Navvye/Olympiad-Problems-CSV huggingface dataset:
2009 Sharygin Geometry Olympiad
Three lines passing through point $ O$ form equal angles by pairs. Points $ A_1$, $ A_2$ on the first line and $ B_1$, $ B_2$ on the second line are such that the common point $ C_1$ of $ A_1B_1$ and $ A_2B_2$ lies on the third line. Let $ C_2$ be the common point of $ A_1B_2$ and $ A_2B_1$. Prove that angle $ C_1OC_2$ is right.
2009 Sharygin Geometry Olympiad
Given triangle $ ABC$ and two points $ X$, $ Y$ not lying on its circumcircle. Let $ A_1$, $ B_1$, $ C_1$ be the projections of $ X$ to $ BC$, $ CA$, $ AB$, and $ A_2$, $ B_2$, $ C_2$ be the projections of $ Y$. Prove that the perpendiculars from $ A_1$, $ B_1$, $ C_1$ to $ B_2C_2$, $ C_2A_2$, $ A_2B_2$, respectively, concur if and only if line $ XY$ passes through the circumcenter of $ ABC$.
2009 Sharygin Geometry Olympiad
Given three parallel lines on the plane. Find the locus of incenters of triangles with vertices lying on these lines (a single vertex on each line).
2009 Sharygin Geometry Olympiad
Given convex $ n$-gon $ A_1\ldots A_n$. Let $ P_i$ ($ i = 1,\ldots , n$) be such points on its boundary that $ A_iP_i$ bisects the area of polygon. All points $ P_i$ don't coincide with any vertex and lie on $ k$ sides of $ n$-gon. What is the maximal and the minimal value of $ k$ for each given $ n$?
2009 Sharygin Geometry Olympiad
Suppose $ H$ and $ O$ are the orthocenter and the circumcenter of acute triangle $ ABC$; $ AA_1$, $ BB_1$ and $ CC_1$ are the altitudes of the triangle. Point $ C_2$ is the reflection of $ C$ in $ A_1B_1$. Prove that $ H$, $ O$, $ C_1$ and $ C_2$ are concyclic.
2009 Sharygin Geometry Olympiad
The opposite sidelines of quadrilateral $ ABCD$ intersect at points $ P$ and $ Q$. Two lines passing through these points meet the side of $ ABCD$ in four points which are the vertices of a parallelogram. Prove that the center of this parallelogram lies on the line passing through the midpoints of diagonals of $ ABCD$.
2009 Sharygin Geometry Olympiad
Construct a quadrilateral which is inscribed and circumscribed, given the radii of the respective circles and the angle between the diagonals of quadrilateral.
2009 Sharygin Geometry Olympiad
Is it true that for each $ n$, the regular $ 2n$-gon is a projection of some polyhedron having not greater than $ n + 2$ faces?
2009 Sharygin Geometry Olympiad
A sphere is inscribed into a quadrangular pyramid. The point of contact of the sphere with the base of the pyramid is projected to the edges of the base. Prove that these projections are concyclic.
2009 Sharygin Geometry Olympiad
Minor base $BC$ of trapezoid $ABCD$ is equal to side $AB$, and diagonal $AC$ is equal to base $AD$. The line passing through B and parallel to $AC$ intersects line $DC$ in point $M$. Prove that $AM$ is the bisector of angle $\angle BAC$.



A.Blinkov, Y.Blinkov
2009 Sharygin Geometry Olympiad
A cyclic quadrilateral is divided into four quadrilaterals by two lines passing through its inner point. Three of these quadrilaterals are cyclic with equal circumradii. Prove that the fourth part also is cyclic quadrilateral and its circumradius is the same.



(A.Blinkov)
2009 Sharygin Geometry Olympiad
Let $AH_a $ and $BH_b$ be the altitudes of triangle $ABC$. Points $P$ and $Q$ are the projections of $H_a$ to $AB$ and $AC$. Prove that line $PQ $ bisects segment $H_aH_b$.



(A.Akopjan, K.Savenkov)
2009 Sharygin Geometry Olympiad
Given is $\triangle ABC$ such that $\angle A = 57^o, \angle B = 61^o$ and $\angle C = 62^o$. Which segment is longer: the angle bisector through $A$ or the median through $B$?



(N.Beluhov)
2009 Sharygin Geometry Olympiad
Given triangle $ABC$. Point $M$ is the projection of vertex $B$ to bisector of angle $C$. $K$ is the touching point of the incircle with side $BC$. Find angle $\angle  MKB$ if $\angle BAC = \alpha$



(V.Protasov)
2009 Sharygin Geometry Olympiad
Can four equal polygons be placed on the plane in such a way that any two of them don't have common interior points, but have a common boundary segment?



(S.Markelov)
2009 Sharygin Geometry Olympiad
Let $s$ be the circumcircle of triangle $ABC, L$ and $W$ be common points of angle's $A$ bisector with side $BC$ and $s$ respectively, $O$ be the circumcenter of triangle $ACL$. Restore triangle $ABC$, if circle $s$ and points $W$ and $O$ are given.



(D.Prokopenko)
2009 Sharygin Geometry Olympiad
A triangle $ABC$ is given, in which the segment $BC$ touches the incircle and the corresponding excircle in points $M$ and $N$. If $\angle BAC = 2 \angle MAN$, show that $BC = 2MN$.



(N.Beluhov)
2009 Sharygin Geometry Olympiad
The midpoint of triangle's side and the base of the altitude to this side are symmetric wrt the touching point of this side with the incircle. Prove that this side equals one third of triangle's perimeter.



(A.Blinkov, Y.Blinkov)
2009 Sharygin Geometry Olympiad
Given a convex quadrilateral $ABCD$. Let $R_a, R_b, R_c$ and $R_d$ be the circumradii of triangles $DAB, ABC, BCD, CDA$. Prove that inequality $R_a < R_b < R_c < R_d$ is equivalent to $180^o -  \angle CDB < \angle CAB < \angle CDB$ .



(O.Musin)
2009 Sharygin Geometry Olympiad
Quadrilateral $ABCD$ is circumscribed, rays $BA$ and $CD$ intersect in point $E$, rays $BC$ and $AD$ intersect in point $F$. The incircle of the triangle formed by lines $AB, CD$ and the bisector of angle $B$, touches $AB$ in point $K$, and the incircle of the triangle formed by lines $AD, BC$ and the bisector of angle $B$, touches $BC$ in point $L$. Prove that lines $KL, AC$ and $EF$ concur.



(I.Bogdanov)
2009 Sharygin Geometry Olympiad
Given regular $17$-gon $A_1 ... A_{17}$. Prove that two triangles formed by lines $A_1A_4, A_2A_{10}, A_{13}A_{14}$ and $A_2A_3, A_4A_6 A_{14}A_{15} $ are equal.



(N.Beluhov)
2009 Sharygin Geometry Olympiad
Let $n$ points lie on the circle. Exactly half of triangles formed by these points are acute-angled. Find all possible $n$.



(B.Frenkin)
2009 Sharygin Geometry Olympiad
Given triangle $ABC$ such that $AB- BC = \frac{AC}{\sqrt2}$ . Let $M$ be the midpoint of $AC$, and $N$ be the base of the bisector from $B$. Prove that $\angle BMC + \angle BNC = 90^o$.



(A.Akopjan)
2009 Sharygin Geometry Olympiad
Given two intersecting circles with centers $O_1, O_2$. Construct the circle touching one of them externally and the second one internally such that the distance from its center to $O_1O_2$ is maximal.



(M.Volchkevich)
2009 Sharygin Geometry Olympiad
Given cyclic quadrilateral $ABCD$. Four circles each touching its diagonals and the circumcircle internally are equal. Is $ABCD$ a square?



(C.Pohoata, A.Zaslavsky)
2009 Sharygin Geometry Olympiad
Let $a, b, c$ be the lengths of some triangle's sides, $p, r$ be the semiperimeter and the inradius of triangle. Prove an inequality $\sqrt{\frac{ab(p- c)}{p}} +\sqrt{\frac{ca(p- b)}{p}} +\sqrt{\frac{bc(p-a)}{p}}   \ge 6r$



(D.Shvetsov)
2009 Sharygin Geometry Olympiad
Given quadrilateral $ABCD$. Its sidelines$ AB$ and $CD$ intersect in point $K$. It's diagonals intersect in point $L$. It is known that line $KL$ pass through the centroid of $ABCD$. Prove that $ABCD$ is trapezoid.



(F.Nilov)
2009 Sharygin Geometry Olympiad
The cirumradius and the inradius of triangle $ABC$ are equal to $R$ and $r, O, I$ are the centers of respective circles. External bisector of angle $C$ intersect $AB$ in point $P$. Point $Q$ is the projection of $P$ to line $OI$. Find distance $OQ.$



(A.Zaslavsky, A.Akopjan)
2009 Sharygin Geometry Olympiad
Three parallel lines $d_a, d_b, d_c$ pass through the vertex of triangle $ABC$. The reflections of $d_a, d_b, d_c$ in $BC, CA, AB$ respectively form triangle $XYZ$. Find the locus of incenters of such triangles.



(C.Pohoata)
2009 Sharygin Geometry Olympiad
Rhombus $CKLN$ is inscribed into triangle $ABC$ in such way that point $L$ lies on side $AB$, point $N$ lies on side $AC$, point $K$ lies on side $BC$. $O_1, O_2$ and $O$ are the circumcenters of triangles $ACL, BCL$ and $ABC$ respectively. Let $P$ be the common point of circles $ANL$ and $BKL$, distinct from $L$. Prove that points $O_1, O_2, O$ and $P$ are concyclic.



(D.Prokopenko)
2009 Sharygin Geometry Olympiad
Let $M, I$ be the centroid and the incenter of triangle $ABC, A_1$ and $B_1$ be the touching points of the incircle with sides $BC$ and $AC, G$ be the common point of lines $AA_1$ and $BB_1$. Prove that angle $\angle CGI$ is right if and only if $GM // AB$.



(A.Zaslavsky)
2009 Sharygin Geometry Olympiad
Given points $O, A_1, A_2, ..., A_n$  on the plane. For any two of these points the square of distance between them is natural number. Prove that there exist two vectors $\vec{x}$ and $\vec{y}$, such that for any point $A_i$, $\vec{OA_i }= k\vec{x}+l \vec{y}$, where $k$ and $l$ are some integer numbers.



(A.Glazyrin)
2009 Sharygin Geometry Olympiad
Can the regular octahedron be inscribed into regular dodecahedron in such way that all vertices of octahedron be the vertices of dodecahedron?



(B.Frenkin)
2008 Sharygin Geometry Olympiad
(B.Frenkin) Does a convex quadrilateral without parallel sidelines exist such that it can be divided into four congruent triangles?
2008 Sharygin Geometry Olympiad
(F.Nilov) Given right triangle $ ABC$ with hypothenuse $ AC$ and $ \angle A = 50^{\circ}$. Points $ K$ and $ L$ on the cathetus $ BC$ are such that $ \angle KAC = \angle LAB = 10^{\circ}$. Determine the ratio $ CK/LB$.
2008 Sharygin Geometry Olympiad
(D.Shnol) Two opposite angles of a convex quadrilateral with perpendicular diagonals are equal. Prove that a circle can be inscribed in this quadrilateral.
2008 Sharygin Geometry Olympiad
(F.Nilov, A.Zaslavsky) Let $ CC_0$ be a median of triangle $ ABC$; the perpendicular bisectors to $ AC$ and $ BC$ intersect $ CC_0$ in points $ A'$, $ B'$; $ C_1$ is the meet of lines $ AA'$ and $ BB'$. Prove that $ \angle C_1CA = \angle C_0CB$.
2008 Sharygin Geometry Olympiad
(A.Zaslavsky) Given two triangles $ ABC$, $ A'B'C'$. Denote by $ \alpha$ the angle between the altitude and the median from vertex $ A$ of triangle $ ABC$. Angles $ \beta$, $ \gamma$, $ \alpha'$, $ \beta'$, $ \gamma'$ are defined similarly. It is known that $ \alpha = \alpha'$, $ \beta = \beta'$, $ \gamma = \gamma'$. Can we conclude that the triangles are similar?
2008 Sharygin Geometry Olympiad
(B.Frenkin) Consider the triangles such that all their vertices are vertices of a given regular 2008-gon. What triangles are more numerous among them: acute-angled or obtuse-angled?
2008 Sharygin Geometry Olympiad
(F.Nilov) Given isosceles triangle $ ABC$ with base $ AC$ and $ \angle B = \alpha$. The arc $ AC$ constructed outside the triangle has angular measure equal to $ \beta$. Two lines passing through $ B$ divide the segment and the arc $ AC$ into three equal parts. Find the ratio $ \alpha / \beta$.
2008 Sharygin Geometry Olympiad
(B.Frenkin, A.Zaslavsky) A convex quadrilateral was drawn on the blackboard. Boris marked the centers of four excircles each touching one side of the quadrilateral and the extensions of two adjacent sides. After this, Alexey erased the quadrilateral. Can Boris define its perimeter?
2008 Sharygin Geometry Olympiad
(A.Zaslavsky) A convex polygon can be divided into 2008 congruent quadrilaterals. Is it true that this polygon has a center or an axis of symmetry?
2008 Sharygin Geometry Olympiad
(F.Nilov) Given quadrilateral $ ABCD$. Find the locus of points such that their projections to the lines $ AB$, $ BC$, $ CD$, $ DA$ form a quadrilateral with perpendicular diagonals.
2008 Sharygin Geometry Olympiad
(R.Pirkuliev) Prove the inequality

$$ \frac1{\sqrt {2\sin A}} + \frac1{\sqrt {2\sin B}} + \frac1{\sqrt {2\sin C}}\leq\sqrt {\frac {p}{r}},

$$

where $ p$ and $ r$ are the semiperimeter and the inradius of triangle $ ABC$.
2008 Sharygin Geometry Olympiad
(F.Nilov, A.Zaslavsky) Let $ CC_0$ be a median of triangle $ ABC$; the perpendicular bisectors to $ AC$ and $ BC$ intersect $ CC_0$ in points $ A_c$, $ B_c$; $ C_1$ is the common point of $ AA_c$ and $ BB_c$. Points $ A_1$, $ B_1$ are defined similarly. Prove that circle $ A_1B_1C_1$ passes through the circumcenter of triangle $ ABC$.
2008 Sharygin Geometry Olympiad
(N.Avilov) Can the surface of a regular tetrahedron be glued over with equal regular hexagons?
2008 Sharygin Geometry Olympiad
(B.Frenkin) Construct the triangle, given its centroid and the feet of an altitude and a bisector from the same vertex.
2008 Sharygin Geometry Olympiad
(A.Zaslavsky) The circumradius of triangle $ ABC$ is equal to $ R$. Another circle with the same radius passes through the orthocenter $ H$ of this triangle and intersect its circumcirle in points $ X$, $ Y$. Point $ Z$ is the fourth vertex of parallelogram $ CXZY$. Find the circumradius of triangle $ ABZ$.
2008 Sharygin Geometry Olympiad
(J.-L.Ayme, France) Points $ P$, $ Q$ lie on the circumcircle $ \omega$ of triangle $ ABC$. The perpendicular bisector $ l$ to $ PQ$ intersects $ BC$, $ CA$, $ AB$ in points $ A'$, $ B'$, $ C'$. Let $ A"$, $ B"$, $ C"$ be the second common points of $ l$ with the circles $ A'PQ$, $ B'PQ$, $ C'PQ$. Prove that $ AA"$, $ BB"$, $ CC"$ concur.
2008 Sharygin Geometry Olympiad
(B.Frenkin) An inscribed and circumscribed $ n$-gon is divided by some line into two inscribed and circumscribed polygons with different numbers of sides. Find $ n$.
2008 Sharygin Geometry Olympiad
(A.Myakishev) Let triangle $ A_1B_1C_1$ be symmetric to $ ABC$ wrt the incenter of its medial triangle. Prove that the orthocenter of $ A_1B_1C_1$ coincides with the circumcenter of the triangle formed by the excenters of $ ABC$.
2008 Sharygin Geometry Olympiad
(V.Yasinsky, Ukraine) Suppose $ X$ and $ Y$ are the common points of two circles $ \omega_1$ and $ \omega_2$. The third circle $ \omega$ is internally tangent to $ \omega_1$ and $ \omega_2$ in $ P$ and $ Q$ respectively. Segment $ XY$ intersects $ \omega$ in points $ M$ and $ N$. Rays $ PM$ and $ PN$ intersect $ \omega_1$ in points $ A$ and $ D$; rays $ QM$ and $ QN$ intersect $ \omega_2$ in points $ B$ and $ C$ respectively. Prove that $ AB = CD$.
2008 Sharygin Geometry Olympiad
(A.Zaslavsky) Given three points $ C_0$, $ C_1$, $ C_2$ on the line $ l$. Find the locus of incenters of triangles $ ABC$ such that points $ A$, $ B$ lie on $ l$ and the feet of the median, the bisector and the altitude from $ C$ coincide with $ C_0$, $ C_1$, $ C_2$.
2008 Sharygin Geometry Olympiad
(I.Bogdanov) A section of a regular tetragonal pyramid is a regular pentagon. Find the ratio of its side to the side of the base of the pyramid.
2008 Sharygin Geometry Olympiad
(B.Frenkin) The product of two sides in a triangle is equal to $ 8Rr$, where $ R$ and $ r$ are the circumradius and the inradius of the triangle. Prove that the angle between these sides is less than $ 60^{\circ}$.
2008 Sharygin Geometry Olympiad
(F.Nilov) Two arcs with equal angular measure are constructed on the medians $ AA'$ and $ BB'$ of triangle $ ABC$ towards vertex $ C$. Prove that the common chord of the respective circles passes through $ C$.
2008 Sharygin Geometry Olympiad
(A.Akopyan, V.Dolnikov) Given a set of points inn the plane. It is known that among any three of its points there are two such that the distance between them doesn't exceed 1. Prove that this set can be divided into three parts such that the diameter of each part does not exceed 1.
2008 Sharygin Geometry Olympiad
(B.Frenkin, 8) Does a regular polygon exist such that just half of its diagonals are parallel to its sides?
2008 Sharygin Geometry Olympiad
(V.Protasov, 8) For a given pair of circles, construct two concentric circles such that both are tangent to the given two. What is the number of solutions, depending on location of the circles?
2008 Sharygin Geometry Olympiad
(A.Zaslavsky, 8) A triangle can be dissected into three equal triangles. Prove that some its angle is equal to $ 60^{\circ}$.
2008 Sharygin Geometry Olympiad
(D.Shnol, 8--9) The bisectors of two angles in a cyclic quadrilateral are parallel. Prove that the sum of squares of some two sides in the quadrilateral equals the sum of squares of two remaining sides.
2008 Sharygin Geometry Olympiad
(Kiev olympiad, 8--9) Reconstruct the square $ ABCD$, given its vertex $ A$ and distances of vertices $ B$ and $ D$ from a fixed point $ O$ in the plane.
2008 Sharygin Geometry Olympiad
(A. Myakishev, 8--9) In the plane, given two concentric circles with the center $ A$. Let $ B$ be an arbitrary point on some of these circles, and $ C$ on the other one. For every triangle $ ABC$, consider two equal circles mutually tangent at the point $ K$, such that one of these circles is tangent to the line $ AB$ at point $ B$ and the other one is tangent to the line $ AC$ at point $ C$. Determine the locus of points $ K$.
2008 Sharygin Geometry Olympiad
(A.Zaslavsky, 8--9) Given a circle and a point $ O$ on it. Another circle with center $ O$ meets the first one at points $ P$ and $ Q$. The point $ C$ lies on the first circle, and the lines $ CP$, $ CQ$ meet the second circle for the second time at points $ A$ and $ B$. Prove that $ AB=PQ$.
2008 Sharygin Geometry Olympiad
(T.Golenishcheva-Kutuzova, B.Frenkin, 8--11)  a) Prove that for $ n > 4$, any convex $ n$-gon

can be dissected into $ n$ obtuse triangles.
2008 Sharygin Geometry Olympiad
(A.Zaslavsky, 9--10) The reflections of diagonal $ BD$ of a quadrilateral $ ABCD$ in the bisectors of angles $ B$ and $ D$ pass

through the midpoint of diagonal $ AC$. Prove that the reflections of diagonal $ AC$ in the bisectors of angles $ A$ and $ C$ pass

through the midpoint of diagonal $ BD$ (There was an error in published condition of this problem).
2008 Sharygin Geometry Olympiad
(A.Zaslavsky, 9--10) Quadrilateral $ ABCD$ is circumscribed arounda circle with center $ I$. Prove that the projections of points $ B$ and $ D$ to the lines $ IA$ and $ IC$ lie on a single circle.
2008 Sharygin Geometry Olympiad
(A.Zaslavsky, 9--10) Given four points $ A$, $ B$, $ C$, $ D$. Any two circles such that one of them contains $ A$ and $ B$, and the other one contains $ C$ and $ D$, meet. Prove that common chords of all these pairs of circles pass through a fixed point.
2008 Sharygin Geometry Olympiad
(A.Myakishev, 9--10) Given a triangle $ ABC$. Point $ A_1$ is chosen on the ray $ BA$ so that segments $ BA_1$ and $ BC$ are equal. Point $ A_2$ is chosen on the ray $ CA$ so that segments $ CA_2$ and $ BC$ are equal. Points  $ B_1$, $ B_2$ and $ C_1$, $ C_2$ are chosen similarly. Prove that lines $ A_1A_2$, $ B_1B_2$, $ C_1C_2$ are parallel.
2008 Sharygin Geometry Olympiad
(A.Myakishev, 9--10) Given triangle $ ABC$. One of its excircles is tangent to the side $ BC$ at point $ A_1$ and to the extensions of two other sides. Another excircle is tangent to side $ AC$ at point $ B_1$. Segments $ AA_1$ and $ BB_1$ meet at point $ N$. Point $ P$ is chosen on the ray $ AA_1$ so that $ AP=NA_1$. Prove that $ P$ lies on the incircle.
2008 Sharygin Geometry Olympiad
(V.Protasov, 9--10) The Euler line of a non-isosceles triangle is parallel to the bisector of one of its angles. Determine this

angle (There was an error in published condition of this problem).
2008 Sharygin Geometry Olympiad
(M.Volchkevich, 9--11) Given two circles and point $ P$ not lying on them. Draw a line through $ P$ which cuts chords of equal length from these circles.
2008 Sharygin Geometry Olympiad
(A.Zaslavsky, 9--11) Given two circles. Their common external tangent is tangent to them at points $ A$ and $ B$. Points $ X$, $ Y$ on these circles are such that some circle is tangent to the given two circles at these points, and in similar way (external or internal). Determine the locus of intersections of lines $ AX$ and $ BY$.
2008 Sharygin Geometry Olympiad
(A.Myakishev, 9--11) Given triangle $ ABC$ and a ruler with two marked intervals equal to $ AC$ and $ BC$. By this ruler only, find the incenter of the triangle formed by medial lines of triangle $ ABC$.
2008 Sharygin Geometry Olympiad
(A.Abdullayev, 9--11) Prove that the triangle having sides $ a$, $ b$, $ c$ and area $ S$ satisfies the inequality

$$ a^2+b^2+c^2-\frac12(|a-b|+|b-c|+|c-a|)^2\geq 4\sqrt3 S.$$
2008 Sharygin Geometry Olympiad
(V.Protasov, 10-11) Given parallelogram $ ABCD$ such that $ AB = a$, $ AD = b$. The first circle has its center at vertex $ A$ and passes through $ D$, and the second circle has its center at $ C$ and passes through $ D$. A circle with center $ B$ meets the first circle at points $ M_1$, $ N_1$, and the second circle at points $ M_2$, $ N_2$. Determine the ratio $ M_1N_1/M_2N_2$.
2008 Sharygin Geometry Olympiad
(A.Zaslavsky, 10--11)  a) Some polygon has the following property:

if a line passes through two points which bisect its perimeter then this line bisects the area of the polygon. Is it true that the polygon is central symmetric?

b) Is it true that any figure with the property from part a) is central symmetric?
2008 Sharygin Geometry Olympiad
(A.Zaslavsky, B.Frenkin, 10--11) In a triangle, one has drawn perpendicular bisectors to its sides and has measured their segments lying inside the triangle.



a) All three segments are equal. Is it true that the triangle is equilateral?



b) Two segments are equal. Is it true that the triangle is isosceles?



c) Can the segments have length 4, 4 and 3?
2008 Sharygin Geometry Olympiad
(A.Khachaturyan, 10--11) a) All vertices of a pyramid lie on the facets of a cube

but not on its edges, and each facet contains at least one vertex. What is the

maximum possible number of the vertices of the pyramid?



b) All vertices of a pyramid lie in the facet planes of a cube but not on the lines

including its edges, and each facet plane contains at least one vertex. What is the

maximum possible number of the vertices of the pyramid?
2008 Sharygin Geometry Olympiad
(V.Protasov, 10--11) In the space, given two intersecting spheres of different radii and a point $ A$ belonging to both spheres. Prove that there is a point $ B$ in the space with the following property:

if an arbitrary circle passes through points $ A$ and $ B$ then the second points of its meet with the given spheres are equidistant from $ B$.
2008 Sharygin Geometry Olympiad
(I.Bogdanov, 11) Let $ h$ be the least altitude of a tetrahedron, and $ d$ the least distance between its opposite edges. For what values of $ t$ the inequality $ d>th$ is possible?
2007 Sharygin Geometry Olympiad
A triangle is cut into several (not less than two) triangles. One of them is isosceles (not equilateral), and all others are equilateral. Determine the angles of the original triangle.
2007 Sharygin Geometry Olympiad
Each diagonal of a quadrangle divides it into two isosceles triangles. Is it true that the quadrangle is a diamond?
2007 Sharygin Geometry Olympiad
Segments connecting an inner point of a convex non-equilateral n-gon to its vertices divide the n-gon into n equal triangles. What is the least possible n?
2007 Sharygin Geometry Olympiad
Does a parallelogram exist such that all pairwise meets of bisectors of its angles are situated outside it?
2007 Sharygin Geometry Olympiad
A non-convex $n$-gon is cut into three parts by a straight line, and two parts are put together so that the resulting polygon is equal to the third part. Can $n$ be equal to:

a) five?

b) four?
2007 Sharygin Geometry Olympiad
a) What can be the number of symmetry axes of a checked polygon, that is, of a polygon whose sides lie on lines of a list of checked paper? (Indicate all possible values.)

b) What can be the number of symmetry axes of a checked polyhedron, that is, of a polyhedron consisting of equal cubes which border one to another by plane facets?
2007 Sharygin Geometry Olympiad
A convex polygon is circumscribed around a circle. Points of contact of its sides with the circle form a polygon with the same set of angles (the order of angles may differ). Is it true that the polygon is regular?
2007 Sharygin Geometry Olympiad
Three circles pass through a point $P$, and the second points of their intersection $A, B, C$ lie on a straight line. Let $A_1 B_1, C_1$ be the second meets of lines $AP, BP, CP$ with the corresponding circles. Let $C_2$ be the intersections of lines $AB_1$ and $BA_1$. Let $A_2, B_2$ be defined similarly. Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ are equal,