Navvye/Olympiad-Problems-CSV huggingface dataset:
2013 Sharygin Geometry Olympiad
Two fixed circles $\omega_1$ and $\omega_2$ pass through point $O$. A circle of an arbitrary radius $R$ centered at $O$ meets $\omega_1$ at points $A$ and $B$, and meets $\omega_2$ at points $C$ and $D$. Let $X$ be the common point of lines $AC$ and $BD$. Prove that all the points X are collinear as $R$ changes.
2013 Sharygin Geometry Olympiad
Three cyclists ride along a circular road with radius $1$ km counterclockwise. Their velocities are constant and different. Does there necessarily exist (in a sufficiently long time) a moment when all the three distances between cyclists are greater than $1$ km?



by V. Protasov
2013 Sharygin Geometry Olympiad
A circle $k$ passes through the vertices $B, C$ of a scalene triangle $ABC$. $k$ meets the extensions of $AB, AC$ beyond $B, C$ at $P, Q$ respectively. Let $A_1$ is the foot the altitude drop from $A$ to $BC$. Suppose $A_1P=A_1Q$. Prove that $\widehat{PA_1Q}=2\widehat{BAC}$.
2013 Sharygin Geometry Olympiad
Let $ABCD$ is a tangential quadrilateral such that $AB=CD>BC$. $AC$ meets $BD$ at $L$. Prove that $\widehat{ALB}$ is acute.





Click to reveal hidden textAccording to the jury, they want to propose a more generalized problem is to prove $(AB-CD)^2 < (AD-BC)^2$, but this problem has appeared some time ago
2013 Sharygin Geometry Olympiad
Let $X$ be a point inside triangle $ABC$ such that $XA.BC=XB.AC=XC.AC$. Let $I_1, I_2, I_3$ be the incenters of $XBC, XCA, XAB$. Prove that $AI_1, BI_2, CI_3$ are concurrent.



Click to reveal hidden textOf course, the most natural way to solve this is the Ceva sin theorem, but there is an another approach that may surprise you;), try not to use the Ceva theorem :))
2013 Sharygin Geometry Olympiad
Given a square cardboard of area $\frac{1}{4}$, and a paper triangle of area $\frac{1}{2}$ such that the square of its sidelength is a positive integer. Prove that the triangle can be folded in some ways such that the squace can be placed inside the folded figure so that both of its faces are completely covered with paper.



Proposed by N.Beluhov, Bulgaria
2013 Sharygin Geometry Olympiad
Let ABCD is a cyclic quadrilateral inscribed in $(O)$. $E, F$ are the midpoints of arcs $AB$ and $CD$ not containing the other vertices of the quadrilateral. The line passing through $E, F$ and parallel to the diagonals of $ABCD$ meet at $E, F, K, L$. Prove that $KL$ passes through $O$.
2013 Sharygin Geometry Olympiad
The altitudes $AA_1, BB_1, CC_1$ of an acute triangle $ABC$ concur at $H$. The perpendicular lines from $H$ to $B_1C_1, A_1C_1$ meet rays $CA, CB$ at $P, Q$ respectively. Prove that the line from $C$ perpendicular to $A_1B_1$ passes through the midpoint of $PQ$.
2013 Sharygin Geometry Olympiad
Given five fixed points in the space. It is known that these points are centers of five spheres, four of which are pairwise externally tangent, and all these point are internally tangent to the fifth one. It turns out that it is impossible to determine which of the marked points is the center of the largest sphere. Find the ratio of the greatest and the smallest radii of the spheres.
2013 Sharygin Geometry Olympiad
Two fixed circles are given on the plane, one of them lies inside the other one. From a point $C$ moving arbitrarily on the external circle, draw two chords $CA, CB$ of the larger circle such that they tangent to the smalaler one. Find the locus of the incenter of triangle $ABC$.
2012 Sharygin Geometry Olympiad
In triangle $ABC$ point $M$ is the midpoint of side $AB$, and point $D$ is the foot of altitude $CD$. Prove that $\angle A = 2\angle B$ if and only if $AC = 2 MD$.
2012 Sharygin Geometry Olympiad
A cyclic $n$-gon is divided by non-intersecting (inside the $n$-gon) diagonals to $n-2$ triangles. Each of these triangles is similar to at least one of the remaining ones. For what $n$ this is possible?
2012 Sharygin Geometry Olympiad
A circle with center $I$ touches sides $AB,BC,CA$ of triangle $ABC$ in points $C_{1},A_{1},B_{1}$. Lines $AI, CI, B_{1}I$ meet $A_{1}C_{1}$ in points $X, Y, Z$ respectively. Prove that $\angle Y B_{1}Z = \angle XB_{1}Z$.
2012 Sharygin Geometry Olympiad
Given triangle $ABC$. Point $M$ is the midpoint of side $BC$, and point $P$ is the projection of $B$ to the perpendicular bisector of segment $AC$. Line $PM$ meets $AB$ in point $Q$. Prove that triangle $QPB$ is isosceles.
2012 Sharygin Geometry Olympiad
On side $AC$ of triangle $ABC$ an arbitrary point is selected $D$. The tangent in $D$ to the circumcircle of triangle $BDC$ meets $AB$ in point $C_{1}$; point $A_{1}$ is defined similarly. Prove that $A_{1}C_{1}\parallel AC$.
2012 Sharygin Geometry Olympiad
Point $C_{1}$ of hypothenuse $AC$ of a right-angled triangle $ABC$ is such that $BC = CC_{1}$.  Point $C_{2}$ on cathetus $AB$ is such that $AC_{2} = AC_{1}$; point $A_{2}$ is defined similarly. Find angle $AMC$, where $M$ is the midpoint of $A_{2}C_{2}$.
2012 Sharygin Geometry Olympiad
In a non-isosceles triangle $ABC$ the bisectors of angles $A$ and $B$ are inversely proportional to the respective sidelengths. Find angle $C$.
2012 Sharygin Geometry Olympiad
Let $BM$ be the median of right-angled triangle $ABC (\angle B = 90^{\circ})$. The incircle of triangle $ABM$ touches sides $AB, AM$ in points $A_{1},A_{2}$; points $C_{1}, C_{2}$ are defined similarly. Prove that lines $A_{1}A_{2}$ and $C_{1}C_{2}$ meet on the bisector of angle $ABC$.
2012 Sharygin Geometry Olympiad
In triangle $ABC$, given lines $l_{b}$ and $l_{c}$ containing the bisectors of angles $B$ and $C$, and the foot $L_{1}$ of the bisector of angle $A$. Restore triangle $ABC$.
2012 Sharygin Geometry Olympiad
In a convex quadrilateral all sidelengths and all angles are pairwise different.

a) Can the greatest angle be adjacent to the greatest side and at the same time the smallest angle be adjacent to the smallest side?

b) Can the greatest angle be non-adjacent to the smallest side and at the same time the smallest angle be non-adjacent to the greatest side?
2012 Sharygin Geometry Olympiad
Given triangle $ABC$ and point $P$. Points $A', B', C'$ are the projections of $P$ to $BC, CA, AB$. A line passing through $P$ and parallel to $AB$ meets the circumcircle of triangle $PA'B'$ for the second time in point $C_{1}$. Points $A_{1}, B_{1}$ are defined similarly. Prove that

a) lines $AA_{1}, BB_{1}, CC_{1}$ concur;

b) triangles $ABC$ and $A_{1}B_{1}C_{1}$ are similar.
2012 Sharygin Geometry Olympiad
Let $O$ be the circumcenter of an acute-angled triangle $ABC$. A line passing through $O$ and parallel to $BC$ meets $AB$ and $AC$ in points $P$ and $Q$ respectively. The sum of distances from $O$ to $AB$ and $AC$ is equal to $OA$. Prove that $PB + QC = PQ$.
2012 Sharygin Geometry Olympiad
Points $A, B$ are given. Find the locus of points $C$ such that $C$, the midpoints of

$AC, BC$ and the centroid of triangle $ABC$ are concyclic.
2012 Sharygin Geometry Olympiad
In a convex quadrilateral $ABCD$ suppose $AC \cap BD = O$ and $M$ is the midpoint of $BC$. Let $MO \cap AD = E$. Prove that $\frac{AE}{ED} = \frac{S_{\triangle ABO}}{S_{\triangle CDO}}$.
2012 Sharygin Geometry Olympiad
Given triangle $ABC$. Consider lines $l$ with the next property: the reflections of $l$ in the sidelines of the triangle concur. Prove that all these lines have a common point.
2012 Sharygin Geometry Olympiad
Given right-angled triangle $ABC$ with hypothenuse $AB$. Let $M$ be the midpoint of $AB$ and $O$ be the center of circumcircle $\omega$ of triangle $CMB$. Line $AC$ meets $\omega$ for the second time in point $K$. Segment $KO$ meets the circumcircle of triangle $ABC$ in point $L$. Prove that segments $AL$ and $KM$ meet on the circumcircle of triangle $ACM$.
2012 Sharygin Geometry Olympiad
A square $ABCD$ is inscribed into a circle. Point $M$ lies on arc $BC$, $AM$ meets $BD$ in point $P$, $DM$ meets $AC$ in point $Q$. Prove that the area of quadrilateral $APQD$ is equal to the half of the area of the square.
2012 Sharygin Geometry Olympiad
A triangle and two points inside it are marked. It is known that one of the triangle’s angles is equal to $58^{\circ}$, one of two remaining angles is equal to $59^{\circ}$, one of two given points is the incenter of the triangle and the second one is its circumcenter. Using only the ruler without partitions determine where is each of the angles and where is each of the centers.
2012 Sharygin Geometry Olympiad
Two circles with radii 1 meet in points $X, Y$, and the distance between these points also is equal to $1$. Point $C$ lies on the first circle, and lines $CA, CB$ are tangents to the second one. These tangents meet the first circle for the second time in points $B', A'$. Lines $AA'$ and $BB'$ meet in point $Z$. Find angle $XZY$.
2012 Sharygin Geometry Olympiad
Point $D$ lies on side $AB$ of triangle $ABC$. Let $\omega_1$ and $\Omega_1,\omega_2$ and $\Omega_2$ be the incircles and the excircles (touching segment $AB$) of triangles $ACD$ and $BCD.$ Prove that the common external tangents to $\omega_1$ and $\omega_2,$ $\Omega_1$ and $\Omega_2$ meet on $AB$.
2012 Sharygin Geometry Olympiad
Two perpendicular lines pass through the orthocenter of an acute-angled triangle. The sidelines of the triangle cut on each of these lines two segments: one lying inside the triangle and another one lying outside it. Prove that the product of two internal segments is equal to the product of two external segments.



Nikolai Beluhov and Emil Kolev
2012 Sharygin Geometry Olympiad
A circle $\omega$ with center $I$ is inscribed into a segment of the disk, formed by an arc and a chord $AB$. Point $M$ is the midpoint of this arc $AB$, and point $N$ is the midpoint of the complementary arc. The tangents from $N$ touch $\omega$ in points $C$ and $D$. The opposite sidelines $AC$ and $BD$ of quadrilateral $ABCD$ meet in point $X$, and the diagonals of $ABCD$ meet in point $Y$. Prove that points $X, Y, I$ and $M$ are collinear.
2012 Sharygin Geometry Olympiad
An arbitrary point is selected on each of twelve diagonals of the faces of a cube.The centroid of these twelve points is determined. Find the locus of all these centroids.
2012 Sharygin Geometry Olympiad
Given are $n$ $(n > 2)$ points on the plane such that no three of them are collinear. In how many ways this set of points can be divided into two non-empty subsets with non-intersecting convex envelops?
2012 Sharygin Geometry Olympiad
Let $M$ be the midpoint of the base $AC$ of an acute-angled isosceles triangle $ABC$. Let $N$ be the reflection of $M$ in $BC$. The line parallel to $AC$ and passing through $N$ meets $AB$ at point $K$. Determine the value of $\angle AKC$.



(A.Blinkov)
2012 Sharygin Geometry Olympiad
In a triangle $ABC$ the bisectors $BB'$ and $CC'$ are drawn. After that, the whole picture except the points $A, B'$, and $C'$ is erased. Restore the triangle using a compass and a ruler.



(A.Karlyuchenko)
2012 Sharygin Geometry Olympiad
A paper square was bent by a line in such way that one vertex came to a side not containing this vertex. Three circles are inscribed into three obtained triangles (see Figure). Prove that one of their radii is equal to the sum of the two remaining ones.



(L.Steingarts)
2012 Sharygin Geometry Olympiad
Let $ABC$ be an isosceles triangle with $\angle B = 120^o$ . Points $P$ and $Q$ are chosen on the prolongations of segments $AB$ and $CB$ beyond point $B$ so that the rays $AQ$ and $CP$ intersect and are perpendicular to each other. Prove that $\angle PQB = 2\angle PCQ$.



(A.Akopyan, D.Shvetsov)
2012 Sharygin Geometry Olympiad
Do there exist a convex quadrilateral and a point $P$ inside it such that the sum of distances from $P$ to the vertices of the quadrilateral is greater than its perimeter?



(A.Akopyan)
2012 Sharygin Geometry Olympiad
Let $\omega$ be the circumcircle of triangle $ABC$. A point $B_1$ is chosen on the prolongation of side $AB$ beyond point B so that $AB_1 = AC$. The angle bisector of $\angle BAC$ meets $\omega$  again at point $W$. Prove that the orthocenter of triangle $AWB_1$ lies on $\omega$ .



(A.Tumanyan)
2012 Sharygin Geometry Olympiad
The altitudes $AA_1$ and $CC_1$ of an acute-angled triangle $ABC$ meet at point $H$. Point $Q$ is the reflection of the midpoint of $AC$ in line $AA_1$, point $P$ is the midpoint of segment $A_1C_1$. Prove that $\angle QPH = 90^o$.



(D.Shvetsov)
2012 Sharygin Geometry Olympiad
A square is divided into several (greater than one) convex polygons with mutually different numbers of sides. Prove that one of these polygons is a triangle.



(A.Zaslavsky)
2012 Sharygin Geometry Olympiad
The altitudes $AA_1$ and $BB_1$ of an acute-angled triangle ABC meet at point $O$. Let $A_1A_2$ and $B_1B_2$ be the altitudes of triangles $OBA_1$ and $OAB_1$ respectively. Prove that $A_2B_2$ is parallel to $AB$.



(L.Steingarts)
2012 Sharygin Geometry Olympiad
Three parallel lines passing through the vertices $A, B$, and $C$ of triangle $ABC$ meet its circumcircle again at points $A_1, B_1$, and $C_1$ respectively. Points $A_2, B_2$, and $C_2$ are the reflections of points $A_1, B_1$, and $C_1$  in $BC, CA$, and $AB$ respectively. Prove that the lines $AA_2, BB_2, CC_2$ are concurrent.



(D.Shvetsov, A.Zaslavsky)
2012 Sharygin Geometry Olympiad
In triangle $ABC$, the bisector $CL$ was drawn. The incircles of triangles $CAL$ and $CBL$ touch $AB$ at points $M$ and $N$ respectively. Points $M$ and $N$ are marked on the picture, and then the whole picture except the points $A, L, M$, and $N$ is erased. Restore the triangle using a compass and a ruler.



(V.Protasov)
2012 Sharygin Geometry Olympiad
Determine all integer $n > 3$ for which a regular $n$-gon can be divided into equal triangles by several (possibly intersecting) diagonals.



(B.Frenkin)
2012 Sharygin Geometry Olympiad
Let $ABC$ be an isosceles right-angled triangle. Point $D$ is chosen on the prolongation of the hypothenuse $AB$ beyond point $A$ so that $AB = 2AD$. Points $M$ and $N$ on side $AC$ satisfy the relation $AM = NC$. Point $K$ is chosen on the prolongation of $CB$ beyond point $B$ so that $CN = BK$. Determine the angle between lines $NK$ and $DM$.



(M.Kungozhin)
2012 Sharygin Geometry Olympiad
Let $ABC$ be an isosceles triangle with $BC = a$ and $AB = AC = b$. Segment $AC$ is the base of an isosceles triangle $ADC$ with $AD = DC = a$ such that points $D$ and $B$ share the opposite sides of AC. Let $CM$ and $CN$ be the bisectors in triangles $ABC$ and $ADC$ respectively. Determine the circumradius of triangle $CMN$.



(M.Rozhkova)
2012 Sharygin Geometry Olympiad
A convex pentagon $P $ is divided by all its diagonals into ten triangles and one smaller pentagon $P'$. Let $N$ be the sum of areas of five triangles adjacent to the sides of $P$ decreased by the area of $P'$. The same operations are performed with the pentagon $P'$, let $N'$ be the similar difference calculated for this pentagon. Prove that $N > N'$.



(A.Belov)
2012 Sharygin Geometry Olympiad
Let $AH$ be an altitude of an acute-angled triangle $ABC$. Points $K$ and $L$ are the projections of $H$ onto sides $AB$ and $AC$. The circumcircle of $ABC$ meets line $KL$ at points $P$ and $Q$, and meets line $AH$ at points $A$ and $T$. Prove that $H$ is the incenter of triangle $PQT$.



(M.Plotnikov)
2012 Sharygin Geometry Olympiad
Determine all integer $n$ such that a surface of an $n \times n \times n$ grid cube can be pasted in one layer by paper $1 \times 2$ rectangles so that each rectangle has exactly five neighbors (by a line segment).



(A.Shapovalov)
2012 Sharygin Geometry Olympiad
We say that a point inside a triangle is good if the lengths of the cevians passing through this point are inversely proportional to the respective side lengths. Find all the triangles for which the number of good points is maximal.



(A.Zaslavsky, B.Frenkin)
2012 Sharygin Geometry Olympiad
Let $M$ and $I$ be the centroid and the incenter of a scalene triangle $ABC$, and let $r$ be its inradius. Prove that $MI = r/3$ if and only if $MI$ is perpendicular to one of the sides of the triangle.



(A.Karlyuchenko)
2012 Sharygin Geometry Olympiad
Consider a square. Find the locus of midpoints of the hypothenuses of rightangled triangles with the vertices lying on three different sides of the square and not coinciding with its vertices.



(B.Frenkin)
2012 Sharygin Geometry Olympiad
A quadrilateral $ABCD$ with perpendicular diagonals is inscribed into a circle $\omega$. Two arcs $\alpha$ and $\beta$ with diameters AB and $CD$ lie outside $\omega$. Consider two crescents formed by the circle $\omega$ and the arcs $\alpha$  and $\beta$ (see Figure). Prove that the maximal radii of the circles inscribed into these crescents are equal.



(F.Nilov)
2012 Sharygin Geometry Olympiad
Consider a tetrahedron $ABCD$. A point $X$ is chosen outside the tetrahedron so that segment $XD$ intersects face $ABC$ in its interior point. Let $A' , B'$ , and $C'$ be the projections of $D$ onto the planes $XBC, XCA$, and $XAB$ respectively. Prove that $A' B'  + B' C'  + C' A'  \le  DA + DB + DC$.



(V.Yassinsky)
2012 Sharygin Geometry Olympiad
Consider a triangle $ABC$. The tangent line to its circumcircle at point $C$ meets line $AB$ at point $D$. The tangent lines to the circumcircle of triangle $ACD$ at points $A$ and $C$ meet at point $K$. Prove that line $DK$ bisects segment $BC$.



(F.Ivlev)
2012 Sharygin Geometry Olympiad
A point $M$ lies on the side $BC$ of square $ABCD$. Let $X$, $Y$ , and $Z$ be the incenters of

triangles $ABM$, $CMD$, and $AMD$ respectively. Let $H_x$, $H_y$, and $H_z$ be the orthocenters

of triangles $AXB$, $CY D$, and $AZD$. Prove that $H_x$, $H_y$, and $H_z$ are collinear.
2011 Sharygin Geometry Olympiad
Does a convex heptagon exist which can be divided into 2011 equal triangles?
2011 Sharygin Geometry Olympiad
Let $ABC$ be a triangle with sides $AB = 4$ and $AC = 6$. Point $H$ is the projection of vertex $B$ to the bisector of angle $A$. Find $MH$, where $M$ is the midpoint of $BC$.
2011 Sharygin Geometry Olympiad
Let $ABC$ be a triangle with $\angle{A} = 60^\circ$. The midperpendicular of segment $AB$ meets line $AC$ at point $C_1$. The midperpendicular of segment $AC$ meets line $AB$ at point $B_1$. Prove that line $B_1C_1$ touches the incircle of triangle $ABC$.
2011 Sharygin Geometry Olympiad
Segments $AA'$, $BB'$, and $CC'$ are the bisectrices of triangle $ABC$. It is known that these lines are also the bisectrices of triangle $A'B'C'$. Is it true that triangle $ABC$ is regular?
2011 Sharygin Geometry Olympiad
Given triangle $ABC$. The midperpendicular of side $AB$ meets one of the remaining sides at point $C'$. Points $A'$ and $B'$ are defined similarly. Find all triangles $ABC$ such that triangle $A'B'C'$ is regular.
2011 Sharygin Geometry Olympiad
Two unit circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. $M$ is an arbitrary point of $\omega_1$, $N$ is an arbitrary point of $\omega_2$. Two unit circles $\omega_3$ and $\omega_4$ pass through both points $M$ and $N$. Let $C$ be the second common point of $\omega_1$ and $\omega_3$, and $D$ be the second common point of $\omega_2$ and $\omega_4$. Prove that $ACBD$ is a parallelogram.
2011 Sharygin Geometry Olympiad
Points $P$ and $Q$ on sides $AB$ and $AC$ of triangle $ABC$ are such that $PB = QC$. Prove that $PQ < BC$.
2011 Sharygin Geometry Olympiad
The incircle of right-angled triangle $ABC$ ($\angle B = 90^o$) touches $AB,BC,CA$ at points $C_1,A_1,B_1$ respectively. Points $A_2, C_2$ are the reflections of $B_1$ in lines $BC, AB$ respectively. Prove that lines $A_1A_2$ and $C_1C_2$ meet on the median of triangle $ABC$.
2011 Sharygin Geometry Olympiad
Let $H$ be the orthocenter of triangle $ABC$. The tangents to the circumcircles of triangles $CHB$ and $AHB$ at point $H$ meet $AC$ at points $A_1$ and $C_1$ respectively. Prove that $A_1H = C_1H$.
2011 Sharygin Geometry Olympiad
The diagonals of trapezoid $ABCD$ meet at point $O$. Point $M$ of lateral side $CD$ and points $P, Q$ of bases $BC$ and $AD$ are such that segments $MP$ and $MQ$ are parallel to the diagonals of the trapezoid. Prove that line $PQ$ passes through point $O$.
2011 Sharygin Geometry Olympiad
The excircle of right-angled triangle $ABC$ ($\angle B =90^o$) touches side $BC$ at point $A_1$ and touches line $AC$ in point $A_2$. Line $A_1A_2$ meets the incircle of $ABC$ for the first time at point $A'$, point $C'$ is defined similarly. Prove that $AC||A'C'$.
2011 Sharygin Geometry Olympiad
Let $AP$ and $BQ$ be the altitudes of acute-angled triangle $ABC$. Using a compass and a ruler, construct a point $M$ on side $AB$ such that $\angle AQM = \angle BPM$.
2011 Sharygin Geometry Olympiad
a) Find the locus of centroids for triangles whose vertices lie on the sides of a given triangle (each side contains a single vertex).

b)  Find the locus of centroids for tetrahedrons whose vertices lie on the faces of a given tetrahedron (each face contains a single vertex).
2011 Sharygin Geometry Olympiad
In triangle $ABC$, the altitude and the median from vertex $A$ form (together with line $BC$) a triangle such that the bisectrix of angle $A$ is the median; the altitude and the median from vertex $B$ form (together with line AC) a triangle such that the bisectrix of angle $B$ is the bisectrix. Find the ratio of sides for triangle $ABC$.
2011 Sharygin Geometry Olympiad
Given a circle with center $O$ and radius equal to $1$. $AB$ and $AC$ are the tangents to this circle from point $A$. Point $M$ on the circle is such that the areas of quadrilaterals $OBMC$ and $ABMC$ are equal. Find $MA$.
2011 Sharygin Geometry Olympiad
Given are triangle $ABC$ and line $\ell$. The reflections of $\ell$ in $AB$ and $AC$ meet at point $A_1$. Points $B_1, C_1$ are defined similarly. Prove that

a) lines $AA_1, BB_1, CC_1$ concur,

b) their common point lies on the circumcircle of $ABC$

c) two points constructed in this way for two perpendicular lines are opposite.
2011 Sharygin Geometry Olympiad
a) Does there exist a triangle in which the shortest median is longer that the longest bisectrix?

b) Does there exist a triangle in which the shortest bisectrix is longer that the longest altitude?
2011 Sharygin Geometry Olympiad
On the plane, given are $n$ lines in general position, i.e. any two of them aren’t parallel and any three of them don’t concur. These lines divide the plane into several parts. What is

a) the minimal,

b) the maximal number of these parts that can be angles?
2011 Sharygin Geometry Olympiad
Does there exist a nonisosceles triangle such that the altitude from one vertex, the bisectrix from the second one and the median from the third one are equal?
2011 Sharygin Geometry Olympiad
Quadrilateral $ABCD$ is circumscribed around a circle with center $I$. Points $M$ and $N$ are the midpoints of diagonals $AC$ and $BD$. Prove that $ABCD$ is cyclic quadrilateral if and only if $IM : AC = IN : BD$.



Nikolai Beluhov and Aleksey Zaslavsky
2011 Sharygin Geometry Olympiad
On a circle with diameter $AC$, let $B$ be an arbitrary point distinct from $A$ and $C$. Points $M, N$ are the midpoints of chords $AB, BC$, and points $P, Q$ are the midpoints of smaller arcs restricted by these chords. Lines $AQ$ and $BC$ meet at point $K$, and lines $CP$ and $AB$ meet at point $L$. Prove that lines $MQ, NP$ and $KL$ concur.
2011 Sharygin Geometry Olympiad
Let $CX, CY$ be the tangents from vertex $C$ of triangle $ABC$ to the circle passing through the midpoints of its sides. Prove that lines $XY , AB$ and the tangent to the circumcircle of $ABC$ at point $C$ concur.
2011 Sharygin Geometry Olympiad
Given are triangle $ABC$ and line $\ell$ intersecting $BC, CA$ and $AB$ at points $A_1, B_1$ and $C_1$ respectively. Point $A'$ is the midpoint of the segment between the projections of $A_1$ to $AB$ and $AC$. Points $B'$ and $C'$ are defined similarly.

(a) Prove that $A', B'$ and $C'$ lie on some line $\ell'$.

(b) Suppose $\ell$  passes through the circumcenter of $\triangle ABC$. Prove that in this case $\ell'$ passes through the center of its nine-points circle.



M. Marinov and N. Beluhov
2011 Sharygin Geometry Olympiad
Given is an acute-angled triangle $ABC$. On sides $BC, CA, AB$, find points $A', B', C'$ such that the longest side of triangle $A'B'C'$ is minimal.