Navvye/Olympiad-Problems-CSV huggingface dataset:
2014 Sharygin Geometry Olympiad
Circles $\omega_1$ and $\omega_2$ meet at points $A$ and $B$. Let points $K_1$ and $K_2 $ of $\omega_1$ and $\omega_2$ respectively be such that $K_1A$ touches $\omega_2$, and $K_2A$ touches $\omega_1$. The circumcircle of triangle $K_1BK_2$ meets lines $AK_1$ and $AK_2$ for the second time at points $L_1$ and $L_2$ respectively. Prove that $L_1$ and $L_2$ are equidistant from line $AB$.
2014 Sharygin Geometry Olympiad
Let $AC$ be a fixed chord of a circle $\omega$ with center $O$. Point $B$ moves along the arc $AC$. A fixed point $P$ lies on $AC$. The line passing through $P$ and parallel to $AO$ meets $BA$ at point $A_1$, the line passing through $P$ and parallel to $CO$ meets $BC$ at point $C_1$. Prove that the circumcenter of triangle $A_1BC_1$ moves along a straight line.
2014 Sharygin Geometry Olympiad
In a given disc, construct a subset such that its area equals the half of the disc area and its intersection with its reflection over an arbitrary diameter has the area equal to the quarter of the disc area.
2014 Sharygin Geometry Olympiad
Let $ABC$ be a non-isosceles triangle. The altitude from $A$, the bisector from $B$ and the median from $C$ concur at point $K$.

a) Which of the sidelengths of the triangle is medial (intermediate in length)?

b) Which of the lengths of segments $AK, BK, CK$ is medial (intermediate in length)?
2014 Sharygin Geometry Olympiad
Given a triangle $ABC$ and an arbitrary point $D$.The lines passing through $D$ and perpendicular to segments $DA, DB, DC$ meet lines $BC, AC, AB$ at points $A_1, B_1, C_1$ respectively. Prove that the midpoints of segments $AA_1, BB_1, CC_1$ are collinear.
2014 Sharygin Geometry Olympiad
Let $AC$ be the hypothenuse of a right-angled triangle $ABC$. The bisector $BD$ is given, and the midpoints $E$ and $F$ of the arcs $BD$ of the circumcircles of triangles $ADB$ and $CDB$ respectively are marked (the circles are erased). Construct the centers of these circles using only a ruler.
2014 Sharygin Geometry Olympiad
Let $I$ be the incenter of a circumscribed quadrilateral $ABCD$. The tangents to circle $AIC$ at points $A, C$ meet at point $X$. The tangents to circle $BID$ at points $B, D$ meet at point $Y$ . Prove that $X, I, Y$ are collinear.
2014 Sharygin Geometry Olympiad
Two circles $\omega_1$ and $\omega_2$ touch externally at point $P$.Let $A$ be a point on $\omega_2$ not lying on the line through the centres of the two circles.Let $AB$ and $AC$ be the tangents to $\omega_1$.Lines $BP$ and $CP$ meet  $\omega_2$  for the second time at points $E$ and $F$.Prove that the line $EF$,the tangent to $\omega_2$ at $A$ and the common tangent at $P$ concur.
2014 Sharygin Geometry Olympiad
A quadrilateral $KLMN$ is given. A circle with center $O$ meets its side $KL$ at points $A$ and $A_1$, side $LM$ at points $B$ and $B_1$, etc. Prove that if the circumcircles of triangles $KDA, LAB, MBC$ and $NCD$ concur at point $P$, then

a) the circumcircles of triangles $KD_1A_1, LA_1B_1, MB_1C_1$ and $NC1D1$ also concur at some point $Q$;

b) point $O$ lies on the perpendicular bisector to $PQ$.
2014 Sharygin Geometry Olympiad
Let $ABCD$ be a circumscribed quadrilateral. Its incircle $\omega$ touches the sides $BC$ and $DA$ at points $E$ and $F$ respectively. It is known that lines $AB,FE$ and $CD$ concur. The circumcircles of triangles $AED$ and $BFC$ meet $\omega$ for the second time at points $E_1$ and $F_1$. Prove that $EF$ is parallel to $E_1 F_1$.
2014 Sharygin Geometry Olympiad
Does there exist a convex polyhedron such that it has diagonals and each of them is shorter than each of its edges?
2014 Sharygin Geometry Olympiad
Let $A, B, C$ and $D$ be a triharmonic quadruple of points, i.e $AB\cdot CD = AC \cdot BD = AD \cdot BC.$ Let $A_1$ be a point distinct from $A$ such that the quadruple $A_1, B, C$ and $D$ is triharmonic. Points $B_1, C_1$ and $D_1$ are defined similarly. Prove that

a) $A, B, C_1, D_1$ are concyclic;

b) the quadruple $A_1, B_1, C_1, D_1$ is triharmonic.
2014 Sharygin Geometry Olympiad
A circumscribed pyramid $ABCDS$ is given. The opposite sidelines of its base meet at points $P$ and $Q$ in such a way that $A$ and $B$ lie on segments $PD$ and $PC$ respectively. The inscribed sphere touches faces $ABS$ and $BCS$ at points $K$ and $L$. Prove that if $PK$ and $QL$ are complanar then the touching point of the sphere with the base lies on $BD$.
2014 Sharygin Geometry Olympiad
The incircle of a right-angled triangle $ABC$ touches its catheti $AC$ and $BC$ at points $B_1$ and $A_1$, the hypotenuse touches the incircle at point $C_1$. Lines $C_1A_1$ and $C_1B_1$ meet $CA$ and $CB$ respectively at points $B_0$ and $A_0$. Prove that $AB_0 = BA_0$.



(J. Zajtseva, D. Shvetsov )
2014 Sharygin Geometry Olympiad
Let $AH_a$ and $BH_b$ be altitudes, $AL_a$ and $BL_b$ be angle bisectors of a triangle $ABC$. It is known that $H_aH_b // L_aL_b$. Is it necessarily true that $AC = BC$?



(B. Frenkin)
2014 Sharygin Geometry Olympiad
Points $M$ and $N$ are the midpoints of sides $AC$ and $BC$ of a triangle $ABC$. It is known that $\angle MAN = 15^o$ and $\angle BAN = 45^o$. Find the value of angle $\angle ABM$.



(A. Blinkov)
2014 Sharygin Geometry Olympiad
Tanya has cut out a triangle from checkered paper as shown in the picture. The lines of the grid have faded. Can Tanya restore them without any instruments only folding the triangle (she remembers the triangle sidelengths)?



(T. Kazitsyna)
2014 Sharygin Geometry Olympiad
A triangle with angles of $30, 70$ and $80$ degrees is given. Cut it by a straight line into two triangles in such a way that an angle bisector in one of these triangles and a median in the other one drawn from two endpoints of the cutting segment are parallel to each other. (It suffices to find one such cutting.)



(A. Shapovalov )
2014 Sharygin Geometry Olympiad
Two circles $k_1$ and $k_2$ with centers $O_1$ and $O_2$ are tangent to each other externally at point $O$. Points $X$ and $Y$ on $k_1$ and $k_2$ respectively are such that rays $O_1X$ and $O_2Y$ are parallel and codirectional. Prove that two tangents from $X$ to $k_2$ and two tangents from $Y$ to $k_1$ touch the same circle passing through $O$.



(V. Yasinsky)
2014 Sharygin Geometry Olympiad
Two points on a circle are joined by a broken line shorter than the diameter of the circle. Prove that there exists a diameter which does not intersect this broken line.



(Folklor )
2014 Sharygin Geometry Olympiad
Let $M$ be the midpoint of the chord $AB$ of a circle centered at $O$. Point $K$ is symmetric to $M$ with respect to $O$, and point $P$ is chosen arbitrarily on the circle. Let $Q$ be the intersection of the line perpendicular to $AB$ through $A$ and the line perpendicular to $PK$ through $P$. Let $H$ be the projection of $P$ onto $AB$. Prove that $QB$ bisects $PH$.



(Tran Quang Hung)
2014 Sharygin Geometry Olympiad
Let ABCD be a cyclic quadrilateral. Prove that $AC > BD$ if and only if $(AD-BC)(AB- CD) > 0$.



(V. Yasinsky)
2014 Sharygin Geometry Olympiad
In a quadrilateral $ABCD$ angles $A$ and $C$ are right. Two circles with diameters $AB$ and $CD$ meet at points $X$ and $Y$ . Prove that line $XY$ passes through the midpoint of $AC$.



(F. Nilov )
2014 Sharygin Geometry Olympiad
An acute angle $A$ and a point $E$ inside it are given. Construct points $B, C$ on the sides of the angle such that $E$ is the center of the Euler circle of triangle $ABC$.



(E. Diomidov)
2014 Sharygin Geometry Olympiad
Let $H$ be the orthocenter of a triangle $ABC$. Given that $H$ lies on the incircle of $ABC$ , prove that three circles with centers $A, B, C$ and radii $AH, BH, CH$ have a common tangent.



(Mahdi Etesami Fard)
2014 Sharygin Geometry Olympiad
In triangle $ABC$ $\angle B = 60^o, O$ is the circumcenter, and $L$ is the foot of an angle bisector of angle $B$. The circumcirle of triangle $BOL$ meets the circumcircle of $ABC$ at point $D \ne B$. Prove that $BD \perp AC$.



(D. Shvetsov)
2014 Sharygin Geometry Olympiad
Let $I$ be the incenter of triangle $ABC$, and $M, N$ be the midpoints of arcs $ABC$ and $BAC$ of its circumcircle. Prove that points $M, I, N$ are collinear if and only if$ AC + BC = 3AB$.



(A. Polyansky)
2014 Sharygin Geometry Olympiad
Nine circles are drawn around an arbitrary triangle as in the figure. All circles tangent to the same side of the triangle have equal radii. Three lines are drawn, each one connecting one of the triangle’s vertices to the center of one of the circles touching the opposite side, as in the figure. Show that the three lines are concurrent.



(N. Beluhov)
2014 Sharygin Geometry Olympiad
A convex polygon $P$ lies on a flat wooden table. You are allowed to drive some nails into the table. The nails must not go through $P$, but they may touch its boundary. We say that a set of nails blocks $P$ if the nails make it impossible to move $P$ without lifting it off the table. What is the minimum number of nails that suffices to block any convex polygon $P$?



(N. Beluhov, S. Gerdgikov)
2014 Sharygin Geometry Olympiad
The vertices and the circumcenter of an isosceles triangle lie on four different sides of a square. Find the angles of this triangle.



(I. Bogdanov, B. Frenkin)
2014 Sharygin Geometry Olympiad
A circle, its chord $AB$ and the midpoint $W$ of the minor arc $AB$ are given. Take an arbitrary point $C$ on the major arc $AB$. The tangent to the circle at $C$ meets the tangents at $A$ and $B$ at points $X$ and $Y$ respectively. Lines $WX$ and WY meet AB at points $N$ and $M$ respectively. Prove that the length of segment $NM$ does not depend on point $C$.



(A. Zertsalov, D. Skrobot)
2014 Sharygin Geometry Olympiad
Do there exist convex polyhedra with an arbitrary number of diagonals (a diagonal is a segment joining two vertices of a polyhedron and not lying on the surface of this polyhedron)?



(A. Blinkov)
2014 Sharygin Geometry Olympiad
Let $ABC$ be a fixed triangle in the plane. Let $D$ be an arbitrary point in the plane. The circle with center $D$, passing through $A$, meets $AB$ and $AC$ again at points $A_b$ and $A_c$ respectively. Points $B_a, B_c, C_a$ and $C_b$ are defined similarly. A point $D$ is called good if the points $A_b, A_c,B_a, B_c, C_a$, and $C_b$ are concyclic. For a given triangle $ABC$, how many good points can there be?



(A. Garkavyj, A. Sokolov )
2014 Sharygin Geometry Olympiad
The altitude from one vertex of a triangle, the bisector from the another one and the median from the remaining vertex were drawn, the common points of these three lines were marked, and after this everything was erased except three marked points. Restore the triangle. (For every two erased segments, it is known which of the three points was their intersection point.)



(A. Zaslavsky)
2014 Sharygin Geometry Olympiad
The incircle of a non-isosceles triangle $ABC$ touches $AB$ at point $C'$. The circle with diameter $BC'$ meets the incircle and the bisector of angle $B$ again at points $A_1$ and $A_2$ respectively. The circle with diameter $AC'$ meets the incircle and the bisector of angle $A$ again at points $B_1$ and $B_2$ respectively. Prove that lines $AB, A_1B_1, A_2B_2$ concur.



(E. H. Garsia)
2014 Sharygin Geometry Olympiad
Prove that the smallest dihedral angle between faces of an arbitrary tetrahedron is not greater than the dihedral angle between faces of a regular tetrahedron.



(S. Shosman, O. Ogievetsky)
2014 Sharygin Geometry Olympiad
Given is a cyclic quadrilateral $ABCD$. The point $L_a$ lies in the interior of $BCD$ and is such that its distances to the sides of this triangle are proportional to the lengths of corresponding sides. The points $L_b, L_c$, and $L_d$ are defined analogously. Given that the quadrilateral $L_aL_bL_cL_d$ is cyclic, prove that the quadrilateral $ABCD$ has two parallel sides.



(N. Beluhov)
2013 Sharygin Geometry Olympiad
Let $ABC$ be an isosceles triangle with $AB = BC$. Point $E$ lies on the side $AB$, and $ED$ is the perpendicular from $E$ to $BC$. It is known that $AE = DE$. Find $\angle DAC$.
2013 Sharygin Geometry Olympiad
Let $ABC$ be an isosceles triangle ($AC = BC$) with $\angle C = 20^\circ$. The bisectors of angles $A$ and $B$ meet the opposite sides at points $A_1$ and $B_1$ respectively. Prove that the triangle $A_1OB_1$ (where $O$ is the circumcenter of $ABC$) is regular.
2013 Sharygin Geometry Olympiad
Let $ABC$ be a right-angled triangle ($\angle B = 90^\circ$). The excircle inscribed into the angle $A$ touches the extensions of the sides $AB$, $AC$ at points $A_1, A_2$ respectively; points $C_1, C_2$ are defined similarly. Prove that the perpendiculars from $A, B, C$ to $C_1C_2, A_1C_1, A_1A_2$ respectively, concur.
2013 Sharygin Geometry Olympiad
Let $ABC$ be a nonisosceles triangle. Point $O$ is its circumcenter, and point $K$ is the center of the circumcircle $w$ of triangle $BCO$. The altitude of $ABC$ from $A$ meets $w$ at a point $P$. The line $PK$ intersects the circumcircle of $ABC$ at points $E$ and $F$. Prove that one of the segments $EP$ and $FP$ is equal to the segment $PA$.
2013 Sharygin Geometry Olympiad
Four segments drawn from a given point inside a convex quadrilateral to its vertices, split the quadrilateral into four equal triangles. Can we assert that this quadrilateral is a rhombus?
2013 Sharygin Geometry Olympiad
Diagonals $AC$ and $BD$ of a trapezoid $ABCD$ meet at $P$. The circumcircles of triangles $ABP$ and $CDP$ intersect the line $AD$ for the second time at points $X$ and $Y$ respectively. Let $M$ be the midpoint of segment $XY$. Prove that $BM = CM$.
2013 Sharygin Geometry Olympiad
Let $BD$ be a bisector of triangle $ABC$. Points $I_a$, $I_c$ are the incenters of triangles $ABD$, $CBD$ respectively. The line $I_aI_c$ meets $AC$ in point $Q$. Prove that $\angle DBQ = 90^\circ$.
2013 Sharygin Geometry Olympiad
Let $X$ be an arbitrary point inside the circumcircle of a triangle $ABC$. The lines $BX$ and $CX$ meet the circumcircle in points $K$ and $L$ respectively. The line $LK$ intersects $BA$ and $AC$ at points $E$ and $F$ respectively. Find the locus of points $X$ such that the circumcircles of triangles $AFK$ and $AEL$ touch.
2013 Sharygin Geometry Olympiad
Let $T_1$ and $T_2$ be the points of tangency of the excircles of a triangle $ABC$ with its sides $BC$ and $AC$ respectively. It is known that the reflection of the incenter of $ABC$ across the midpoint of $AB$ lies on the circumcircle of triangle $CT_1T_2$. Find $\angle BCA$.
2013 Sharygin Geometry Olympiad
The incircle of triangle $ABC$ touches the side $AB$ at point $C'$; the incircle of triangle $ACC'$ touches the sides $AB$ and $AC$ at points $C_1, B_1$; the incircle of triangle $BCC'$ touches the sides $AB$ and $BC$ at points $C_2$, $A_2$. Prove that the lines $B_1C_1$, $A_2C_2$, and $CC'$ concur.
2013 Sharygin Geometry Olympiad
a) Let $ABCD$ be a convex quadrilateral and $r_1 \le r_2 \le r_3 \le r_4$ be the radii of the incircles of triangles $ABC, BCD, CDA, DAB$. Can the inequality $r_4 > 2r_3$ hold?



b) The diagonals of a convex quadrilateral $ABCD$ meet in point $E$. Let $r_1 \le r_2 \le r_3 \le r_4$ be the radii of the incircles of triangles $ABE, BCE, CDE, DAE$. Can the inequality $r_2 > 2r_1$ hold?
2013 Sharygin Geometry Olympiad
On each side of triangle $ABC$, two distinct points are marked. It is known that these points are the feet of the altitudes and of the bisectors.



a) Using only a ruler determine which points are the feet of the altitudes and which points are the feet of the bisectors.



b) Solve p.a) drawing only three lines.
2013 Sharygin Geometry Olympiad
Let $A_1$ and $C_1$ be the tangency points of the incircle of triangle $ABC$ with $BC$ and $AB$ respectively, $A'$ and $C'$ be the tangency points of the excircle inscribed into the angle $B$ with the extensions of $BC$ and $AB$ respectively. Prove that the orthocenter $H$ of triangle $ABC$ lies on $A_1C_1$ if and only if the lines $A'C_1$ and $BA$ are orthogonal.
2013 Sharygin Geometry Olympiad
Let $M$, $N$ be the midpoints of diagonals $AC$, $BD$ of a right-angled trapezoid $ABCD$ ($\measuredangle A=\measuredangle D = 90^\circ$).

The circumcircles of triangles $ABN$, $CDM$ meet the line $BC$ in the points $Q$, $R$.

Prove that the distances from $Q$, $R$ to the midpoint of $MN$ are equal.
2013 Sharygin Geometry Olympiad
(a) Triangles $A_1B_1C_1$ and $A_2B_2C_2$ are inscribed into triangle $ABC$ so that $C_1A_1 \perp BC$, $A_1B_1 \perp CA$, $B_1C_1 \perp AB$, $B_2A_2 \perp BC$, $C_2B_2 \perp CA$, $A_2C_2 \perp AB$.  Prove that these triangles are equal.



(b) Points $A_1$, $B_1$, $C_1$, $A_2$, $B_2$, $C_2$ lie inside a triangle $ABC$ so that $A_1$ is on segment $AB_1$, $B_1$ is on segment $BC_1$, $C_1$ is on segment $CA_1$, $A_2$ is on segment $AC_2$, $B_2$ is on segment $BA_2$, $C_2$ is on segment $CB_2$, and the angles $BAA_1$, $CBB_2$, $ACC_1$, $CAA_2$, $ABB_2$, $BCC_2$ are equal.  Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ are equal.
2013 Sharygin Geometry Olympiad
The incircle of triangle $ABC$ touches $BC$, $CA$, $AB$ at points $A_1$, $B_1$, $C_1$, respectively.  The perpendicular from the incenter $I$ to the median from vertex $C$ meets the line $A_1B_1$ in point $K$. Prove that $CK$ is parallel to $AB$.
2013 Sharygin Geometry Olympiad
An acute angle between the diagonals of a cyclic quadrilateral is equal to $\phi$. Prove that an acute angle between the diagonals of any other quadrilateral having the same sidelengths is smaller than $\phi$.
2013 Sharygin Geometry Olympiad
Let $AD$ be a bisector of triangle $ABC$. Points $M$ and $N$ are projections of $B$ and $C$ respectively to $AD$. The circle with diameter $MN$ intersects $BC$ at points $X$ and $Y$. Prove that $\angle BAX = \angle CAY$.
2013 Sharygin Geometry Olympiad
a) The incircle of a triangle $ABC$ touches $AC$ and $AB$ at points $B_0$ and $C_0$ respectively. The bisectors of angles $B$ and $C$ meet the perpendicular bisector to the bisector $AL$ in points $Q$ and $P$ respectively. Prove that the lines $PC_0, QB_0$ and $BC$ concur.



b) Let $AL$ be the bisector of a triangle $ABC$. Points $O_1$ and $O_2$ are the circumcenters of triangles $ABL$ and $ACL$ respectively. Points $B_1$ and $C_1$ are the projections of $C$ and $B$ to the bisectors of angles $B$ and $C$ respectively. Prove that the lines $O_1C_1, O_2B_1,$ and $BC$ concur.



c) Prove that the two points obtained in pp. a) and b) coincide.
2013 Sharygin Geometry Olympiad
Let $C_1$ be an arbitrary point on the side $AB$ of triangle $ABC$. Points $A_1$ and $B_1$ on the rays $BC$ and $AC$ are such that $\angle AC_1B_1 = \angle BC_1A_1 = \angle ACB$. The lines $AA_1$ and $BB_1$ meet in point $C_2$. Prove that all the lines $C_1C_2$ have a common point.
2013 Sharygin Geometry Olympiad
Chords $BC$ and $DE$ of circle $\omega$ meet at point $A$.  The line through $D$ parallel to $BC$ meets $\omega$ again at $F$, and $FA$ meets $\omega$ again at $T$.  Let $M = ET \cap BC$ and let $N$ be the reflection of $A$ over $M$.  Show that $(DEN)$ passes through the midpoint of $BC$.
2013 Sharygin Geometry Olympiad
The common perpendiculars to the opposite sidelines of a nonplanar quadrilateral are mutually orthogonal. Prove that they intersect.
2013 Sharygin Geometry Olympiad
Two convex polytopes $A$ and $B$ do not intersect. The polytope $A$ has exactly $2012$ planes of symmetry. What is the maximal number of symmetry planes of the union of $A$ and $B$, if $B$ has a) $2012$, b) $2013$ symmetry planes?



c) What is the answer to the question of p.b), if the symmetry planes are replaced by the symmetry axes?
2013 Sharygin Geometry Olympiad
Let $ABCDE$ be a pentagon with right angles at vertices $B$ and $E$ and such that $AB = AE$ and $BC = CD = DE$. The diagonals $BD$ and $CE$ meet at point $F$. Prove that $FA = AB$.
2013 Sharygin Geometry Olympiad
Two circles with centers $O_1$ and $O_2$ meet at points $A$ and $B$. The bisector of angle $O_1AO_2$ meets the circles for the second time at points $C $and $D$. Prove that the distances from the circumcenter of triangle $CBD$ to $O_1$ and to $O_2$ are equal.
2013 Sharygin Geometry Olympiad
Each vertex of a convex polygon is projected to all nonadjacent sidelines. Can it happen that each of these projections lies outside the corresponding side?
2013 Sharygin Geometry Olympiad
The diagonals of a convex quadrilateral $ABCD$ meet at point $L$. The orthocenter $H$ of the triangle $LAB$ and the circumcenters $O_1, O_2$, and $O_3$ of the triangles $LBC, LCD$, and $LDA$ were marked. Then the whole configuration except for points $H, O_1, O_2$, and $O_3$ was erased. Restore it using a compass and a ruler.
2013 Sharygin Geometry Olympiad
The altitude $AA'$, the median $BB'$, and the angle bisector $CC'$ of a triangle $ABC$ are concurrent at point $K$. Given that $A'K = B'K$, prove that $C'K = A'K$.
2013 Sharygin Geometry Olympiad
Dear Mathlinkers,



1. A, B the end points of an arch circle

2. (O) a circle tangent to AB intersecting the arch in question

3. T the point of contact of (O) and AB

4. C, D the points of intersection of (O) with the arch in the order A, D, C, B

5. E, F the points of intersection of AC and DT, BD and CT.



Prove :	EF is parallel to AB.



Sincerely

Jean-Louis
2013 Sharygin Geometry Olympiad
In the plane, four points are marked. It is known that these points are the centers of four circles, three of which are pairwise externally tangent, and all these three are internally tangent to the fourth one. It turns out, however, that it is impossible to determine which of the marked points is the center of the fourth (the largest) circle. Prove that these four points are the vertices of a rectangle.
2013 Sharygin Geometry Olympiad
Let P be an arbitrary point on the arc $AC$ of the circumcircle of a fixed triangle $ABC$, not containing $B$. The bisector of angle $APB$ meets the bisector of angle $BAC$ at point $P_a$ the bisector of angle $CPB$ meets the bisector of angle $BCA$ at point $P_c$. Prove that for all points $P$, the circumcenters of triangles $PP_aP_c$ are collinear.



by I. Dmitriev
2013 Sharygin Geometry Olympiad
All angles of a cyclic pentagon $ABCDE$ are obtuse. The sidelines $AB$ and $CD$ meet at point $E_1$, the sidelines $BC$ and $DE$ meet at point $A_1$. The tangent at $B$ to the circumcircle of the triangle $BE_1C$ meets the circumcircle $\omega$ of the pentagon for the second time at point $B_1$. The tangent at $D$ to the circumcircle of the triangle $DA_1C$ meets  $\omega$  for the second time at point $D_1$. Prove that $B_1D_1 // AE$
2013 Sharygin Geometry Olympiad
Two circles $\omega_1$ and $\omega_2$ with centers $O_1$ and $O_2$ meet at points $A$ and $B$. Points $C$ and $D$ on $\omega_1$ and $\omega_2$, respectively, lie on the opposite sides of the line $AB$ and are equidistant from this line. Prove that $C$ and $D$ are equidistant from the midpoint of $O_1O_2$.
2013 Sharygin Geometry Olympiad
Each sidelength of a convex quadrilateral $ABCD$ is not less than $1$ and not greater than $2$. The diagonals of this quadrilateral meet at point $O$. Prove that $S_{AOB}+ S_{COD} \le  2(S_{AOD}+ S_{BOC})$.
2013 Sharygin Geometry Olympiad
A point $F$ inside a triangle $ABC$ is chosen so that $\angle AFB = \angle BFC = \angle CFA$. The line passing through $F$ and perpendicular to $BC$ meets the median from $A$ at point $A_1$. Points $B_1$ and $C_1$ are defined similarly. Prove that the points $A_1, B_1$, and $C_1$ are three vertices of some regular hexagon, and that the three remaining vertices of that hexagon lie on the sidelines of $ABC$.
2013 Sharygin Geometry Olympiad
Points $E$ and $F$ lie on the sides $AB$ and $AC$ of a triangle $ABC$. Lines $EF$ and $BC$ meet at point $S$. Let $M$ and $N$ be the midpoints of $BC$ and $EF$, respectively. The line passing through $A$ and parallel to $MN$ meets $BC$ at point $K$. Prove that $\frac{BK}{CK}=\frac{FS}{ES}$ .

.
2013 Sharygin Geometry Olympiad
A line $\ell$ passes through the vertex $B$ of a regular triangle $ABC$. A circle $\omega_a$ centered at $I_a$ is tangent to $BC$ at point $A_1$, and is also tangent to the lines $\ell$  and $AC$. A circle $\omega_c$ centered at $I_c$ is tangent to $BA$ at point $C_1$, and is also tangent to the lines $\ell$  and $AC$.  Prove that the orthocenter of triangle $A_1BC_1$ lies on the line $I_aI_c$.