Poland 2nd Round 1993 - 2021 59p

geometry problems from the Second Round of Polish Mathematical Olympiads
with aops links in the names

Polska Olimpiada Matematyczna

1993 - 2021

1993 Polish 2nd Round P2

Let be given a circle with center $O$ and a point $P$ outside the circle. A line $l$ passes through $P$ and cuts the circle at $A$ and $B$. Let $C$ be the point symmetric to $A$ with respect to $OP$, and let $m$ be the line $BC$. Prove that all lines $m$ have a common point as $l$ varies.

1993 Polish 2nd Round P3

A tetrahedron $OA_1B_1C_1$ is given. Let $A_2,A_3 \in OA_1, A_2,A_3 \in OA_1, A_2,A_3 \in OA_1$ be points such that the planes $A_1B_1C_1,A_2B_2C_2$ and $A_3B_3C_3$ are parallel and $OA_1 > OA_2 > OA_3 > 0$. Let $V_i$ be the volume of the tetrahedron $OA_iB_iC_i$ ($i = 1,2,3$) and $V$ be the volume of $OA_1B_2C_3$. Prove that $V_1 +V_2 +V_3 \ge 3V$.

1993 Polish 2nd Round P5

Let $D,E,F$ be points on the sides $BC,CA,AB$ of a triangle $ABC$, respectively. Suppose that the inradii of the triangles $AEF,BFD,CDE$ are all equal to $r_1$. If $r_2$ and $r$ are the inradii of triangles $DEF$ and $ABC$ respectively, prove that $r_1 +r_2 =r$.

1994 Polish 2nd Round P3
A plane passing through the center of a cube intersects the cube in a cyclic hexagon. Show that this hexagon is regular.

1994 Polish 2nd Round P5

The incircle $\omega$ of a triangle $ABC$ is tangent to the sides $AB$ and $BC$ at $P$ and $Q$ respectively. The angle bisector at $A$ meets $PQ$ at point $S$. Prove $\angle ASC = 90^o$

1995 Polish 2nd Round P2
Let $ABCDEF$ be a convex hexagon with $AB = BC, CD = DE$ and $EF = FA$.
Prove that the lines through $C,E,A$ perpendicular to $BD,DF,FB$ are concurrent.

1995 Polish 2nd Round P5
The incircles of the faces $ABC$ and $ABD$ of a tetrahedron $ABCD$ are tangent to the edge $AB$ in the same point. Prove that the points of tangency of these incircles to the edges $AC,BC,AD,BD$ are concyclic.

1996 Polish 2nd Round P2
A circle with center $O$ inscribed in a convex quadrilateral $ABCD$ is tangent to the lines $AB, BC, CD, DA$ at points $K, L, M, N$ respectively. Assume that the lines $KL$ and $MN$ are not parallel and intersect at the point $S$. Prove that $BD$ is perpendicular $OS$.

1996 Polish 2nd Round P6
Prove that every interior point of a parallelepiped with edges $a,b,c$ is on the distance at most $\frac12 \sqrt{a^2 +b^2 +c^2}$ from some vertex of the parallelepiped.

1997 Polish 2nd Round P2
Let $P$ be a point inside triangle $ABC$ such that $3<ABP = 3<ACP = <ABC + <ACB$. Prove that $AB/(AC + PB) = AC/(AB + PC)$.

1997 Polish 2nd Round P6
Let eight points be given in a unit cube. Prove that two of these points are on a distance not greater than $1$.

1998 Polish 2nd Round P2
In triangle $ABC$, the angle $\angle BCA$ is obtuse and $\angle BAC = 2\angle ABC\,.$ The line through $B$ and perpendicular to $BC$ intersects line $AC$ in $D$. Let $M$ be the midpoint of $AB$. Prove that $\angle AMC=\angle BMD$.

1998 Polish 2nd Round P6
Prove that the edges $AB$ and $CD$ of a tetrahedron $ABCD$ are perpendicular if and only if there exists a parallelogram $CDPQ$ such that $PA = PB = PD$ and $QA = QB = QC$.

1999 Polish 2nd Round P3
Let $ABCD$ be a cyclic quadrilateral and let $E$ and $F$ be the points on the sides $AB$ and $CD$ respectively such that $AE : EB = CF : FD$. Point $P$ on the segment EF satsfies $EP : PF = AB : CD$. Prove that the ratio of the areas of $\vartriangle APD$ and $\vartriangle BPC$ does not depend on the choice of $E$ and $F$.

1999 Polish 2nd Round P4
Let $P$ be a point inside a triangle $ABC$ such that $\angle PAB = \angle PCA$ and $\angle PAC = \angle PBA$. If $O \ne P$ is the circumcenter of $\triangle ABC$, prove that $\angle APO$ is right.

2000 Polish 2nd Round P2
Bisector of angle $BAC$ of triangle $ABC$ intersects circumcircle of this triangle in point $D \neq A$. Points $K$ and $L$ are orthogonal projections on line $AD$ of points $B$ and $C$, respectively. Prove that $AD \ge BK + CL$.

2000 Polish 2nd Round P4
Point $I$ is incenter of triangle $ABC$ in which $AB \neq AC$. Lines $BI$ and $CI$ intersect sides $AC$ and $AB$ in points $D$ and $E$, respectively. Determine all measures of angle $BAC$, for which may be $DI = EI$.

2001 Polish 2nd Round P2
Points $A,B,C$ with $AB<BC$ lie in this order on a line. Let $ABDE$ be a square. The circle with diameter $AC$ intersects the line $DE$ at points $P$ and $Q$ with $P$ between $D$ and $E$. The lines $AQ$ and $BD$ intersect at $R$. Prove that $DP=DR$.

2001 Polish 2nd Round P5
In a triangle $ABC$, $I$ is the incentre and $D$ the intersection point of $AI$ and $BC$. Show that $AI+CD=AC$ if and only if $\angle B=60^{\circ}+\frac{_1}{^3}\angle C$.

2002 Polish 2nd Round P2
In a convex quadrilateral $ABCD$, both $\angle ADB=2\angle ACB$ and $\angle BDC=2\angle BAC$. Prove that $AD=CD$.

2002 Polish 2nd Round P5
Triangle $ABC$ with $\angle BAC=90^{\circ}$ is the base of the pyramid $ABCD$. Moreover, $AD=BD$ and $AB=CD$. Prove that $\angle ACD\ge 30^{\circ}$.

2003 Polish 2nd Round P2
The quadrilateral $ABCD$ is inscribed in the circle $o$. Bisectors of angles $DAB$ and $ABC$ intersect at point $P$, and bisectors of angles $BCD$ and $CDA$ intersect in point $Q$. Point $M$ is the center of this arc $BC$ of the circle $o$ which does not contain points $D$ and $A$. Point $N$ is the center of the arc $DA$ of the circle $o$, which does not contain points $B$ and $C$. Prove that the points $P$ and $Q$ lie on the line perpendicular to $MN$.

2003 Polish 2nd Round P5
Point $A$ lies outside circle $o$ of center $O$. From point $A$ draw two lines tangent to a circle $o$ in points $B$ and $C$. A tangent to a circle $o$ cuts segments $AB$ and $AC$ in points $E$ and $F$, respectively. Lines $OE$ and $OF$ cut segment $BC$ in points $P$ and $Q$, respectively. Prove that from line segments $BP$, $PQ$, $QC$ can construct triangle similar to triangle $AEF$.

2004 Polish 2nd Round P2
In convex hexagon $ABCDEF$ all sides have equal length and $\angle A+\angle C+\angle E=\angle B+\angle D+\angle F$. Prove that the diagonals $AD,BE,CF$ are concurrent.

2004 Polish 2nd Round P5
Points $D$ and $E$ are taken on sides $BC$ and $CA$ of a triangle $BD=AE$. Segments $AD$ and $BE$ meet at $P$. The bisector of $\angle ACB$ intersects $AD$ and $BE$ at $Q$ and $R$ respectively. Prove that $\frac{PQ}{PR}=\frac{AD}{BE}$.

2005 Polish 2nd Round P2
In a convex quadrilateral $ABCD$, point $M$ is the midpoint of the diagonal $AC$. Prove that if $\angle BAD=\angle BMC=\angle CMD$, then a circle can be inscribed in quadrilateral $ABCD$.

2005 Polish 2nd Round P5
A rhombus $ABCD$ with $\angle BAD=60^{\circ}$ is given. Points $E$ on side $AB$ and $F$ on side $AD$ are such that $\angle ECF=\angle ABD$. Lines $CE$ and $CF$ respectively meet line $BD$ at $P$ and $Q$. Prove that $\frac{PQ}{EF}=\frac{AB}{BD}$.

2006 Polish 2nd Round P2 (IMO ISL 2005)
Given a triangle $ABC$ satisfying $AC+BC=3\cdot AB$. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Let $K$ and $L$ be the reflections of the points $D$ and $E$ with respect to $I$. Prove that the points $A$, $B$, $K$, $L$ lie on one circle.

by Dimitris Kontogiannis, Greec

2006 Polish 2nd Round P5
Point $C$ is a midpoint of $AB$. Circle $o_1$ which passes through $A$ and $C$ intersect circle $o_2$ which passes through $B$ and $C$ in two different points $C$ and $D$. Point $P$ is a midpoint of arc $AD$ of circle $o_1$ which doesn't contain $C$. Point $Q$ is a midpoint of arc $BD$ of circle $o_2$ which doesn't contain $C$. Prove that $PQ \perp CD$.

2007 Polish 2nd Round P2
$ABCDE$ is a convex pentagon and $BC=CD, \;\;\; DE=EA, \;\;\; \angle BCD=\angle DEA=90^{\circ}$
Prove, that it is possible to build a triangle from segments $AC$, $CE$, $EB$. Find the value of its angles if $\angle ACE=\alpha$ and $\angle BEC=\beta$.

2007 Polish 2nd Round P5
We are given a cyclic quadrilateral $ABCD \quad AB\not=CD$. Quadrilaterals $AKDL$ and $CMBN$ are rhombuses with equal sides. Prove, that $KLMN$ is cyclic

2008 Polish 2nd Round P2
In the convex pentagon $ABCDE$ following equalities holds: $\angle ABD= \angle ACE, \angle ACB=\angle ACD, \angle ADC=\angle ADE$ and $\angle ADB=\angle AEC$. The point $S$ is the intersection of the segments $BD$ and $CE$. Prove that lines $AS$ and $CD$ are perpendicular.

2008 Polish 2nd Round P5
We are given a triangle $ABC$ such that $AC = BC$. There is a point $D$ lying on the segment $AB$, and $AD < DB$. The point $E$ is symmetrical to $A$ with respect to $CD$. Prove that: $\frac {AC}{CD} = \frac {BE}{BD - AD}$

2009 Polish 2nd Round P3
Disjoint circles $o_1, o_2$, with centers $I_1, I_2$ respectively, are tangent to the line $k$ at $A_1, A_2$ respectively and they lie on the same side of this line. Point $C$ lies on segment $I_1I_2$ and $\angle A_1CA_2 = 90^{\circ}$. Let $B_1$ be the second intersection of $A_1C$ with $o_1$, and let $B_2$ be the second intersection of $A_2C$ with $o_2$. Prove that $B_1B_2$ is tangent to the circles $o_1, o_2$.

2009 Polish 2nd Round P4
$ABCD$ is a cyclic quadrilateral inscribed in the circle $\Gamma$ with $AB$ as diameter. Let $E$ be the intersection of the diagonals $AC$ and $BD$. The tangents to $\Gamma$ at the points $C,D$ meet at $P$. Prove that $PC=PE$.

2010 Polish 2nd Round P2
The orthogonal projections of the vertices $A, B, C$ of the tetrahedron $ABCD$ on the opposite faces are denoted by $A', B', C'$ respectively. Suppose that point $A'$ is the circumcenter of the triangle $BCD$, point $B'$ is the incenter of the triangle $ACD$ and $C'$ is the centroid of the triangle $ABD$. Prove that tetrahedron $ABCD$ is regular.

2010 Polish 2nd Round P4
In the convex pentagon $ABCDE$ all interior angles have the same measure. Prove that the perpendicular bisector of segment $EA$, the perpendicular bisector of segment $BC$ and the angle bisector of $\angle CDE$ intersect in one point.

2011 Polish 2nd Round P2
The convex quadrilateral $ABCD$ is given, $AB<BC$ and $AD<CD$. $P,Q$ are points on $BC$ and $CD$ respectively such that $PB=AB$ and $QD=AD$. $M$ is midpoint of $PQ$. We assume that $\angle BMD=90^{\circ}$, prove that $ABCD$ is cyclic.

2011 Polish 2nd Round P4
Points $A,B,C,D,E,F$ lie in that order on semicircle centered at $O$, we assume that $AD=BE=CF$. $G$ is a common point of $BE$ and $AD$, $H$ is a common point of $BE$ and $CD$. Prove that: $\angle AOC=2\angle GOH.$

2012 Polish 2nd Round P2
Prove that for tetrahedron $ABCD$; vertex $D$, center of insphere and centroid of $ABCD$ are collinear iff areas of triangles $ABD,BCD,CAD$ are equal.

2012 Polish 2nd Round P5
Let $ABC$ be a triangle with $\angle A=60^{\circ}$ and $AB\neq AC$, $I$-incenter, $O$-circumcenter. Prove that perpendicular bisector of $AI$, line $OI$ and line $BC$ have a common point.

2013 Polish 2nd Round P2
Circles $o_1$ and $o_2$ with centers in $O_1$ and $O_2$, respectively, intersect in two different points $A$ and $B$, wherein angle $O_1AO_2$ is obtuse. Line $O_1B$ intersects circle $o_2$ in point $C \neq B$. Line $O_2B$ intersects circle $o_1$ in point $D \neq B$. Show that point $B$ is incenter of triangle $ACD$.

2013 Polish 2nd Round P6
Decide, whether exist tetrahedrons $T$, $T'$ with walls $S_1$, $S_2$, $S_3$, $S_4$ and $S_1'$, $S_2'$, $S_3'$, $S_4'$, respectively, such that for $i = 1, 2, 3, 4$ triangle $S_i$ is similar to triangle $S_i'$, but despite this, tetrahedron $T$ is not similar to tetrahedron $T'$.

2014 Polish 2nd Round P2
Distinct points $A$, $B$ and $C$ lie on a line in this order. Point $D$ lies on the perpendicular bisector of the segment $BC$. Denote by $M$ the midpoint of the segment  $BC$. Let $r$ be the radius of the incircle of the triangle $ABD$ and let $R$ be the radius of the circle with center lying outside the triangle $ACD$, tangent to $CD$, $AC$ and $AD$. Prove that $DM=r+R$.

2014 Polish 2nd Round P5
Circles $o_1$ and $o_2$ tangent to some line at points $A$ and $B$, respectively, intersect at points $X$ and $Y$ ($X$ is closer to the line $AB$). Line $AX$ intersects $o_2$ at point  $P\neq X$. Tangent to $o_2$ at point $P$ intersects line $AB$ at point $Q$. Prove that $\sphericalangle XYB = \sphericalangle BYQ$.

2015 Polish 2nd Round P1
Points $E, F, G$ lie, and on the sides $BC, CA, AB$, respectively of a triangle  $ABC$, with $2AG=GB, 2BE=EC$ and $2CF=FA$. Points $P$ and $Q$ lie on segments $EG$ and $FG$, respectively such that $2EP = PG$ and $2GQ=QF$. Prove that the quadrilateral $AGPQ$ is a parallelogram.

2015 Polish 2nd Round P6
Let $ABC$ be a triangle. Let $K$ be a midpoint of $BC$ and $M$ be a point on the segment $AB$. $L=KM \cap AC$ and $C$ lies on the segment $AC$ between $A$ and $L$. Let $N$ be a midpoint of $ML$. $AN$ cuts circumcircle of $\Delta ABC$ in $S$ and $S \neq N$. Prove that circumcircle of $\Delta KSN$ is tangent to $BC$.

2016 Polish 2nd Round P1
Point $P$ lies inside triangle of sides of length $3, 4, 5$. Show that if distances between $P$ and vertices of triangle are rational numbers then distances from $P$ to sides of triangle are rational numbers too.

2016 Polish 2nd Round P2
In acute triangle $ABC$ bisector of angle $BAC$ intersects side $BC$ in point $D$. Bisector of line segment $AD$ intersects circumcircle of triangle $ABC$ in points  $E$ and $F$. Show that circumcircle of triangle $DEF$ is tangent to line $BC$.

2016 Polish 2nd Round P5
Quadrilateral $ABCD$ is inscribed in circle. Points $P$ and $Q$ lie respectively on rays $AB^{\rightarrow}$ and $AD^{\rightarrow}$ such that $AP = CD$, $AQ = BC$. Show that middle point of line segment $PQ$ lies on the line $AC$.

2017 Polish 2nd Round P2
In an acute triangle $ABC$ the bisector of $\angle BAC$ crosses $BC$ at $D$. Points $P$ and $Q$ are orthogonal projections of $D$ on lines $AB$ and $AC$. Prove that $[APQ]=[BCQP]$ if and only if the circumcenter of $ABC$ lies on $PQ$.

2017 Polish 2nd Round P4
Incircle of a triangle $ABC$ touches $AB$ and $AC$ at $D$ and $E$, respectively. Point $J$ is the excenter of $A$. Points $M$ and $N$ are midpoints of $JD$ and $JE$. Lines  $BM$ and $CN$ cross at point $P$. Prove that $P$ lies on the circumcircle of $ABC$.

2018 Polish 2nd Round P3
Bisector of side $BC$ intersects circumcircle of triangle $ABC$ in points $P$ and $Q$. Points $A$ and $P$ lie on the same side of line $BC$. Point $R$ is an orthogonal projection of point $P$ on line $AC$. Point $S$ is middle of line segment $AQ$. Show that points $A, B, R, S$ lie on one circle.

2018 Polish 2nd Round P4
Let $ABCD$ be a trapezoid with bases $AB$ and $CD$. Circle of diameter $BC$ is tangent to line $AD$. Prove, that circle of diameter $AD$ is tangent to line $BC$.

2019 Polish 2nd Round P1
A cyclic quadrilateral $ABCD$ is given. Point $K_1, K_2$ lie on the segment $AB$, points $L_1, L_2$ on the segment $BC$, points $M_1, M_2$ on the segment $CD$ and points $N_1, N_2$ on the segment $DA$. Moreover, points $K_1, K_2, L_1, L_2, M_1, M_2, N_1, N_2$ lie on a circle $\omega$ in that order. Denote by $a, b, c, d$ the lengths of the arcs $N_2K_1, K_2L_1, L_2M_1, M _2N_1$ of the circle $\omega$ not containing points $K_2, L_2, M_2, N_2$, respectively. Prove that $a+c=b+d.$

2019 Polish 2nd Round P6
Let $X$ be a point lying in the interior of the acute triangle $ABC$ such that $\sphericalangle BAX = 2\sphericalangle XBA$ and $\sphericalangle XAC = 2\sphericalangle ACX.$  Denote by $M$ the midpoint of the arc $BC$ of the circumcircle $(ABC)$ containing $A$. Prove that $XM=XA$.

2020 Polish 2nd Round P3
Let $M$ be the midpoint of the side $BC$ of a acute triangle $ABC$. Incircle of the triangle $ABM$ is tangent to the side $AB$ at the point $D$. Incircle of the triangle $ACM$ is tangent to the side $AC$ at the point $E$. Let $F$ be the such point, that the quadrilateral $DMEF$ is a parallelogram. Prove that $F$ lies on the bisector of $\angle BAC$.

2020 Polish 2nd Round P4
Let $ABCDEF$ be a such convex hexagon that
$AB=CD=EF\; \text{and} \; BC=DE=.FA$ Prove that if $\sphericalangle FAB + \sphericalangle ABC=\sphericalangle FAB + \sphericalangle EFA = 240^{\circ}$, then $\sphericalangle FAB+\sphericalangle CDE=240^{\circ}$.

2021 Polish 2nd Round P2

The point P lies on the side $CD$ of the parallelogram $ABCD$ with $\angle DBA = \angle CBP$. Point $O$ is the center of the circle passing through the points $D$ and $P$ and tangent to the straight line $AD$ at point $D$. Prove that $AO = OC$.