(O_2), (O_2)$ and $(O_3), (O_3)$ and $(O_4), (O_4)$ and $(O_1)$.
(a) Prove that $X, Y,Z, T$ are on the same circle of radius $I$.
(b) Prove that the midpoints of the line segments $O_1O_3,O_2O_4,OI$ are collinear.
Nguyễn Văn Linh
Ukraine Correspondence MO - geometry, 2020.11
The diagonals of the cyclic quadrilateral $ABCD$ intersect at the point $E$. Let $P$ and $Q$ are the centers of the circles circumscribed around the triangles $BCE$ and $DCE$, respectively. A straight line passing through the point $P$ parallel to $AB$, and a straight line passing through the point $Q$ parallel to $AD$, intersect at the point $R$. Prove that the point $R$ lies on segment $AC$.
2008 Indonesia TST, 1
Let $ABCD$ be a cyclic quadrilateral, and angle bisectors of $\angle BAD$ and $\angle BCD$ meet at point $I$. Show that if $\angle BIC = \angle IDC$, then $I$ is the incenter of triangle $ABD$.