Navvye/Olympiad-Problems-CSV huggingface dataset:
2016 Sharygin Geometry Olympiad
A trapezoid $ABCD$ with bases $AD$ and $BC$ is such that $AB = BD$. Let $M$ be the midpoint of $DC$. Prove that $\angle MBC$ = $\angle BCA$.
2016 Sharygin Geometry Olympiad
Mark three nodes on a cellular paper so that the semiperimeter of the obtained triangle would be equal to the sum of its two smallest medians.



(Proposed by L.Emelyanov)
2016 Sharygin Geometry Olympiad
Let $AH_1$, $BH_2$ be two altitudes of an acute-angled triangle $ABC$ , $D$ be the projection of $H_1$ to $AC$, $E$ be the projection of $D$ to $AB$, $F$ be the common point of $ED$ and $AH_1$.

Prove that $H_2F \parallel BC$.



(Proposed by E.Diomidov)
2016 Sharygin Geometry Olympiad
In quadrilateral $ABCD$, $\angle B = \angle D = 90$ and $AC = BC + DC$. Point $P$ of ray $BD$ is such that $BP = AD$. Prove that line $CP$ is parallel to the bisector of angle $ABD$.



(Proposed by A.Trigub)
2016 Sharygin Geometry Olympiad
In quadrilateral $ABCD$, $AB = CD$, $M$ and $K$ are the midpoints of $BC$ and $AD$.Prove that the angle between $MK$ and $AC$ is equal to the half-sum of angles $BAC$ and $DCA$



(Proposed by M.Volchkevich)
2016 Sharygin Geometry Olympiad
Let $M$ be the midpoint of side $AC$ of triangle $ABC$, $MD$ and $ME$ be the perpendiculars from $M$ to $AB$ and $BC$ respectively. Prove that the distance between the circumcenters of triangles $ABE$ and $BCD$ is equal to $AC/4$



(Proposed by M.Volchkevich)
2016 Sharygin Geometry Olympiad
Let all distances between the vertices of a convex $n$-gon ($n > 3$) be

different.

a) A vertex is called uninteresting if the closest vertex is adjacent to it. What is the

minimal possible number of uninteresting vertices (for a given $n$)?

b) A vertex is called unusual if the farthest vertex is adjacent to it. What is the maximal

possible number of unusual vertices (for a given $n$)?



(Proposed by B.Frenkin)
2016 Sharygin Geometry Olympiad
Let $ABCDE$ be an inscribed pentagon such that $\angle B +\angle E = \angle C +\angle D$.Prove that $\angle CAD < \pi/3 < \angle A$.



(Proposed by B.Frenkin)
2016 Sharygin Geometry Olympiad
Let $ABC$ be a right-angled triangle and $CH$ be the altitude from its right angle $C$. Points $O_1$ and $O_2$ are the incenters of triangles $ACH$ and $BCH$ respectively,  $P_1$ and $P_2$ are the touching points of their incircles with $AC$ and $BC$. Prove that lines $O_1P_1$ and $O_2P_2$ meet on $AB$.
2016 Sharygin Geometry Olympiad
Point $X$ moves along side $AB$ of triangle $ABC$, and point $Y$ moves along its circumcircle in such a way that line $XY$ passes through the midpoint of arc $AB$. Find the locus of the circumcenters of triangles $IXY$ , where I is the incenter of $ ABC$.
2016 Sharygin Geometry Olympiad
Restore a triangle $ABC$ by vertex $B$, the centroid and the common point of the symmedian from $B$ with the circumcircle.
2016 Sharygin Geometry Olympiad
Let $BB_1$ be the symmedian of a nonisosceles acute-angled triangle $ABC$. Ray $BB_1$ meets the circumcircle of $ABC$ for the second time at point $L$. Let $AH_A, BH_B, CH_C$ be the altitudes of triangle $ABC$. Ray $BH_B$ meets the circumcircle of $ABC$ for the second time at point $T$. Prove that $H_A, H_C, T, L$ are concyclic.
2016 Sharygin Geometry Olympiad
$L$ is a Line that intersect with the side $AB,BC,AC$ of triangle $ABC$ at $F,D,E$

The line perpendicular to $BC$ from $D$ intersect $AB,AC$ at $A_{1},A_{2}$ respectively

Name $B_{1},B_{2},C_{1},C_{2}$ similarly

Prove that the circumcenters of $AA_{1}A_{2},BB_{1}B_{2},CC_{1}C_{2}$ are collinear
2016 Sharygin Geometry Olympiad
Let a triangle $ABC$ be given. Consider the circle touching its circumcircle at $A$ and touching externally its incircle at some point $A_1$. Points $B_1$ and $C_1$ are defined similarly.

a) Prove that lines $AA_1, BB_1$ and $CC1$ concur.

b) Let $A_2$ be the touching point of the incircle with $BC$. Prove that lines $AA_1$ and $AA_2$ are symmetric about the bisector of angle $\angle A$.
2016 Sharygin Geometry Olympiad
Let $O, M, N$ be the circumcenter, the centroid and the Nagel point of a triangle. Prove that angle $MON$ is right if and only if one of the triangle’s angles is equal to $60^o$.
2016 Sharygin Geometry Olympiad
Let $BB_1$ and $CC_1$ be altitudes of triangle $ABC$. The tangents to the circumcircle of $AB_1C_1$ at $B_1$ and $C_1$ meet AB and $AC$ at points $M$ and $N$ respectively. Prove that the common point of circles $AMN$ and $AB_1C_1$ distinct from $A$ lies on the Euler line of $ABC$.
2016 Sharygin Geometry Olympiad
Let $D$ be an arbitrary point on side $BC$ of triangle $ABC$. Circles $\omega_1$ and $\omega_2$ pass through $A$ and $D$ in such a way that $BA$ touches $\omega_1$ and $CA$ touches $\omega_2$. Let $BX$ be the second tangent from $B$ to $\omega_1$, and $CY$ be the second tangent from $C$ to $\omega_2$. Prove that the circumcircle of triangle $XDY$ touches $BC$.
2016 Sharygin Geometry Olympiad
Let $ABC$ be a triangle with $\angle C=90^{\circ}$, and $K, L $ be the midpoints of the minor arcs AC and BC of its circumcircle. Segment $KL$ meets $AC$,at point $N$. Find angle $NIC$ where $I$is the incenter of $ABC$.
2016 Sharygin Geometry Olympiad
Let $ABCDEF$ be a regular hexagon. Points $P$ and $Q$ on tangents to its circumcircle at $A$ and $D$ respectively are such that $PQ$ touches the minor arc $EF$ of this circle. Find the angle between $PB$ and $QC$.
2016 Sharygin Geometry Olympiad
The incircle $\omega$ of a triangle $ABC$ touches $BC, AC$ and $AB$ at points $A_0, B_0$ and $C_0$ respectively. The bisectors of angles $B$ and $C$ meet the perpendicular bisector to segment $AA_0$ at points $Q$ and $P$ respectively. Prove that $PC_0$ and $QB_0$ meet on $\omega$ .
2016 Sharygin Geometry Olympiad
The areas of rectangles $P$ and $Q$ are equal, but the diagonal of $P$ is greater. Rectangle $Q$ can be covered by two copies of $P$. Prove that $P$ can be covered by two copies of $Q$.
2016 Sharygin Geometry Olympiad
Let $M_A, M_B, M_C$ be the midpoints of the sides $BC, CA, AB$ respectively of a non-isosceles triangle $ABC$. Points $H_A, H_B, H_C$ lie on the corresponding sides, different from $M_A, M_B, M_C$ such that $M_AH_B=M_AH_C, $ $M_BH_A=M_BH_C,$ and $M_CH_A=M_CH_B$. Prove that $H_A, H_B, H_C$ are the feet of the corresponding altitudes.
2016 Sharygin Geometry Olympiad
A sphere touches all edges of a tetrahedron. Let $a, b, c$ and d be the segments of the tangents to the sphere from the vertices of the tetrahedron. Is it true that that some of these segments necessarily form a triangle?



(It is not obligatory to use all segments. The side of the triangle can be formed by two segments)
2016 Sharygin Geometry Olympiad
A sphere is inscribed into a prism $ABCA'B'C'$ and touches its lateral faces $BCC'B', CAA'C', ABB'A' $ at points $A_o, B_o, C_o$ respectively. It is known that $\angle A_oBB' = \angle B_oCC' =\angle C_oAA'$.

a) Find all possible values of these angles.

b) Prove that segments $AA_o, BB_o, CC_o$ concur.

c) Prove that the projections of the incenter to $A'B', B'C', C'A'$ are the vertices of a regular triangle.
2015 Sharygin Geometry Olympiad
In trapezoid $ABCD$ angles $A$ and $B$ are right, $AB = AD, CD = BC + AD, BC < AD$. Prove that $\angle ADC = 2\angle ABE$, where $E$ is the midpoint of segment $AD$.



(V. Yasinsky)
2015 Sharygin Geometry Olympiad
A circle passing through $A, B$ and the orthocenter of triangle $ABC$ meets sides $AC, BC$ at their inner points. Prove that $60^o < \angle C < 90^o$ .



(A. Blinkov)
2015 Sharygin Geometry Olympiad
In triangle $ABC$ we have $AB = BC, \angle B = 20^o$. Point $M$ on $AC$ is such that $AM : MC = 1 : 2$, point $H$ is the projection of $C$ to $BM$. Find angle $AHB$.



(M. Yevdokimov)
2015 Sharygin Geometry Olympiad
Prove that an arbitrary convex quadrilateral can be divided into five polygons having symmetry axes.



(N. Belukhov)
2015 Sharygin Geometry Olympiad
Two equal hard triangles are given. One of their angles is equal to $ \alpha$ (these angles are marked). Dispose these triangles on the plane in such a way that the angle formed by some three vertices would be equal to $ \alpha / 2$.

(No instruments are allowed, even a pencil.)



(E. Bakayev, A. Zaslavsky)
2015 Sharygin Geometry Olympiad
Lines $b$ and $c$ passing through vertices $B$ and $C$ of triangle $ABC$ are perpendicular to sideline $BC$. The perpendicular bisectors to $AC$ and $AB$ meet $b$ and $c$ at points $P$ and $Q$ respectively. Prove that line $PQ$ is perpendicular to median $AM$ of triangle $ABC$.



(D. Prokopenko)
2015 Sharygin Geometry Olympiad
Point $M$ on side $AB$ of quadrilateral $ABCD$ is such that quadrilaterals $AMCD$ and $BMDC$ are circumscribed around circles centered at $O_1$ and $O_2$ respectively. Line $O_1O_2$ cuts an isosceles triangle with vertex M from angle $CMD$. Prove that $ABCD$ is a cyclic quadrilateral.



(M. Kungozhin)
2015 Sharygin Geometry Olympiad
Points $C_1, B_1$ on sides $AB, AC$ respectively of triangle $ABC$ are such that $BB_1 \perp  CC_1$. Point $X$ lying inside the triangle is such that $\angle XBC = \angle B_1BA, \angle XCB = \angle C_1CA$. Prove that $\angle B_1XC_1 =90^o- \angle A$.



(A. Antropov, A. Yakubov)
2015 Sharygin Geometry Olympiad
Circles $\alpha$ and  $\beta$ pass through point $C$. The tangent to $\alpha$  at this point meets $\beta$ at point $B$, and the tangent to  $\beta$ at $C$ meets $\alpha$  at point $A$ so that $A$ and $B$ are distinct from $C$ and angle $ACB$ is obtuse. Line $AB$ meets $\alpha$  and $\beta$ for the second time at points $N$ and $M$ respectively. Prove that $2MN < AB$.



(D. Mukhin)
2015 Sharygin Geometry Olympiad
A convex quadrilateral is given. Using a compass and a ruler construct a point such that its projections to the sidelines of this quadrilateral are the vertices of a parallelogram.



(A. Zaslavsky)
2015 Sharygin Geometry Olympiad
Let $100$ discs lie on the plane in such a way that each two of them have a common point. Prove that there exists a point lying inside at least $15$ of these discs.



(M. Kharitonov, A. Polyansky)
2015 Sharygin Geometry Olympiad
A fixed triangle $ABC$ is given. Point $P$ moves on its circumcircle so that segments $BC$ and $AP$ intersect. Line $AP$ divides triangle $BPC$ into two triangles with incenters $I_1$ and $I_2$. Line $I_1I_2$ meets $BC$ at point $Z$. Prove that all lines $ZP$ pass through a fixed point.



(R. Krutovsky, A. Yakubov)
2015 Sharygin Geometry Olympiad
Let $BM$ be a median of nonisosceles right-angled triangle $ABC$ ($\angle B = 90^o$), and $Ha, Hc$ be the orthocenters of triangles $ABM, CBM$ respectively. Prove that lines $AH_c$ and $CH_a$ meet on the medial line of triangle $ABC$.



(D. Svhetsov)
2015 Sharygin Geometry Olympiad
The diagonals of convex quadrilateral $ABCD$ are perpendicular. Points $A' , B' , C' , D' $ are the circumcenters of triangles $ABD, BCA, CDB, DAC$ respectively. Prove that lines $AA' , BB' , CC' , DD' $ concur.



(A. Zaslavsky)
2015 Sharygin Geometry Olympiad
Let $ABC$ be an acute-angled, nonisosceles triangle. Altitudes $AA'$ and $BB' $meet at point $H$, and the medians of triangle $AHB$ meet at point $M$. Line $CM$ bisects segment $A'B'$. Find angle $C$.



(D. Krekov)
2015 Sharygin Geometry Olympiad
A perpendicular bisector of side $BC$ of triangle $ABC$ meets lines $AB$ and $AC$ at points $A_B$ and $A_C$ respectively. Let $O_a$ be the circumcenter of triangle $AA_BA_C$. Points $O_b$ and $O_c$ are defined similarly. Prove that the circumcircle of triangle $O_aO_bO_c$ touches the circumcircle of the original triangle.
2015 Sharygin Geometry Olympiad
Let $K$ be an arbitrary point on side $BC$ of triangle $ABC$, and $KN$ be a bisector of triangle $AKC$. Lines $BN$ and $AK$ meet at point $F$, and lines $CF$ and $AB$ meet at point $D$. Prove that $KD$ is a bisector of triangle $AKB$.
2015 Sharygin Geometry Olympiad
Prove that an arbitrary triangle with area $1$ can be covered by an isosceles triangle with area less than $\sqrt{2}$.
2015 Sharygin Geometry Olympiad
Let $A_1$, $B_1$ and $C_1$ be the midpoints of sides $BC$, $CA$ and $AB$ of triangle $ABC$, respectively. Points $B_2$ and $C_2$ are the midpoints of segments $BA_1$ and $CA_1$ respectively. Point $B_3$ is symmetric to $C_1$ wrt $B$, and $C_3$ is symmetric to $B_1$ wrt $C$.

Prove that one of common points of circles $BB_2B_3$ and $CC_2C_3$ lies on the circumcircle of triangle $ABC$.
2015 Sharygin Geometry Olympiad
Let $AA_1$, $BB_1$, $CC_1$ be the altitudes of an acute-angled, nonisosceles triangle $ABC$, and $A_2$, $B_2$, $C_2$ be the touching points of sides $BC$, $CA$, $AB$ with the correspondent excircles. It is known that line $B_1C_1$ touches the incircle of $ABC$.

Prove that $A_1$ lies on the circumcircle of $A_2B_2C_2$.
2015 Sharygin Geometry Olympiad
Let $BM$ be a median of right-angled nonisosceles triangle $ABC$ ($\angle B = 90$), and $H_a$, $H_c$ be the orthocenters of triangles $ABM$, $CBM$ respectively. Lines $AH_c$ and $CH_a$ meet at point $K$.

Prove that $\angle MBK = 90$.
2015 Sharygin Geometry Olympiad
Let $H$ and $O$ be the orthocenter and the circumcenter of triangle $ABC$. The circumcircle of triangle $AOH$ meets the perpendicular bisector of $BC$ at point $A_1 \neq O$. Points $B_1$ and $C_1$ are defined similarly. Prove that lines $AA_1$, $BB_1$ and $CC_1$ concur.
2015 Sharygin Geometry Olympiad
Let $SABCD$ be an inscribed pyramid, and $AA_1$, $BB_1$, $CC_1$, $DD_1$ be the perpendiculars from $A$, $B$, $C$, $D$ to lines $SC$, $SD$, $SA$, $SB$ respectively. Points $S$, $A_1$, $B_1$, $C_1$, $D_1$ are distinct and lie on a sphere.

Prove that points $A_1$, $B_1$, $C_1$ and $D_1$ are coplanar.
2015 Sharygin Geometry Olympiad
Does there exist a rectangle which can be divided into a regular hexagon with sidelength $1$ and several congruent right-angled triangles with legs $1$ and $\sqrt{3}$?
2015 Sharygin Geometry Olympiad
Tanya cut out a convex polygon from the paper, fold it several times and obtained a two-layers quadrilateral. Can the cutted polygon be a heptagon?
2015 Sharygin Geometry Olympiad
Let $O$ and $H$ be the circumcenter and the orthocenter of a triangle $ABC$. The line passing through the midpoint of $OH$ and parallel to $BC$ meets $AB$ and $AC$ at points $D$ and $E$. It is known that $O$ is the incenter of triangle $ADE$. Find the angles of $ABC$.
2015 Sharygin Geometry Olympiad
The side $AD$ of a square $ABCD$ is the base of an obtuse-angled isosceles triangle $AED$ with vertex $E$ lying inside the square. Let $AF$ be a diameter of the circumcircle of this triangle, and $G$ be a point on $CD$ such that $CG = DF$. Prove that angle $BGE$ is less than half of angle $AED$.
2015 Sharygin Geometry Olympiad
In a parallelogram $ABCD$ the trisectors of angles $A$ and $B$ are drawn. Let $O$ be the common points of the trisectors nearest to $AB$. Let $AO$ meet the second trisector of angle $B$ at point $A_1$, and let $BO$ meet the second trisector of angle $A$ at point $B_1$. Let $M$ be the midpoint of $A_1B_1$. Line $MO$ meets $AB$ at point $N$ Prove that triangle $A_1B_1N$ is equilateral.
2015 Sharygin Geometry Olympiad
Let a triangle $ABC$ be given. Two circles passing through $A$ touch $BC$ at points $B$ and $C$ respectively. Let $D$ be the second common point of these circles ($A$ is closer to $BC$ than $D$). It is known that $BC = 2BD$. Prove that $\angle DAB = 2\angle ADB.$
2015 Sharygin Geometry Olympiad
Let $AA', BB'$ and $CC'$ be the altitudes of an acute-angled triangle $ABC$. Points $C_a, C_b$ are symmetric to $C' $ wrt $AA'$ and $BB'$. Points $A_b, A_c, B_c, B_a$ are defined similarly. Prove that lines $A_bB_a, B_cC_b$ and $C_aA_c$ are parallel.
2015 Sharygin Geometry Olympiad
The altitudes $AA_1$ and $CC_1$ of a triangle $ABC$ meet at point $H$. Point $H_A$ is symmetric to $H$ about $A$. Line $H_AC_1$ meets $BC$ at point $C' $, point $A' $ is defined similarly. Prove that $A' C' // AC$.
2015 Sharygin Geometry Olympiad
Diagonals of an isosceles trapezoid $ABCD$ with bases $BC$ and $AD$ are perpendicular. Let $DE$ be the perpendicular from $D$ to $AB$, and let $CF$ be the perpendicular from $C$ to $DE$. Prove that angle $DBF$ is equal to half of angle $FCD$.
2015 Sharygin Geometry Olympiad
Let $ABC$ be an acute-angled triangle. Construct points $A', B', C'$ on its sides $BC, CA, AB$ such that:

- $A'B'  // AB$,

- $C'C$ is the bisector of angle $A'C'B'$,

- $A'C' + B'C'= AB$.
2015 Sharygin Geometry Olympiad
The diagonals of a convex quadrilateral divide it into four similar triangles. Prove that is possible to inscribe a circle into this quadrilateral
2015 Sharygin Geometry Olympiad
Let $H$ be the orthocenter of an acute-angled triangle A$BC$. The perpendicular bisector to segment $BH$ meets $BA$ and $BC$ at points $A_0, C_0$ respectively. Prove that the perimeter of triangle $A_0OC_0$ ($O$ is the circumcenter of \triangle $ABC$) is equal to $AC$.
2015 Sharygin Geometry Olympiad
Find the maximal number of discs which can be disposed on the plane so that each two of them have a common point and no three have it
2015 Sharygin Geometry Olympiad
Let $AH_1, BH_2$ and $CH_3$ be the altitudes of a triangle $ABC$. Point $M$ is the midpoint of $H_2H_3$. Line $AM$ meets $H_2H_1$ at point $K$. Prove that $K$ lies on the medial line of $ABC$ parallel to $AC$.
2015 Sharygin Geometry Olympiad
Let $ABC$ be an acute-angled, nonisosceles triangle. Point $A_1, A_2$ are symmetric to the feet of the internal and the external bisectors of angle $A$ wrt the midpoint of $BC$. Segment $A_1A_2$ is a diameter of a circle $\alpha$. Circles $\beta$ and $\gamma$  are defined similarly. Prove that these three circles have two common points.
2015 Sharygin Geometry Olympiad
The sidelengths of a triangle $ABC$ are not greater than $1$. Prove that $p(1 -2Rr)$ is not greater than $1$, where $p$ is the semiperimeter, $R$ and $r$ are the circumradius and the inradius of $ABC$.
2015 Sharygin Geometry Olympiad
The diagonals of a convex quadrilateral divide it into four triangles. Restore the quadrilateral by the circumcenters of two adjacent triangles and the incenters of two mutually opposite triangles
2015 Sharygin Geometry Olympiad
Let $O$ be the circumcenter of a triangle $ABC$. The projections of points $D$ and $X$ to the sidelines of the triangle lie on lines $\ell $ and $L $ such that $\ell // XO$. Prove that the angles formed by $L$ and by the diagonals of quadrilateral $ABCD$ are equal.
2015 Sharygin Geometry Olympiad
Let $ABCDEF$ be a cyclic hexagon, points $K, L, M, N$ be the common points of lines $AB$ and $CD$, $AC$ and $BD$, $AF$ and $DE$, $AE$ and $DF$ respectively. Prove that if three of these points are collinear then the fourth point lies on the same line.
2015 Sharygin Geometry Olympiad
Let $L$ and $K$ be the feet of the internal and the external bisector of angle $A$ of a triangle $ABC$. Let $P$ be the common point of the tangents to the circumcircle of the triangle at $B$ and $C$. The perpendicular from $L$ to $BC$ meets $AP$ at point $Q$. Prove that $Q$ lies on the medial line of triangle $LKP$.
2015 Sharygin Geometry Olympiad
Given are a circle and an ellipse lying inside it with focus $C$. Find the locus of the circumcenters of triangles $ABC$, where $AB$ is a chord of the circle touching the ellipse.
2015 Sharygin Geometry Olympiad
A quadrilateral $ABCD$ is inscribed into a circle $\omega$ with center $O$. Let $M_1$ and $M_2$ be the midpoints of segments $AB$ and $CD$ respectively. Let $\Omega$ be the circumcircle of triangle $OM_1M_2$. Let $X_1$ and $X_2$ be the common points of $\omega$  and  $\Omega$ and $Y_1$ and $Y_2$ the second common points of $\Omega$  with the circumcircles of triangles $CDM_1$ and $ABM_2$. Prove that $X_1X_2  // Y_1Y_2$.
2015 Sharygin Geometry Olympiad
The faces of an icosahedron are painted into $5$ colors in such a way that two faces painted into the same color have no common points, even a vertices. Prove that for any point lying inside the icosahedron the sums of the distances from this point to the red faces and the blue faces are equal.
2015 Sharygin Geometry Olympiad
A tetrahedron $ABCD$ is given. The incircles of triangles $ ABC$ and $ABD$ with centers $O_1, O_2$, touch $AB$ at points $T_1, T_2$. The plane $\pi_{AB}$ passing through the midpoint of $T_1T_2$ is perpendicular to $O_1O_2$. The planes $\pi_{AC},\pi_{BC}, \pi_{AD}, \pi_{BD}, \pi_{CD}$ are defined similarly. Prove that these six planes have a common point.
2015 Sharygin Geometry Olympiad
The insphere of a tetrahedron ABCD with center $O$ touches its faces at points $A_1,B_1,C_1$ and  $D_1$.

a) Let $P_a$ be a point such that its reflections in lines $OB,OC$ and $OD$ lie on plane $BCD$.

Points $P_b, P_c$ and $P_d$ are defined similarly. Prove that lines $A_1P_a,B_1P_b,C_1P_c$ and $D_1P_d$ concur at some point $ P$.

b) Let $I$ be the incenter of $A_1B_1C_1D_1$ and $A_2$ the common point of line $A_1I $ with plane $B_1C_1D_1$. Points $B_2, C_2, D_2$ are defined similarly. Prove that $P$ lies inside $A_2B_2C_2D_2$.
2014 Sharygin Geometry Olympiad
A right-angled triangle $ABC$ is given.  Its catheus $AB$ is the base of a regular triangle $ADB$ lying in the exterior of $ABC$, and its hypotenuse $AC$ is the base of a regular triangle $AEC$ lying in the interior of $ABC$.  Lines $DE$ and $AB$ meet at point $M$.  The whole configuration except points $A$ and $B$ was erased.  Restore the point $M$.
2014 Sharygin Geometry Olympiad
A paper square with sidelength $2$ is given.  From this square, can we cut out a $12$-gon having all sidelengths equal to $1$ and all angles divisible by $45^\circ$?
2014 Sharygin Geometry Olympiad
Let $ABC$ be an isosceles triangle with base $AB$.  Line $\ell$ touches its circumcircle at point $B$.  Let $CD$ be a perpendicular from $C$ to $\ell$, and $AE$, $BF$ be the altitudes of $ABC$.  Prove that $D$, $E$, and $F$ are collinear.
2014 Sharygin Geometry Olympiad
A square is inscribed into a triangle (one side of the triangle contains two vertices and each of two remaining sides contains one vertex.  Prove that the incenter of the triangle lies inside the square.
2014 Sharygin Geometry Olympiad
In an acute-angled triangle $ABC$, $AM$ is a median, $AL$ is a bisector and $AH$ is an altitude ($H$ lies between $L$ and $B$).  It is known that $ML=LH=HB$.  Find the ratios of the sidelengths of $ABC$.
2014 Sharygin Geometry Olympiad
Given a circle with center $O$ and a point $P$ not lying on it, let $X$ be an arbitrary point on this circle and $Y$ be a common point of the bisector of angle $POX$ and the perpendicular bisector to segment $PX$.  Find the locus of points $Y$.
2014 Sharygin Geometry Olympiad
A parallelogram $ABCD$ is given.  The perpendicular from $C$ to $CD$ meets the perpendicular from $A$ to $BD$ at point $F$, and the perpendicular from $B$ to $AB$ meets the perpendicular bisector to $AC$ at point $E$.  Find the ratio in which side $BC$ divides segment $EF$.
2014 Sharygin Geometry Olympiad
Let $ABCD$ be a rectangle.  Two perpendicular lines pass through point $B$.  One of them meets segment $AD$ at point $K$, and the second one meets the extension of side $CD$ at point $L$.  Let $F$ be the common point of $KL$ and $AC$.  Prove that $BF\perp KL$.
2014 Sharygin Geometry Olympiad
Two circles $\omega_1$ and $\omega_2$ touching externally at point $L$ are inscribed into angle $BAC$.  Circle $\omega_1$ touches ray $AB$ at point $E$, and circle $\omega_2$ touches ray $AC$ at point $M$.  Line $EL$ meets $\omega_2$ for the second time at point $Q$.  Prove that $MQ\parallel AL$.
2014 Sharygin Geometry Olympiad
Two disjoint circles $\omega_1$ and $\omega_2$ are inscribed into an angle.  Consider all pairs of parallel lines $l_1$ and $l_2$ such that $l_1$ touches $\omega_1$ and $l_2$ touches $\omega_2$ ($\omega_1$, $\omega_2$ lie between $l_1$ and $l_2$).  Prove that the medial lines of all trapezoids formed by $l_1$ and $l_2$ and the sides of the angle touch some fixed circle.
2014 Sharygin Geometry Olympiad
Points $K, L, M$ and $N$ lying on the sides $AB, BC, CD$ and $DA$ of a square $ABCD$ are vertices of another square. Lines $DK$ and $N M$ meet at point $E$, and lines $KC$ and $LM$ meet at point $F$ . Prove that $EF\parallel AB$.