geometry problems from Saint Petersburg Mathematical Olympiads
with aops links in the names
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.
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 $XY$.
$O$ -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 $\angle EMC= 90^o$
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$,
$CX = AY = CA$?
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.
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/
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.
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
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 $AB \perp CD$
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 P5Streets 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 $\triangle MON$
$O$ -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 $AC$
$ABC$ 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
$SABCD$ 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 $BE=BC$
$ABCD$ - 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.
$ABC$-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.
$ABCD$ - 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 $\angle BDA + \angle CAB = 90^o$ .
$ABCD$ - convex quadrilateral. $M$ -midpoint $AC$ and $\angle MCB=\angle CMD =\angle MBA=\angle MBC -\angle MDC$. Prove that $AD=DC+AB$.
$ABCD$ 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 $XN=YN$.
$ABC$ 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 $\angle KPL=2\alpha$
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 $XYZ$ .
$ABCD$ 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 $\angle DMP=\angle DMQ$.
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.
$ABC$ 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 $\angle A- \angle C$ ?
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.
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
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
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
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 $\angle B$ .
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.
$D$ 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 $AB>AC$.
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.
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 $AL \leq \frac{AC+AC_1}{2}$.
$I$ - 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 $\angle AIP=90^o$.
$AB=CD,AD \parallel BC$ and $AD>BC$. $\Omega$ is circumcircle of $ABCD$. Point $E$ is on $\Omega$ such that $BE \perp AD$. Prove that $AE+BC>DE$.
$ABCD$ 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.
$ABCD$ - convex quadrilateral. Bisectors of angles $A$ and $D$ intersect in $K$, Bisectors of angles $B$ and $C$ intersect in $L$. Prove $2KL \geq |AB-BC+CD-DA|$.
$ABCDE$ is convex pentagon. $\angle BCA=\angle BEA = \frac{\angle BDA}{2}, \angle BDC =\angle EDA$. Prove, that $\angle DEB=\angle DAC$.
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
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 $.
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 $\angle ACB + \angle PRQ = 180^o$.
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$
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 $P$.
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.
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 $\angle ABC$
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 $\angle ABC$
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 $\angle{DHE}$.
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 $BC>2AL$.
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.
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.
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$,
$CX = AY = CA$?
$ABCD$ 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 $AM=CM$
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 $TX=TY$
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 $CD$.
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 $AB / 2$
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 $\angle AKD = \angle MKC$.
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 $CB_1C'$.
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 $BC$.
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.
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.)
[9] Prove that $BE+CE<AD$.
[10] Prove that $\triangle BCE$ is isosceles. (Here the condition that $\angle B$ is acute is not necessary.)
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 $L$
$BB_1$ 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.
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 $B_1 O_b +C_1O_c > \frac{BC}{4}$
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.
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.
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 $ED_1 = EA_1$
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 $\ell$
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 $3A_1C_1>BD$
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.
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 $AX$.
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.
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?
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