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239 Open MO 1994 - 2021 (-97,-98,-07,-18,-20) 84p

geometry problems from  the Open Math Olympiad of 239 Presidential Physics and Mathematics Lyceum  in Saint-Petersburg, with aops links

Открытая олимпиада ФМЛ №239.

239 MO aops contest collection is here
junior (grades 8-9) collected inside aops here
senior (grades 10-11) collected inside aops here


1994-95, 1997,  1999- 2006, 2008-17, 2019, 2021 so far

 Find a point in a triangle, the sum of the distances from which to the vertices and the midpoints of the sides would be the smallest.

In an equilateral triangle $ABC$ with center $O$ on side $BC$ is taken point $M$. $MK$ and $ML$ are perpendiculars to $AB$ and $AC$, respectively. Prove that the line $OM$ bisects $KL$.

In triangle $ABC$, angle $B$ is $30^\circ$, $N$ is a point, symmetric to the orthocenter with respect to the midpoint of side $AC$. Prove that $BN=2AC$.

In an acute triangle $ABC$ ,  $AA_1$, $BB_1$ and $CC_1$ are aktitudes, and $H$, $H_1$, $H_2$, $H_3$ are the orthocenters of triangles $ABC$, $AB_1C_1$, $BC_1A_1$, $CA_1B_1$ respectively. Prove that if $H$ is the incircle center of triangle $H_1H_2H_3$, then $ABC$ is equilateral triangle.


In a convex quadrilateral $ ABCD$ ,  $\angle A + \angle D = 120 ^\circ $ and $ AB = BC = CD $. Prove that the intersection point of the diagonals is equidistant from the vertices $ A $ and $ D $. 

The diagonals of a convex quadrangle $ ABCD $ intersect at point $ O $. , $ O_1 $, $ O_2 $, $ O_3 $ and $ O_4 $ are centers of circumscribed circles of triangles $AOB $, $BOC $, $COD $ and $DOA $ respectively. Prove that $ 2S_{O_1O_2O_3O_4} \ge S_{ABCD} $.

An isosceles right triangle has a leg of length $1$. It is allowed to replace one vertex of the triangle any other vertex symmetric wrt to it . As a result after several such operations, a triangle was obtained with sides $a, b$ and $c$, where $c$ is the longest side. Prove that $a+b-c\leq 2-\sqrt{2}$.

Two pyramids have a common base, which is convex polygon, with one lying inside other. Prove that the sum of the flat angles at the vertex the outer pyramid is smaller than at the top of the inner. 

A point $O$ is chosen inside a convex quadrilateral $ABCD$ not lying on the diagonal $BD$ such that $\angle ODC=\angle CAB$ , $\angle OBC=\angle CAD$. Prove that $\angle ACB=\angle OCD$

In the trapezoid $ABCD$ on the sides $AB$ and $CD,$ you can choose points $K$ and $L$ so that the segment $KL$ is not parallel to the bases and is divided by diagonals into three equal parts. Find the ratio of the bases of the trapezoid.

The altitudes $AA_1$, $CC_1$ of triangle $ABC$ intersect at point $H$, and the circumscribed circles of triangles $ABC$ and $A_1BC_1$ intersect at a point $M$ other than $B$. Prove that line $MH$ divides side $AC$ in half.

A convex hexagon is symmetrical about the point $O$. The point $O $ was reflected wrt the midpoints of the small diagonals of the hexagon, and none of the images fell on the side of the hexagon. How many images could get inside?



In the triangle $ABC$, $K \in AB$, $N \in BC$, $M$ is the midpoint of $AC$. It is known that $\angle BKM = \angle BNM$. Prove that the perpendiculars on the sides of the original triangle at points $K, N, M$ intersect at one point.
(S. Berlov)
The bisectors of an inscribed quadrilateral form in crossing a convex quadrilateral. Prove that the diagonals of the resulting quadrilateral are perpendicular.
(S. Berlov)
A convex $2n$-gon is given on a unit lattice. Prove, that its area is not less than
a) $n(n - 1)/2$.
b) $n^3/100$

(a for juniors, b for seniors )
(S. Ivanov) 
Point $I$ is the center of the inscribed circle of triangle $ABC$. A circle with center at $I$ intersects side $BC$ at points $A_1$ and $A_2$, side $C A$ at points $B_1$ and $B_2$, side AB at points $C_1$ and $C_2$. Received points are located on the circle in the order $A_1$, $A_2$, $B_1$, $B_2$, $C_1$, $C_2$. Points $A_3$, $B_3$, $C_3$ are the midpoints of arcs $A_1A_2$, $B_1B_2$, $C_1C_2$ respectively. Lines $A_2A3$ and $B_1B_3$ intersect at point $C_4$, lines $B_2B_3$ and $C_1C_3$ intersect at point $A_4$, and lines $C_2C_3$ and $A_1A_3$ intersect at the point $B_4$. Prove that the segments $A_3A_4$, $B_3B_4$, $C_3C_4$ intersect at one point.
(S. Berlov)


1999 239 MO VIII-IX p2
In the inscribed quadrangle $ ABCD $, let $O$ be the intersection point of the diagonals. The point $ O '$ is symmetric $ O $ with respect to $ AD $ and lies on the circumscribed circle. Prove that $ O'O $ is the bisector of the angle $ BO'C $.

Let $ ABCD $ be an isosceles trapezoid ($ BC \parallel AD $), $ E $ be point of the arc $ AD $ of the circumscribed circle. From points $ A $ and $ D $ draw the perpendiculars on lines $ BE $ and $ CE $. Prove that the feet of the perpendiculars lie on the same circle.

1999 239 MO X-XI p1
In the triangle $ ABC $ on the bisector $ BB_1 $, the point $ O $ is chosen so that that $ \angle OCA = \angle BAC + \angle ABC $. $ AO $ and $ CO $ cross the sides of $ BC $ and $ AB $ at points $ A_1 $ and $ C_1 $ respectively. Prove that the angle $\angle A_1B_1C_1$ is right.

2000 239 MO VIII-IX p3
Let $ AA_1 $ and $ CC_1 $ be  the heights of the acute-angled triangle $ ABC $. A line passing through the centers of the inscribed circles the triangles $ AA_1C $ and $ CC_1A $ intersect the sides of $ AB $ and $ BC $ triangle $ ABC $ at points $ X $ and   $ Y $. Prove that $ BX = BY $.

Let ABCD be a convex quadrilateral, and let M and N be the midpoints of its sides AD and BC, respectively. Assume that the points A, B, M, N are concyclic, and the circumcircle of triangle BMC touches the line AB. Show that the circumcircle of triangle AND touches the line AB, too.
S. Berlov
The perpendicular bisectors of the sides AB and BC of a triangle ABC meet the lines BC and AB at the points X and Z, respectively. The angle bisectors of the angles XAC and ZCA intersect at a point B'. Similarly, define two points C' and A'. Prove that the points A', B', C' lie on one line through the incenter I of triangle ABC.
S. Berlov, upgrade of 7 for 8-9 form 
2001 239 MO VIII-IX p2
In a convex quadrangle $ ABCD $, the rays $ DA $ and $ CB $ intersect at point  $ Q $, and the rays $ BA $ and $ CD $  at the point  $ P $. It turned out that $ \angle AQB = \angle APD $. The bisectors of the angles $ \angle AQB $ and $ \angle APD $ intersect the sides quadrangle at points $ X $, $ Y $ and $ Z $, $ T $ respectively. Circumscribed circles of triangles $ ZQT $ and $ XPY $ intersect at  $ K $ inside quadrangle. Prove that $ K $ lies on the diagonal  $ AC $.

2001 239 MO VIII-IX p5
The circles $ S_1 $ and $ S_2 $ intersect at points $ A $ and $ B $. Circle $ S_3 $ externally touches $ S_1 $ and $ S_2 $ at points $ C $ and $ D $ respectively. Let $ K $  be the midpoint of the chord cut by the line $ AB $ on circles $ S_3 $. Prove that $ \angle CKA = \angle DKA $.

2001 239 MO X-XI p3
The circles $ S_1 $ and $ S_2 $ intersect at points $ A $ and $ B $. Circle $ S_3 $ externally touches $ S_1 $ and $ S_2 $ at points $ C $ and $ D $ respectively. Let $ PQ $  be a chord cut by the line $ AB $ on circle $ S_3 $, and $ K $  be the midpoint of  $ CD $. Prove that $ \angle PKC = \angle QKC $.

The quadrangle $ ABCD $ contains two circles of radii $ R_1 $ and  $ R_2 $ tangent externally. The first circle touches the sides of $ DA $,$ AB $ and $ BC $, moreover, the sides of $ AB $ at the point  $ E $. The second circle touches sides $ BC $, $ CD $ and $ DA $, and sides $ CD $ at  $ F $. Diagonals of the quadrangle intersect at  $ O $. Prove that $ OE + OF \leq 2 (R_1 + R_2) $.

F.  Bakharev, S.  Berlov
2002 239 MO VIII-IX p2
A circle concentric with the inscribed circle of a triangle $ ABC $, intersects the sides of the triangle at six points forming a convex hexagon $ A_1A_2B_1B_2C_1C_2 $, where $ A_i $ lie on the side of $ BC $, etc. Prove that if the line $ A_1B_1 $ is parallel to the bisector of the angle $ \angle B $, then the line $ A_2C_2 $ is parallel to the bisector of the angle $ \angle C $.

2002 239 MO VIII-IX p5
On the sides $ BC $, $ AD $ and $ AB $ of the rhombus $ ABCD $, the points $ P $, $ Q $ and $ R $ are selected respectively, so that $ DP = DQ $ and $ \angle BRD = \angle PDR $. Prove that the lines $ DR, PQ $ and $ AC $ pass through one point.

2002 239 MO X-XI p2
Two triangles have a common inscribed and a common circumscribed circle. The sides of one of them touch the inscribed circle at the points $ K $, $ L $ and $ M $, the sides of the other at points $ K_1 $, $ L_1 $ and $ M_1 $. Prove that the triangles $ KLM $ and $ K_1L_1M_1 $ have a common orthocenter.

On the arc $ AC $ of the circumscribed circle of triangle $ ABC $, the point $ P $ is chosen. The lines $ AP $ and $ CP $ intersect the extensions of the sides $ AB $ and $ BC $ at points  $ C_1 $ and $ A_1 $ respectively, and the line $ BP $ intersects the side of $ AC $ at point  $ B_1 $. The lines $ C_1B_1 $ and $ A_1B_1 $ intersect the sides $ BC $ and $ AB $ at points $ X $ and $ Y $ respectively. Prove that the line $ XY $ passes through the Lemoine point of the triangle.

2003 239 MO VIII-IX p3
Let $ABC$ be a triangle with $ AL $  bisector and  $ M $  midpoint of segment $ CL $. Point $ K $ lies on the side $ AB $  such that bisector of $ AL $ bisects the segment $ MK $  at the point $ O $. Prove that the angle $ \angle AOC $ is obtuse.

2003 239 MO VIII-IX p5
From the point  $ A $ to the given circle  $ S $, draw tangents $ AB $ and $ AC $. On the midline of the triangle $ ABC $, parallel to side $ BC $, arbitrary points $ X $ and  $ Y $ are selected. Segments of tangents from points $ X $ and  $ Y $ to $ S $ intersect at the point $Z $. Prove that the quadrangle $ AXZY $ is tangential.

2003 239 MO X-XI p1
Given a convex quadrangle $ ABCD $. Prove that the $9$-point circles of the triangles $ ABC $, $ ABD $, $ ACD $ and $ BCD $ have a common point.

2003 239 MO X-XI p4
Let  $ ABC $ be a triangle. Circle  $ \omega_1 $ with  center on the segment $ AB $ passes through $ A $ and crosses again the segments $ AB $ and $ AC $ at  points $ A_1 $ and $ A_2 $ respectively. Circle  $ \omega_2 $ with  center on the segment $ BC $ passes through $ C $ and crosses the segments $ BC $ and $ AC $ again at points $ C_1 $ and $ C_2 $ respectively. It is known that circles are tangent at  point  $ K $. Prove that each of the lines $ A_1K $, $ A_2K $,$ C_1K $ and $ C_2K $ pass through a fixed point independent of circles.
 L.  Emelyanov
2004 239 MO VIII-IX p2 (also here)
The incircle of a triangle $ABC$ has centre $I$ and touches sides $AB, BC, CA$ in points $C_1, A_1, B_1$ respectively. Denote by $L$ the foot of a bissector of angle $B$, and by $K$ the point of intersecting of lines  $B_1I$ and $A_1C_1$. Prove that $KL\parallel BB_1$.
 L. Emelyanov, S. Berlov
2004 239 MO VIII-IX p7
Given an isosceles triangle $ABC\ (AB=BC)$. A point $X$ is chosen on a side $AC$. The circle passes through $X$, touches the side $AC$ and intersects the circumcircle of triangle $ABC$ in points $M$ and $N$ such that the segment $MN$ bisects $BX$ and intersects sides $AB$ and $BC$ in points $P$ and $Q$. Prove that the circumcircle of triangle $PBQ$ passes through the circumcentre of triangle $ABC$.
S. Berlov
The incircle of triangle $ABC$ touches its sides $AB, BC, CA$ in points $C_1, A_1, B_1$ respectively. The point $B_2$ is symmetric to $B_1$ with respect to line $A_1C_1$, lines $BB_2$ and $AC$ meet in point $B_3$. points $A_3$ and $C_3$ may be defined analogously. Prove that points $A_3, B_3$ and $C_3$ lie on a line, which passes through the circumcentre of a triangle $ABC$.
 L. Emelyanov
2004 239 MO X-XI p8 (also here)
Given a triangle $ABC$. A point $X$ is chosen on a side $AC$. Some circle passes through $X$, touches the side $AC$ and intersects the circumcircle of triangle $ABC$ in points $M$ and $N$ such that the segment $MN$ bisects $BX$ and intersects sides $AB$ and $BC$ in points $P$ and $Q$. Prove that the circumcircle of triangle $PBQ$ passes through a fixed point different from $B$.

Sergej Berlov
The diagonals of the inscribed quadrilateral $ABCD$ intersect at point $P$. The centers of the circumscribed circles of triangles $APB$ and $CPD$ lie on the circumscribed circle of $ABCD$. Prove that $AC + BD = 2(BC + AD)$.

Inside the inscribed $n$-gon $A_1A_2...A_n$, was found a point $P$ such that $\angle PA_1A_2 = \angle P A_2A_3 = ... = \angle PA_nA_1$. Prove that inside this polygon there is a point $Q$ such that $\angle QA_2A_1 = \angle QA_3A_2 = ... = \angle QA_1A_n$.

In the convex quadrilateral  is chosen  point $M$.The rays $AB$ and $DC$ intersecting at the point $K$,the rays $BC$ and $AD$ at point $L$.It is known that $\angle{AMB}=70^{0}$ 
$\angle{BMK}=40^{0}$,$\angle{KMC}=\angle{CMD}=60^{0}$.Find $\angle{LMD}$.

Let $K$ is the point of intersection the diagonals of cyclic quadrilateral $ABCD$. In the triangle $AKD$ exists the point $P$, such that $\angle APC=\angle ADC+90^\circ$ and $\angle BPD=\angle BAD+90^\circ$.  Prove that the diagonals of the convex quadrilateral, formed the foots of perpendiculars from $P$ on sides of $ABCD$, are perpendicular.

2006 239 MO VIII-IX p3
The point $ P $ lies inside the acute-angled triangle $ ABC $. Prove that the bases of the perpendiculars from $ P $ to the sides of $ AB $ and $ AC $ equidistant from the midpoint of the side of $ BC $ if and only if when the points are symmetric $ P $ with respect to $ BC $ and the bisector of the angle $ A $, lie on the same line with the point $ A $. 
R. Sakhipov
Let $ ABCD $  be convex quadrangle in which $ \angle DBC + \angle ADC = 90 ^\circ $, and $ \angle ACB + 2 \angle ACD = 180 ^\circ $. Let the circumscribed circle of the triangle $ ABC $ intersect the segment $ BD $ at $ T $. Prove that $ TA = TD $. 

F. Bakharev
Given an inscribed quadrangle $ ABCD $, let $ P $  be the intersection point of its diagonals, $ M $ be the midpoint  of the arc $ AD $. It turned out that lines $ AB, CD, PM $ intersect at one point. Prove that $ AD \perp PM $.
R. Sakhipov
Given a circle $ S $ and a point $ P $ outside it. A chord $ AB $ of constant length slides around the circumference , the middle of which is $ M $. Via point $ B $ draw a line parallel to $ PM $ that crosses circle at points $ B $ and $ C $. Prove that all lines $ AC $ pass through one point, independent of the position of the chord. 
S. Berlov
2007 missing

2008 239 MO VIII-IX p4
Point $P$ is located inside an acute-angled triangle $ABC$. $A_1$, $B_1$, $C_1$ are points symmetric to $P$ with respect to the sides of triangle $ABC$. It turned out that the hexagon $AB_1CA_1BC_1$ is inscribed. Prove that $P$ is the Torricelli point of triangle $ABC$.

$AB$ is the chord of the circle $S$. Circles $S_1$ and $S_2$ touch the circle $S$ at points $P$ and $Q$, respectively, and the segment $AB$ at point $K$. It turned out that $\angle{PBA}=\angle{QBA}$. Prove that $AB$ is the diameter of the circle $S$.

A circumscribed quadrilateral $ABCD$ is given. $E$ and $F$ are the intersection points of opposite sides of the $ABCD$. It turned out that the radii of the inscribed circles of the triangles $AEF$ and $CEF$ are equal. Prove that $AC \bot BD$.

In the triangle $ABC$, $\angle{B} = 120^{\circ}$, point $M$ is the midpoint of side $AC$. On the sides $AB$ and $BC$, the points $K$ and $L$ are chosen such that $KL \parallel AC$. Prove that $MK + ML \ge MA$.

On the sides $AB, BC$ and $CA$ of triangle $ABC$, points $K, L$ and $M$ are selected, respectively, such that $AK = AM$ and $BK = BL$. If $\angle{MLB} = \angle{CAB}$, Prove that $ML = KI$, where $I$ is the incenter of triangle $CML$.

In the triangle $ABC$, the cevians $AA_1$, $BB_1$ and $CC_1$ intersect at the point $O$. It turned out that $AA_1$ is the bisector, and the point $O$ is closer to the straight line $AB$ than to the straight lines $A_1C_1$ and $B_1A_1$. Prove that $\angle{BAC} > 120^{\circ}$

The Feuerbach point (the tangent point of the inscribed circle and the nine-point circle of triangle $ABC$) $F$ is marked in triangle $ABC$. $A_1$ is on the side $BC$ such that $AA_1$ is the altitude of triangle $ABC$. Prove that the line symmetric to $FA_1$ with respect to $BC$ is perpendicular to $IO$, where $O$ is the center of the circumcircle of the triangle $ABC$ and $I$ is the center of its incircle.

The incircle of the triangle $ABC$ touches the sides $AC$ and $BC$ at points $K$ and $L$, respectively. the $B$-excircle touches the side $AC$ of this triangle at point $P$. Line $KL$ intersects with the line passing through $A$ and parallel to $BC$ at point $M$. Prove that $PL = PM$.

In a convex quadrilateral $ABCD$, We have $\angle{B} = \angle{D} = 120^{\circ}$. Points $A'$, $B'$ and $C'$ are symmetric to $D$ relative to $BC$, $CA$ and $AB$, respectively. Prove that lines $AA'$, $BB'$ and $CC'$ are concurrent.

2010 239 MO X-XI p2
The incircle of the triangle $ABC$ touches the sides $AC$ and $BC$ at points $K$ and $L$, respectively. the $B$-excircle touches the side $AC$ of this triangle at point $P$. The segment $AL$ intersects the inscribed circle for the second time at point $S$. Line $KL$ intersects the circumscribed circle of triangle $ASK$ for the second at point $M$. Prove that $PL = PM$.

Consider three pairwise intersecting circles $\omega_1$, $\omega_2$ and $\omega_3$. Let their three common chords intersect at point $R$. We denote by $O_1$ the center of the circumcircle of a triangle formed by some triple common points of $\omega_1$ & $\omega_2$, $\omega_2$ & $\omega_3$ and $\omega_3$ & $\omega_1$. and we denote by $O_2$ the center of the circumcircle of the triangle formed by the second intersection points of the same pairs of circles. Prove that points $R$, $O_1$ and $O_2$ are collinear.

In the acute triangle $ABC$ on $AC$ point $P$ is chosen such that $2AP=BC$. Points $X$ and $Y$ are symmetric to $P$ wrt $A$ and $C$ respectively. It turned out that $BX=BY$. Find angle $C$.

Rhombus ABCD with acute angle $B$ is given. $O$ is a circumcenter of $ABC$. Point $P$ lies on line $OC$ beyond $C$. $PD$ intersect the line that goes through $O$ and parallel to $AB$ at $Q$. Prove that $\angle AQO=\angle PBC$

In acute quadrilateral $ABCD$, where $AB=AD$, on $BD$ point $K$ is chosen. On $KC$ point $L$ is such that $\bigtriangleup BAD \sim \bigtriangleup BKL$. Line parallel to $DL$ and passes through $K$, intersect $CD$ at $P$. Prove that $\angle APK = \angle LBC$.

On the hypotenuse $AB$ of the right-angled triangle $ABC$, a point $K$ is chosen such that $BK = BC$. Let $P$ be a point on the perpendicular line from point $K$ to the line $CK$, equidistant from the points $K$ and $B$. Also let $L$ denote the midpoint of the segment $CK$. Prove that line $AP$ is tangent to the circumcircle of the triangle $BLP$.

2012 239 MO VIII-IX p7
A circumscribed quadrilateral $ABCD$ is given. It is known that $\angle{ACB} = \angle{ACD}$. On the angle bisector of $\angle{C}$, a point $E$ is marked such that $AE \bot BD$. Point $F$ is the foot of the perpendicular line from point $E$ to the side $BC$. Prove that $AB = BF$.

Point $M$ is the midpoint of the base $AD$ of trapezoid $ABCD$ inscribed in circle $S$. Rays $AB$ and $DC$ intersect at point $P$, and ray $BM$ intersects $S$ at point $K$. The circumscribed circle of triangle $PBK$ intersects line $BC$ at point $L$. Prove that $\angle{LDP} = 90^{\circ}$.

The altitudes $AA_1$ and $CC_1$ of an acute-angled triangle $ABC$ intersect at point $H$. A straight line passing through $H$ parallel to line $A_1C_1$ intersects the circumscribed circles of triangles $AHC_1$ and $CHA_1$ at points $X$ and $Y$, respectively. Prove that points $X$ and $Y$ are equidistant from the midpoint of segment $BH$.

Inside a regular triangle $ABC$, points $X$ and $Y$ are chosen such that $\angle{AXC} = 120^{\circ}$, $2\angle{XAC} + \angle{YBC} = 90^{\circ}$and $XY = YB = \frac{AC}{\sqrt{3}}$. Prove that point $Y$ lies on the incircle of triangle $ABC$.

Point $M$ is the midpoint of side $BC$ of convex quadrilateral $ABCD$. If $\angle{AMD} < 120^{\circ}$. Prove that
$$(AB+AM)^2 + (CD+DM)^2 > AD \cdot BC + 2AB \cdot CD$$

Given a tangential quadrilateral $ABCD$. Points $O_A, O_B, O_C, O_D$ are the centers of the circumscribed circles of triangles $BCD$, $CDA$, $DAB$, $ABC$, respectively. It is known that $O_AO_BO_CO_D$  is a convex quadrilateral whose three sides are equal. Prove that $ABCD$ is a rhombus.

$AA'$ is the diameter of the circumscribed circle of the acute triangle $ABC$. The inscribed circle of this triangle with center $I$ touches its sides $AB$, $AC$, $BC$ at points $P, Q, R$, respectively. The line $A'I$ intersects the segment $PQ$ at the point $T$. Prove that $RT \perp PQ$.

The median $CM$ of the triangle $ABC$ is equal to the bisector $BL$, also $\angle BAC=2\angle ACM$. prove that the triangle is right.

A circle $\omega$ is strictly inside triangle $ABC$. The tangents from $A$ to $\omega$ intersect $BC$ in $A_1,A_2$ define $B_1,B_2,C_1,C_2$ similarly. Prove that if five of six points $A_1,A_2,B_1,B_2,C_1,C_2$ lie on a circle the sixth one lie on the circle too.

Let the incircle of triangle $ABC$ touches the sides $AB,BC,CA$ in $C_1,A_1,B_1$ respectively. If $A_1C_1$ cuts the parallel to $BC$ from $A$ at $K$ prove that $\angle KB_1A_1=90.$

On a circle $4$ points are chosen and for each point we wrote the multiple of its distances to the rest. Could the written numbers be $1,2,3, 4$ in some order?

On a circle $100$ points are chosen and for each point we wrote the multiple of its distances to the rest. Could the written numbers be $1,2,\dots, 100$ in some order?

In a convex quadrilateral $ABCD$ rays $AB$ and $DC$ intersect at point $P$, and rays $BC$ and $AD$ at point $Q$. There is a point $T$ on the diagonal $AC$ such that the triangles $BTP$ and $DTQ$ are similar, in that order. Prove that $BD \Vert PQ$.

Triangle $ABC$ in which $AB <BC$, is inscribed in a circle $\omega$ and circumscribed about a circle $\gamma$ with center $I$. The line $\ell$ parallel to $AC$, touches the circle $\gamma$ and intersects the arcs $BAC$ and $BCA$ at points $P$ and $Q$, respectively. It is known that $PQ = 2BI$. Prove that $AP + 2PB = CP$.

In triangle $ABC$, the incircle touches sides $AB$ and $BC$ at points $P$ and $Q$, respectively. Median of triangle $ABC$ from vertex $B$ meets segment $P Q$ at point $R$. Prove that angle $ARC$ is obtuse.

Through point $ P $ inside triangle $ ABC $, straight lines were drawn, parallel to the sides, until they intersect with the sides. In the three resulting parallelograms, diagonals that do not contain point $ P $, are drawn. Points $ A_1 $, $ B_1 $ and $ C_1 $ are the intersection points of the lines containing these diagonals such that $A_1$ and $A$ are in different sides of line $BC$ and $B_1$ and $C_1$ are similar. Prove that if hexagon $ AC_1BA_1CB_1 $ is inscribed and convex, then point $ P $ is the orthocenter of triangle $ A_1B_1C_1 $.

Inside the circle $\omega$ through points $A, B$ point $C$ is chosen. An arbitrary point $X$ is selected on the segment $BC$. The ray $AX$ cuts the circle in $Y$. Prove that all circles $CXY$ pass through a two fixed points that is they intersect and are coaxial, independent of the position of $X$.

Given a quadrilateral $ABCD$ in which$$\sqrt{2}(BC-BA)=AC.$$Let $X$ be the midpoint of $AC$. Prove that $2\angle BXD=\angle DAB - \angle DCB.$

On the side $AC$ of triangle $ABC$ point $D$ is chosen. Let $I_1, I_2, I$ be the incenters of triangles $ABD, BCD, ABC$ respectively. It turned out that $I$ is the orthocentre of triangle $I_1I_2B$. Prove that $BD$ is an altitude of triangle $ABC$.

Given a quadrilateral $ABCD$ in which $\sqrt{2}(BC-BA)=AC$/ Let $X$ be the midpoint of $AC$ and $Y$ a point on the angle bisector of $B$ such that $XD$ is the angle bisector of $BXY$. Prove that $BD$ is tangent to the circumcircle of $DXY$.

 2018 missing

Circle $\omega$ touches the side $AC$ of the equilateral triangle $ABC$ at point $D$, and its circumcircle at the point $E$ lying on the arc $BC$. Prove that with segments $AD$, $BE$ and $CD$, you can form a triangle, in which the difference of two of its angles is $60^{\circ}$.

2019 239 MO VIII-IX p7,  X-XI. p3
The radius of the circumscribed circle of an acute-angled triangle is $23$ and the radius of its Inscribed circle is $9$. Common external tangents to its ex-circles, other than straight lines containing the sides of the original triangle, form a triangle. Find the radius of its inscribed circle.
Circle $\Gamma$ touches the circumcircle of triangle $ABC$ at point $R$, and it touches the sides $AB$ and $AC$ at points $P$ and $Q$, respectively. Rays $PQ$ and $BC$ intersect at point $X$. The tangent line at point $R$ to the circle $\Gamma$ meets the segment $QX$ at point $Y$. The line segment $AX$ intersects the circumcircle of triangle $APQ$ at point $Z$. Prove that the circumscribed circles of triangles $ABC$ and $XY Z$ are tangent.

Points $X$ and $Y$ are the midpoints of arcs $AB$ and $BC$ of the circumscribed circle of triangle $ABC$. Point $T$ lies on side $AC$. It turned out that the bisectors of the angles $ATB$ and $BTC$ pass through points $X$ and $Y$ respectively. What angle $B$ can be in triangle $ABC$?

The median $AD$ is drawn in triangle $ABC$. Point $E$ is selected on segment $AC$, and on the ray $DE$ there is a point $F$, and $\angle ABC = \angle AED$ and $AF // BC$. Prove that from segments $BD, DF$ and $AF$, you can make a triangle, the area of which is not less half the area of triangle $ABC$.

A triangle $ABC$ with an obtuse angle at the vertex $C$ is inscribed in a circle with a center at point $O$. Circumcircle of triangle $AOB$ centered at point $P$ intersects line $AC$ at points $A$ and $A_1$, line $BC$ at points $B$ and $B_1$, and the perpendicular bisector of the segment $PC$ at points $D$ and $E$. Prove that points $D$ and $E$ together with the centers of the circumscribed circles of triangles $A_1OC$ and $B_1OC$ lie on one circle.

Symedians of an acute-angled non-isosceles triangle $ABC$ intersect at a point at point $L$, and $AA_1$, $BB_1$ and $CC_1$ are its altitudes. Prove that you can construct equilateral triangles $A_1B_1C'$, $B_1C_1A'$ and $C_1A_1B'$ not lying in the plane $ABC$, so that lines $AA' , BB'$ and $CC'$ and also perpendicular to the plane $ABC$ at point $L$ intersected at one point.


                                                  other years under construction

Let $ABCD$ be an arbitrary quadrilateral. The bisectors of external angles $A$ and $C$ of the quadrilateral intersect at $P$; the bisectors of external angles $B$ and $D$ intersect at $Q$. The lines $AB$ and $CD$ intersect at $E$, and the lines $BC$ and $DA$ intersect at $F$. Now we have two new angles: $E$ (this is the angle $\angle{AED}$) and $F$ (this is the angle $\angle{BFA}$). We also consider a point $R$ of intersection of the external bisectors of these angles. Prove that the points $P$, $Q$ and $R$ are collinear.

2007 239 MO (also Zhautykov 2011 p6)
Diagonals of a cyclic quadrilateral $ABCD$  intersect at point $K.$ The midpoints of diagonals $AC$ and $BD$ are $M$ and $N,$ respectively. The circumscribed circles $ADM$ and $BCM$ intersect at points $M$ and $L.$ Prove that the points $K ,L ,M,$ and $ N$ lie on a circle. (all points are supposed to be different.)
P.Sahipov
239 MO (at most 2019)
Through the point $P$ inside the triangle $ABC$ drawn lines parallel to sides up to intersect with ones. In three obtained parallelograms drawn the diagonals which not containing $P$. Points $A_1,B_1,C_1$ are respective points of intersection. Prove that if hexagon $AC_1BA_1CB_1$ is inscribed in a circle and is convex then $P$ is the orthocenter of triangle $A_1B_1C_1$.

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