St. Petersburg City MO 2008-21 IX-XI (Russia) 72p

geometry problems from Saint Petersburg Mathematical Olympiads
with aops links in the names

 started in 1934 as Leningrad MO, 

renamed in 1992 as St. Petersburg

(Leningrad shall be collected here)

2008 - 2021

2008 St. Petersburg  MO grade IX P3, grade XI P2
Pentagon $ABCDE$ has circle $S$ inscribed into it. Side $BC$ is tangent to $S$ at point $K$. If $AB=BC=CD$, prove that angle $EKB$ is a right angle.

2008 St. Petersburg  MO grade X P2

Point $O$ is the center of the circle into which quadrilateral $ABCD$ is inscribed. If angles $AOC$ and $BAD$ are both equal to $110$ degrees and angle $ABC$ is greater than angle $ADC$, prove that $AB+AD>CD$.

2008 St. Petersburg  MO grade X P5
In cyclic quadrilateral $ABCD$ rays $AB$ and $DC$ intersect at point $E$, while segments $AC$ and $BD$ intersect at $F$. Point $P$ is on ray $EF$ such that angles $BPE$ and $CPE$ are congruent. Prove that angles $APB$ and $DPC$ are also equal.

2008 St. Petersburg  MO grade XI P5

All faces of the tetrahedron $ABCD$ are acute-angled triangles.$AK$ and $AL$ -are altitudes in faces $ABC$ and $ABD$. Points $C,D,K,L$ lies on circle. Prove, that $AB \perp CD$

2009 St. Petersburg  MO grade IX P4

Points $A_1$ and $C_1$ are on $BC$ and $AB$ of acute-angled triangle $ABC$ . $AA_1$ and $CC_1$ intersect in $K$. Circumcircles of $AA_1B,CC_1B$ intersect in $P$ - incenter of $AKC$.
Prove that $P$ - orthocenter of $ABC$ .

2009 St. Petersburg  MO grade IX P5
$ABC$ is acute-angled triangle. $AA_1,BB_1,CC_1$ are altitudes. $X,Y$ - midpoints of $AC_1,A_1C$. $XY=BB_1$. Prove that one side of $ABC$ in $\sqrt{2}$ greater than other side.

2009 St. Petersburg  MO grade X P2 
$ABCD$ is convex quadrilateral with $AB=CD$. $AC$ and $BD$ intersect in $O$. $X,Y,Z,T$ are midpoints of $BC,AD,AC,BD$. Prove, that circumcenter of $OZT$ lies on $XY$.

2009 St. Petersburg  MO grade X P4 , grade XI P3
Streets of Moscow are some circles (rings) with common center $O$ and some straight lines from center $O$ to external ring. Point $A,B$ - two crossroads on external ring. Three friends want to move from $A$ to $B$. Dima goes by external ring, Kostya goes from $A$ to $O$ then to $B$. Sergey says, that there is another way, that is shortest. Prove, that he is wrong.

2009 St. Petersburg  MO grade X P7 
Points $Y,X$ lies on $AB,BC$ of $\triangle ABC$ and $X,Y,A,C$ are concyclic. $AX$ and $CY$ intersect in $O$. Points $M,N$ are midpoints of $AC$ and $XY$. Prove, that $BO$ is tangent to circumcircle of $\triangle MON$

2009 St. Petersburg  MO grade XI P5
$O$ -circumcenter of $ABCD$. $AC$ and $BD$ intersect in $E$, $AD$ and $BC$ in $F$. $X,Y$ - midpoints of $AD$ and $BC$. $O_1$ -circumcenter of $EXY$. Prove that $OF \parallel O_1E$

2010 St. Petersburg  MO grade IX P2, grade X P3
$M,N$ are midpoints of $AB$ and $CD$ for convex quadrilateral $ABCD$. Points $X$ and $Y$ are on $AD$ and $BC$ and $XD=3AX,YC=3BY$. $\angle MXA=\angle MYB = 90^o$.  Prove that $\angle XMN=\angle ABC$

2010 St. Petersburg  MO grade IX P7
Incircle  of $ABC$ tangent $AB,AC,BC$ in $C_1,B_1,A_1$. $AA_1$ intersect incircle in $E$. $N$ is midpoint $B_1A_1$. $M$ is symmetric to $N$ relatively $AA_1$. Prove that $\angle EMC= 90^o$

2010 St. Petersburg  MO grade X P5, grade XI P2
$ABC$ is triangle with $AB=BC$. $X,Y$ are midpoints of $AC$ and $AB$. $Z$  is base of perpendicular from $B$ to $CY$. Prove, that circumcenter of $XYZ$ lies on $AC$

2010 St. Petersburg  MO grade XI P5
$SABCD$ is quadrangular pyramid. Lateral faces are acute triangles with orthocenters lying in one plane. $ABCD$ is base of pyramid and $AC$ and $BD$ intersects at $P$, where $SP$ is height of pyramid. Prove that $AC \perp BD$

2011 St. Petersburg  MO grade IX  P3

Point $D$ is inside $\triangle ABC$ and $AD=DC$. $BD$ intersect $AC$ in $E$. $\frac{BD}{BE}=\frac{AE}{EC}$. Prove, that $BE=BC$

2011 St. Petersburg  MO grade IX P5

$ABCD$ - convex quadrilateral. $\angle A+ \angle D=150^o, \angle B<150^o, \angle C<150^o$ Prove, that area $ABCD$ is greater than $\frac{1}{4}(AB\cdot CD+AB\cdot BC+BC\cdot CD)$

2011 St. Petersburg  MO grade X P3, grade XI P2
$ABC$-triangle with circumcenter $O$ and $\angle B=30$. $BO$ intersect $AC$ at $K$. $L$ - midpoint of arc $OC$ of circumcircle $KOC$, that does not contains $K$. Prove, that $A,B,L,K$ are concyclic.

2011 St. Petersburg  MO grade X P7

$ABCD$ - convex quadrilateral.  $P$ is such point on $AC$ and inside $\triangle ABD$, that $\angle ACD+\angle BDP = \angle ACB+ \angle DBP = 90^o-\angle BAD$. Prove that $\angle BAD+ \angle BCD =90^o$ or $\angle BDA + \angle CAB = 90^o$ .

2011 St. Petersburg  MO grade XI  P6

$ABCD$ - convex quadrilateral. $M$ -midpoint $AC$ and $\angle MCB=\angle CMD =\angle MBA=\angle MBC -\angle MDC$.  Prove that $AD=DC+AB$.

2012 St. Petersburg  MO grade IX P3

$ABCD$ is inscribed. Bisector of angle between diagonals intersect $AB$ anc $CD$ at $X$ and $Y$. $M,N$ are midpoints of $AD,BC$. $XM=YM$ Prove, that $XN=YN$.

2012 St. Petersburg  MO grade IX P6

$ABC$ is triangle. Point $L$ is inside $ABC$ and lies on bisector of $\angle B$. $K$ is on $BL$. $\angle KAB=\angle LCB= \alpha$. Point $P$ inside triangle is such, that $AP=PC$ and $\angle APC=2\angle AKL$.  Prove that $\angle KPL=2\alpha$

2012 St. Petersburg  MO grade X P2

Points $C,D$ are on side $BE$ of triangle $ABE$, such that $BC=CD=DE$. Points $X,Y,Z,T$ are circumcenters of $ABE,ABC,ADE,ACD$. Prove, that $T$ - centroid of $XYZ$ .

2012 St. Petersburg  MO grade X P6

$ABCD$ is parallelogram. Line $l$ is perpendicular to $BC$ at $B$. Two circles passes through $D,C$, such that $l$ is tangent in points $P$ and $Q$. $M$ - midpoint $AB$. Prove that $\angle DMP=\angle DMQ$.

2012 St. Petersburg  MO grade XI  P3

At the base of the pyramid $SABCD$ lies a convex quadrilateral $ABCD$, such that $BC \cdot AD = BD \cdot AC$. Also $\angle ADS =\angle BDS ,\angle ACS =\angle BCS$. Prove that the plane $SAB$ is perpendicular to the plane of the base.

2013 St. Petersburg  MO grade IX  P3

$ABC$ is triangle. $l_1$- line passes through $A$ and parallel to $BC$, $l_2$ -  line passes through $C$ and parallel to $AB$. Bisector of $\angle B$ intersect $l_1$ and $l_2$ at $X,Y$. $XY=AC$. What  value can take $\angle A- \angle C$ ?

2013 St. Petersburg  MO grade IX P6, grade X P5

Given quadrilateral $ABCD$ with $AB=BC=CD$. Let $AC\cap BD=O$, $X,Y$ are symmetry points of $O$ respect to midpoints of $BC$, $AD$, and $Z$ is intersection point of lines, which perpendicular bisects of  $AC$, $BD$. Prove that $X,Y,Z$ are collinear.

2013 St. Petersburg  MO grade X P2

In a convex quadrilateral $ABCD$ , $M,N$ are midpoints of $BC,AD$ respectively. If $AM=BN$ and  $DM=CN$ then prove that $AC=BD$.

by S. Berlov

2013 St. Petersburg  MO grade XI P3

Let $M$ and $N$ are midpoint of edges $AB$ and $CD$ of the tetrahedron $ABCD$, $AN=DM$ and $CM=BN$. Prove that $AC=BD$.

by S. Berlov

2013 St. Petersburg  MO grade XI P6

Let $(I_b)$, $(I_c)$ are excircles of a triangle $ABC$. Given a circle $\omega$ passes through $A$ and externally tangents to the circles $(I_b)$ and $(I_c)$ such that it intersects with $BC$ at points $M$, $N$. Prove that $\angle BAM=\angle CAN$.

by A. Smirnov 

2014 St. Petersburg  MO grade IX P2

All angles of $ABC$ are in $(30,90)$. Circumcenter of $ABC$ is $O$ and circumradius is $R$. Point $K$ is projection of $O$ to angle bisector of $\angle B$, point $M$ is midpoint $AC$. It is known, that $2KM=R$. Find $\angle B$ .

2014 St. Petersburg  MO grade IX P6

Points $A,B$ are on circle $\omega$. Points $C$ and $D$ are moved on the arc $AB$, such that $CD$ has constant length. $I_1,I_2$ - incenters of $ABC$ and $ABD$. Prove that line $I_1I_2$ is tangent to some fixed circle.

2014 St. Petersburg  MO grade X  P3

$D$ is inner point of triangle $ABC$. $E$ is on $BD$ and $CE=BD$. $\angle ABD=\angle ECD=10^o,\angle BAD=40^o,\angle CED=60^o$ . Prove that $AB>AC$.

2014 St. Petersburg  MO grade X P5

Incircle $\omega$ of $ABC$ touch $AC$ at $B_1$. Point $E,F$ on the $\omega$ such that $\angle AEB_1=\angle B_1FC=90$. Tangents to $\omega$ at $E,F$ intersects in $D$, and $B$ and $D$ are on different sides for line $AC$.  $M$- midpoint of $AC$. Prove, that $AE,CF,DM$ intersects at one point.

2014 St. Petersburg  MO grade XI P4

Points $B_1,C_1$  are on $AC$ and $AB$ and $B_1C_1 \parallel BC$. Circumcircle of $ABB_1$ intersect $CC_1$ at $L$. Circumcircle $CLB_1$ is tangent to $AL$. Prove $AL \leq \frac{AC+AC_1}{2}$.

2014 St. Petersburg  MO grade  XI P7

$I$ - incenter , $M$- midpoint of arc $BAC$ of circumcircle, $AL$ - angle bisector of triangle $ABC$. $MI$ intersect circumcircle in $K$. Circumcircle of $AKL$ intersect $BC$ at $L$ and $P$.

 Prove that $\angle AIP=90^o$.

2015 St. Petersburg  MO grade IX P2

$AB=CD,AD \parallel BC$ and $AD>BC$. $\Omega$ is circumcircle of $ABCD$. Point $E$ is on $\Omega$ such that $BE \perp AD$. Prove that $AE+BC>DE$.

2015 St. Petersburg  MO grade IX P5, grade XI P4

$ABCD$ is convex quadrilateral. Circumcircle of $ABC$ intersect $AD$ and $DC$ at points $P$ and $Q$. Circumcircle of $ADC$ intersect $AB$ and $BC$ at points $S$ and $R$. Prove that if $PQRS$ is parallelogram then $ABCD$ is parallelogram.

2015 St. Petersburg  MO grade X P3

$ABCD$ - convex quadrilateral. Bisectors of angles $A$ and $D$ intersect in $K$,  Bisectors of angles $B$ and $C$ intersect in $L$. Prove $2KL \geq |AB-BC+CD-DA|$.

2015 St. Petersburg  MO grade X P5

$ABCDE$ is convex pentagon. $\angle BCA=\angle BEA = \frac{\angle BDA}{2}, \angle BDC =\angle EDA$.  Prove, that $\angle DEB=\angle DAC$.

2015 St. Petersburg  MO grade XI P7

Let $BL$ be angle bisector of acute triangle $ABC$.Point $K$ choosen on $BL$ such that $\measuredangle AKC-\measuredangle ABC=90º$.point $S$ lies on the extention of $BL$ from $L$ such that $\measuredangle ASC=90º$.Point $T$ is diametrically opposite the point $K$  on the circumcircle of $\triangle AKC$.Prove that $ST$ passes through midpoint of arc $ABC$.

by S. Berlov

2016 St. Petersburg  MO grade IX P3

On the side $AB$ of the non-isosceles triangle $ABC$, let the points $P$ and $Q$ be so that $AC = AP$ and $BC = BQ$. The perpendicular bisector of the segment $PQ$ intersects the angle bisector of the $\angle C$ at the point $R$ (inside the triangle). Prove that $\angle ACB +  \angle PRQ = 180^o$.

2016 St. Petersburg  MO grade IX P6
Incircle of $\triangle ABC$ touch $AC$ at $D$. $BD$ intersect incircle at $E$. Points $F,G$ on incircle are such points, that $FE \parallel BC,GE \parallel AB$. $I_1,I_2$ are incenters of $DEF,DEG$. Prove that angle bisector of $\angle GDF$ passes though the midpoint of $I_1I_2$.

2016 St. Petersburg  MO grade X P3
The circle inscribed in the triangle $ABC$ is tangent to side $AC$ at point $B_1$, and to side $BC$ at point $A_1$. On the side $AB$ there is a point $K$ such that $AK = KB_1, BK = KA_1$. Prove that $\angle ACB\ge 60$

2016 St. Petersburg  MO grade X P5
Points $A$ and $P$ are marked in the plane not lying on the line $\ell$. For all right triangles $ABC$ with hypotenuse on $\ell$, show that the circumcircle of triangle $BPC$ passes through a fixed point other than $P$.

2016 St. Petersburg  MO grade XI P3
In a tetrahedron, the midpoints of all the edges lie on the same sphere. Prove that it's altitudes intersect at one point.

2016 St. Petersburg  MO grade XI P5
Incircle of $\triangle ABC$ touch $AC$ at $D$. $BD$ intersect incircle at $E$. Points $F,G$ on incircle are such points, that $FE \parallel BC,GE \parallel AB$. $I_1,I_2$ are incenters of $DEF,DEG$. Prove that $I_1I_2 \perp$ bisector of $\angle ABC$ 

2017 St. Petersburg  MO grade IX  P2

Given a triangle $ABC$, there’s a point $X$ on the side $AB$ such that $2BX = BA + BC$. Let $Y$ be the point symmetric to the incenter $I$ of triangle $ABC$, with respect to point $X$. Prove that $YI_B\perp AB$ where $I_B$ is the $B$-excenter of triangle $ABC$.

2017 St. Petersburg  MO grade IX P5
Given a scalene triangle $ABC$ with $\angle B=130^{\circ}$. Let $H$ be the foot of altitude from $B$. $D$ and $E$ are points on the sides $AB$ and $BC$, respectively, such that $DH=EH$ and $ADEC$ is a cyclic quadrilateral. Find $\angle{DHE}$.

2017 St. Petersburg  MO grade  X P3
Let $ABC$ be an acute triangle, with median $AM$, height $AH$ and internal angle bisector $AL$. Suppose that $B, H, L, M, C$ are collinear in that order, and $LH<LM$. Prove that $BC>2AL$.

2017 St. Petersburg  MO grade X P6
In acute-angled triangle $ABC$, the height $AH$ and median $BM$ were drawn. Point $D$ lies on the circumcircle of triangle $BHM$ such that $AD \parallel BM$ and $B, D$ are on opposite sides of line $AC$. Prove that $BC=BD$.

2017 St. Petersburg  MO grade XI P2
A circle passing through vertices $A$ and $B$ of triangle $ABC$ intersects the sides $AC$ and $BC$ again at points $P$ and $Q$, respectively. Given that the median from vertex $C$ bisect the arc $PQ$ of the circle. Prove that $ABC$ is an isosceles triangle.

2017 St. Petersburg  MO grade XI P5

Given a tetrahedron $PABC$, draw the height $PH$ from vertex $P$ to $ABC$. From point $H$, draw perpendiculars $HA’,HB’,HC’$ to the lines $PA,PB,PC$. Suppose the planes $ABC$ and $A’B’C’$ intersects at line $\ell$. Let $O$ be the circumcenter of triangle $ABC$. Prove that $OH\perp \ell$.

2018 St. Petersburg  MO grade IX  P3
$ABC$ is acuteangled triangle. Variable point $X$ lies on segment $AC$, and variable point $Y$ lies on the ray $BC$ but not segment $BC$, such that $\angle ABX+\angle CXY =90^o$. $T$ is projection of $B$ on the $XY$. Prove that all points $T$ lies on the line.

2018 St. Petersburg  MO grade IX  P5
Can we draw $\triangle ABC$ and points $X,Y$, such that $AX=BY=AB$, $BX = CY = BC$,
$CX = AY = CA$?

2018 St. Petersburg  MO grade X  P5

$ABCD$ is inscribed quadrilateral. Line, that perpendicular to $BD$ intersects segments $AB$ and $BC$ and rays $DA,DC$ at $P,Q,R,S$ . $PR=QS$. $M$ is midpoint of $PQ$. Prove that $AM=CM$

2018 St. Petersburg  MO grade XI  P3

Point $T$ lies on the bisector of $\angle B$ of acuteangled $\triangle ABC$. Circle $S$ with diameter $BT$ intersects $AB$ and $BC$ at points $P$ and $Q$. Circle, that goes through point $A$ and tangent to $S$ at $P$ intersects line $AC$ at $X$.  Circle, that goes through point $C$ and tangent to $S$ at $Q$ intersects line $AC$ at $Y$. Prove, that $TX=TY$

2018 St. Petersburg  MO grade XI  P7

Points $A,B$ lies on the circle $S$. Tangent lines to $S$ at $A$ and $B$ intersects at $C$. $M$ -midpoint of $AB$. Circle $S_1$ goes through $M,C$ and intersects $AB$ at $D$ and $S$ at $K$ and $L$. Prove, that tangent lines to $S$ at $K$ and $L$ intersects at point on the segment $CD$.

2019 St. Petersburg  MO grade IX  P3

Prove that the distance between the midpoint of side $BC$ of triangle $ABC$ and the midpoint of the arc $ABC$ of its circumscribed circle is not less than $AB / 2$

2019 St. Petersburg  MO grade IX  P6

The bisectors $BB_1$ and $CC_1$ of the acute triangle $ABC$ intersect in point $I$. On the extensions of the segments $BB_1$ and $CC_1$, the points $B'$ and $C'$ are marked, respectively So, the quadrilateral $AB'IC'$ is a parallelogram. Prove that if $\angle BAC = 60^o$, then the straight line $B'C'$ passes through the intersection point of the circumscribed circles of the triangles $BC_1B'$ and $CB_1C'$.

2019 St. Petersburg  MO grade X  P6

Given a convex quadrilateral $ABCD$. The medians of the triangle $ABC$ intersect at point $M$, and the medians of the triangle $ACD$ at point$N$. The circle, circumscibed around the triangle $ACM$, intersects the segment $BD$ at the point $K$ lying inside the triangle $AMB$ . It is known that $\angle MAN = \angle ANC = 90^o$. Prove that $\angle AKD = \angle MKC$.

2019 St. Petersburg  MO grade XI  P4

A non-equilateral triangle $\triangle ABC$ of perimeter $12$ is inscribed in circle $\omega$ .Points $P$ and $Q$ are arc midpoints of arcs $ABC$ and $ACB$ , respectively. Tangent to $\omega$ at $A$ intersects line $PQ$ at $R$. It turns out that the midpoint of segment $AR$ lies on line $BC$ . Find the length of the segment $BC$.

2020 St. Petersburg  MO grade IX  P5

Point $I_a$ is the $A$-excircle center of $\triangle ABC$ which is tangent to $BC$ at $X$. Let $A'$ be diametrically opposite point of $A$ with respect to the circumcircle of $\triangle ABC$. On the segments $I_aX, BA'$ and $CA'$ are chosen respectively points $Y,Z$ and $T$ such that $I_aY=BZ=CT=r$ where $r$ is the inradius of $\triangle ABC$. Prove that the points $X,Y,Z$ and $T$ are concyclic.

2020 St. Petersburg  MO grade IX  P3, X P3

On the side $AD$ of the convex quadrilateral $ABCD$ with an acute angle at $B$, a point $E$ is marked. It is known that $\angle CAD = \angle ADC=\angle ABE =\angle DBE$.
[9] Prove that $BE+CE<AD$.

[10] Prove that $\triangle BCE$ is isosceles. (Here the condition that $\angle B$ is acute is not necessary.)

2020 St. Petersburg  MO grade X  P5

Rays $\ell, \ell_1, \ell_2$ have the same starting point $O$, such that the angle between $\ell$ and $\ell_2$ is acute and the ray $\ell_1$ lies inside this angle. The ray $\ell$ contains a fixed point of $F$ and an arbitrary point $L$. Circles passing through $F$ and $L$ and tangent to $\ell_1$ at $L_1$, and passing through $F$ and $L$ and tangent to $\ell_2$ at $L_2$. Prove that the circumcircle of $\triangle FL_1L_2$ passes through a fixed point other than $F$ independent on $L$

2020 St. Petersburg  MO grade XI  P3

$BB_1$ is the angle bisector of $\triangle ABC$, and $I$ is its incenter. The perpendicular bisector of segment $AC$ intersects the circumcircle of $\triangle AIC$ at $D$ and $E$. Point $F$ is on the segment $B_1C$ such that $AB_1=CF$.Prove that the four points $B, D, E$ and $F$ are concyclic.

2020 St. Petersburg  MO grade X  P5

The altitudes $BB_1$ and $CC_1$ of the acute triangle $\triangle ABC$ intersect at $H$. The circle centered at $O_b$ passes through points $A,C_1$, and the midpoint of $BH$. The circle centered at $O_c$ passes through $A,B_1$ and the midpoint of $CH$. Prove that $B_1 O_b +C_1O_c > \frac{BC}{4}$

2021 St. Petersburg  MO grade IX P3

In the pyramid $SA_1A_2 \cdots A_n$, all sides are equal. Let point $X_i$ be the midpoint of arc $A_iA_{i+1}$ in the circumcircle of $\triangle SA_iA_{i+1}$ for $1 \le i \le n$ with indices taken mod $n$. Prove that the circumcircles of $X_1A_2X_2, X_2A_3X_3, \cdots, X_nA_1X_1$ have a common point.

2021 St. Petersburg  MO grade IX P6

Point $M$ is the midpoint of base $AD$ of an isosceles trapezoid $ABCD$ with circumcircle $\omega$. The angle bisector of $ABD$ intersects $\omega$ at $K$. Line $CM$ meets $\omega$ again at $N$. From point $B$, tangents $BP, BQ$ are drawn to $(KMN)$. Prove that $BK, MN, PQ$ are concurrent.

2021 St. Petersburg  MO grade X P3

Given a convex pentagon $ABCDE$, points $A_1, B_1, C_1, D_1, E_1$ are such that $$AA_1 \perp BE, BB_1 \perp AC, CC_1 \perp BD, DD_1 \perp CE, EE_1 \perp DA.$$ In addition, $AE_1 = AB_1, BC_1 = BA_1, CB_1 = CD_1$ and $DC_1 = DE_1$. Prove that $ED_1 = EA_1$

2021 St. Petersburg  MO grade X P6

A line $\ell$ passes through vertex $C$ of the rhombus $ABCD$ and meets the extensions of $AB, AD$ at points $X,Y$. Lines $DX, BY$ meet $(AXY)$ for the second time at $P,Q$. Prove that the circumcircle of $\triangle PCQ$ is tangent to $\ell$

2021 St. Petersburg  MO grade XI P3

Given is cyclic quadrilateral $ABCD$ with∠$A = 3$∠$B$. On the $AB$ side is chosen point $C_1$, and on side $BC$ - point $A_1$ so that $AA_1 = AC = CC_1$. Prove that $3A_1C_1>BD$

2021 St. Petersburg  MO grade XI P5

Given is an isosceles trapezoid $ABCD$, such that $AD$ and $BC$ are bases and $AD=2AB$, and it is inscribed in a circle $c$. Points $E$ and $F$ are selected on a circle $c$ so that $AC$ || $DE$ and $BD$ || $AF$. The line $BE$ intersects lines $AC$ and $AF$ at points $X$ and $Y$, respectively. Prove that the circumcircles of triangles $BCX$ and $EFY$ are tangent to each other.

oldies

1997 
Let $B'$ be the antipode of $B$ on the circumcircle of triangle $ABC,$ let $I$ be the incenter of triangle $ABC,$ and let $M$ be the point where the incircle touches $AC.$ The points $K$ and $L$ are chosen on sides $AB$ and $BC,$ respectively, so that $KB = MC,$ $LB=AM.$ Prove that the lines $B'I$ and $KL$ are perpendicular.

2000

The line S is tangent to the circumcircle of acute triangle ABC at B. Let K be the projection of the
orthocenter of triangle ABC onto line S (i.e. K is the foot of perpendicular from the orthocenter of triangle ABC to S). Let L be the midpoint of side AC. Show that triangle BKL is isosceles.


2002
Let $ABC$ be a triangle. The incircle of triangle $ABC$ touches the sides $BC$, $CA$, $AB$ at the points $A_{1}$, $B_{1}$, $C_{1}$ respectively. The perpendicular to the line $AA_{1}$ through the point $A_{1}$ intersects the line $B_{1}C_{1}$ at a point $X$. Prove that the line $BC$ bisects the segment $AX$.

Year Unknown
The point $I$ is the incenter of triangle $ABC$. A circle centered at $I$ meets $BC$ at $A_{1}$ and $A_{2}$, $CA$ at $B_{1}$ and $B_{2}$, and $AB$ at $C_{1}$ and $C_{2}$, where the points occur around the circle in the order $A_{1}$, $A_{2}$, $B_{1}$, $B_{2}$, $C_{1}$, $C_{2}$. Let $A_{3}$, $B_{3}$, $C_{3}$ be the midpounts of the arcs $A_{1}A_{2}$, $B_{1}B_{2}$, $C_{1}C_{2}$, respectively. The lines $A_{2}A_{3}$ and $B_{1}B_{3}$ meet at $C_{4}$, $B_{2}B_{3}$ and $C_{1}C_{3}$ meet at $A_{4}$, and $C_{2}C_{3}$ and $A_{1}A_{3}$ meet at $B_{4}$. Prove that the lines $A_{3}A_{4}$, $B_{3}B_{4}$, $C_{3}C_{4}$ are concurrent.

Year Unknown2
Let ABCD be an isosceles trapezoid with bases AD and BC. Let a circle which is tangent to both AB and AC intersect segment BC at points M and N. Now, consider $\omega$, the incircle of triangle BCD. Let X and Y be the intersections (closer to D) of DM and DN with $\omega$. Prove that XY is parallel to BC.

Year Unknown3
Let S be a square, Q be the perimeter of the square, and P be the perimeter of a quadrilateral T inscribed within S such that each of its vertices lies on a different edge of S. What is the smallest possible ratio of P to Q?

official page: www.pdmi.ras.ru/~olymp/