Navvye/Olympiad-Problems-CSV huggingface dataset:
2007 Sharygin Geometry Olympiad
Suppose two convex quadrangles are such that the sides of each of them lie on the middle perpendiculars to the sides of the other one. Determine their angles,
2007 Sharygin Geometry Olympiad
Find the locus of centers of regular triangles such that three given points $A, B, C$ lie respectively on three lines containing sides of the triangle.
2007 Sharygin Geometry Olympiad
A boy and his father are standing on a seashore. If the boy stands on his tiptoes, his eyes are at a height of $1$ m above sea-level, and if he seats on father’s shoulders, they are at a height of $2$ m. What is the ratio of distances visible for him in two eases?

(Find the answer to $0,1$, assuming that the radius of Earth equals $6000$ km.)
2007 Sharygin Geometry Olympiad
A rectangle $ABCD$ and a point $P$ are given. Lines passing through $A$ and $B$ and perpendicular to $PC$ and $PD$ respectively, meet at a point $Q$. Prove that $PQ \perp AB$.
2007 Sharygin Geometry Olympiad
On the side $AB$ of a triangle $ABC$, two points $X, Y$ are chosen so that $AX = BY$. Lines $CX$ and $CY$ meet the circumcircle of the triangle, for the second time, at points $U$ and $V$. Prove that all lines $UV$ (for all $X, Y$, given $A, B, C$) have a common point.
2007 Sharygin Geometry Olympiad
In a trapezium with bases $AD$ and $BC$, let $P$ and $Q$ be the middles of diagonals $AC$ and $BD$ respectively. Prove that if $\angle DAQ = \angle CAB$ then $\angle PBA = \angle DBC$.
2007 Sharygin Geometry Olympiad
In a triangle $ABC$, let $AA', BB'$ and $CC'$ be the bisectors. Suppose $A'B' \cap CC' =P$ and $A'C' \cap BB'= Q$. Prove that $\angle PAC = \angle QAB$.
2007 Sharygin Geometry Olympiad
On two sides of an angle, points $A, B$ are chosen. The middle $M$ of the segment $AB$ belongs to two lines such that one of them meets the sides of the angle at points $A_1, B_1$, and the other at points $A_2, B_2$. The lines $A_1B_2$ and $A_2B_1$ meet $AB$ at points $P$ and $Q$. Prove that $M$ is the middle of $PQ$.
2007 Sharygin Geometry Olympiad
What triangles can be cut into three triangles having equal radii of circumcircles?
2007 Sharygin Geometry Olympiad
Determine the locus of vertices of triangles which have prescribed orthocenter and center of circumcircle.
2007 Sharygin Geometry Olympiad
Into an angle $A$ of size $a$, a circle is inscribed tangent to its sides at points $B$ and $C$. A line tangent to this circle at a point M meets the segments $AB$ and $AC$ at points $P$ and $Q$ respectively. What is the minimum $a$ such that the inequality $S_{PAQ}<S_{BMC}$ is possible?
2007 Sharygin Geometry Olympiad
The base of a pyramid is a regular triangle having side of size $1$. Two of three angles at the vertex of the pyramid are right. Find the maximum value of the volume of the pyramid.
2007 Sharygin Geometry Olympiad
There are two pipes on the plane (the pipes are circular cylinders of equal size, $4$ m around). Two of them are parallel and, being tangent one to another in the common generatrix, form a tunnel over the plane. The third pipe is perpendicular to two others and cuts out a chamber in the tunnel. Determine the area of the surface of this chamber.
2007 Sharygin Geometry Olympiad
Determine on which side is the steering wheel disposed in the car depicted in the figure.
2007 Sharygin Geometry Olympiad
By straightedge and compass, reconstruct a right triangle $ABC$ ($\angle C = 90^o$), given the vertices $A, C$ and a point on the bisector of angle $B$.
2007 Sharygin Geometry Olympiad
The diagonals of a convex quadrilateral dissect it into four similar triangles.

Prove that this quadrilateral can also be dissected into two congruent triangles.
2007 Sharygin Geometry Olympiad
Determine the locus of orthocenters of triangles, given the midpoint of a side and the bases of the altitudes drawn to two other sides.
2007 Sharygin Geometry Olympiad
Medians $AA'$ and $BB'$ of triangle $ABC$ meet at point $M$, and $\angle AMB = 120^o$.

Prove that angles $AB'M$ and $BA'M$ are neither both acute nor both obtuse.
2007 Sharygin Geometry Olympiad
Two non-congruent triangles are called analogous  if they can be denoted as $ABC$ and $A'B'C'$ such that $AB = A'B', AC = A'C'$ and $\angle B = \angle B'$ . Do there exist three mutually analogous triangles?
2007 Sharygin Geometry Olympiad
Given a circumscribed quadrilateral $ABCD$.

Prove that its inradius is smaller than the sum of the inradii of triangles $ABC$ and $ACD$.
2007 Sharygin Geometry Olympiad
Points $E$ and $F$ are chosen on the base side $AD$ and the lateral side $AB$ of an isosceles trapezoid $ABCD$, respectively. Quadrilateral $CDEF$ is an isosceles trapezoid as well. Prove that $AE \cdot ED = AF \cdot FB$.
2007 Sharygin Geometry Olympiad
Given a hexagon $ABCDEF$ such that $AB=BC$, $CD=DE$ , $EF=FA$ and $\angle A = \angle C = \angle E $ Prove that $AD, BE, CF$ are concurrent.
2007 Sharygin Geometry Olympiad
Given a triangle $ABC$. An arbitrary point $P$ is chosen on the circumcircle of triangle $ABH$ ($H$ is the orthocenter of triangle $ABC$). Lines $AP$ and $BP$ meet the opposite sidelines of the triangle at points $A' $ and $B'$, respectively. Determine the locus of midpoints of segments $A'B'$.
2007 Sharygin Geometry Olympiad
Reconstruct a triangle, given the incenter, the midpoint of some side and the base of the altitude drawn to this side.
2007 Sharygin Geometry Olympiad
A cube with edge length $2n+ 1$ is dissected into small cubes of size  $1\times 1\times 1$ and bars of size $2\times 2\times 1$. Find the least possible number of cubes in such a dissection.
2007 Sharygin Geometry Olympiad
In an acute triangle $ABC$, altitudes at vertices $A$ and $B$ and bisector line at angle $C$ intersect the circumcircle again at points $A_1, B_1$ and $C_0$. Using the straightedge and compass, reconstruct the triangle by points $A_1, B_1$ and $C_0$.
2007 Sharygin Geometry Olympiad
Points $A', B', C'$ are the bases of the altitudes $AA', BB'$ and $CC'$ of an acute triangle $ABC$. A circle with center $B$ and radius $BB'$ meets line $A'C'$ at points $K$ and $L$ (points $K$ and $A$ are on the same side of line $BB'$). Prove that the intersection point of lines $AK$ and $CL$ belongs to line $BO$ ($O$ is the circumcenter of triangle $ABC$).
2007 Sharygin Geometry Olympiad
Given two circles intersecting at points $P$ and $Q$. Let C be an arbitrary point distinct from $P$ and $Q$ on the former circle. Let lines $CP$ and $CQ$ intersect again the latter circle at points A and B, respectively. Determine the locus of the circumcenters of triangles $ABC$.
2007 Sharygin Geometry Olympiad
A quadrilateral A$BCD$ is inscribed into a circle with center $O$. Points $C', D'$ are the reflections of the orthocenters of triangles $ABD$ and $ABC$ at point $O$. Lines $BD$ and $BD'$ are symmetric with respect to the bisector of angle $ABC$. Prove that lines $AC$ and $AC'$ are symmetric with respect to the bisector of angle $DAB$.
2007 Sharygin Geometry Olympiad
Each edge of a convex polyhedron is shifted such that the obtained edges form the frame of another convex polyhedron. Are these two polyhedra necessarily congruent?
2007 Sharygin Geometry Olympiad
Given are two concentric circles $\Omega$ and $\omega$. Each of the circles $b_1$ and $b_2$ is externally tangent to $\omega$ and internally tangent to $\Omega$, and $\omega$ each of the circles $c_1$ and $c_2$ is internally tangent to both $\Omega$ and $\omega$. Mark each point where one of the circles $b_1, b_2$ intersects one of the circles $c_1$ and $c_2$. Prove that there exist two circles distinct from $b_1, b_2, c_1, c_2$ which contain all $8$ marked points. (Some of these new circles may appear to be lines.)
2006 Sharygin Geometry Olympiad
Two straight lines intersecting at an angle of $46^o$  are the axes of symmetry of the figure $F$ on the plane. What is the smallest number of axes of symmetry this figure can have?
2006 Sharygin Geometry Olympiad
Points $A, B$ move with equal speeds along two equal circles.

Prove that the perpendicular bisector of  $AB$ passes through a fixed point.
2006 Sharygin Geometry Olympiad
The map shows sections of three straight roads connecting the three villages, but the villages themselves are located outside the map. In addition, the fire station located at an equal distance from the three villages is not indicated on the map, although its location is within the map. Is it possible to find this place with the help of a compass and a ruler, if the construction is carried out only within the map?
2006 Sharygin Geometry Olympiad
a) Given two squares $ABCD$ and $DEFG$, with point $E$ lying on the segment $CD$, and points$ F,G$ outside the square $ABCD$. Find the angle between lines $AE$ and $BF$.

b) Two regular pentagons $OKLMN$ and $OPRST$ are given, and the point $P$ lies on the segment $ON$, and the points $R, S, T$ are outside the pentagon $OKLMN$. Find the angle between straight lines $KP$ and $MS$.
2006 Sharygin Geometry Olympiad
a) Fold a $10 \times 10$ square from a $1 \times 118$ rectangular strip.

b) Fold a $10 \times 10$ square from a $1 \times (100+9\sqrt3)$ rectangular strip (approximately $1\times 115.58$).

The strip can be bent, but not torn.
2006 Sharygin Geometry Olympiad
a) Given a segment $AB$ with a point $C$ inside it, which is the chord of a circle of radius $R$.

Inscribe in the formed segment a circle tangent to point $C$ and to the circle of radius $R$.

b)  Given a segment $AB$ with a point $C$ inside it, which is the point of tangency of a circle of radius $r$.

Draw through $A$ and $B$ a circle tangent to a circle of radius $r$.
2006 Sharygin Geometry Olympiad
The point $E$ is taken inside the square $ABCD$, the point $F$ is taken outside, so that the triangles $ABE$ and $BCF$ are congruent . Find the angles of the triangle $ABE$, if it is known that$EF$ is equal to the side of the square, and the angle $BFD$ is right.
2006 Sharygin Geometry Olympiad
The segment $AB$ divides the square into two parts, in each of which a circle can be inscribed.

The radii of these circles are equal to $r_1$ and $r_2$ respectively, where $r_1> r_2$.

Find the length of $AB$.
2006 Sharygin Geometry Olympiad
$L(a)$ is the line connecting the points of the unit circle corresponding to the angles $a$ and $\pi - 2a$.

Prove that if $a + b + c = 2\pi$, then the lines $L (a), L (b)$ and $L (c)$ intersect at one point.
2006 Sharygin Geometry Olympiad
At what $n$ can a regular $n$-gon be cut by disjoint diagonals into $n- 2$ isosceles (including equilateral) triangles?
2006 Sharygin Geometry Olympiad
In the triangle $ABC, O$ is the center of the circumscribed circle, $A ', B', C '$ are the symmetrics  of $A, B, C$ with respect to opposite sides, $ A_1, B_1, C_1$ are the intersection points of the lines $OA'$ and $BC, OB'$ and $AC, OC'$ and $AB$. Prove that the lines $A A_1, BB_1, CC_1$ intersect at one point.
2006 Sharygin Geometry Olympiad
In the triangle $ABC$, the bisector of angle $A$ is equal to the half-sum of the height and median drawn from vertex $A$. Prove that if $\angle A$ is obtuse, then $AB = AC$.
2006 Sharygin Geometry Olympiad
Two straight lines $a$ and $b$ are given and also points $A$ and $B$. Point $X$ slides along the line $a$, and point $Y$ slides along the line $b$, so that $AX // BY$. Find the locus of the intersection point of $AY$ with $XB$.
2006 Sharygin Geometry Olympiad
Given a circle and a fixed point $P$  not lying on it. Find the geometrical locus of the orthocenters of the triangles $ABP$, where $AB$ is the diameter of the circle.
2006 Sharygin Geometry Olympiad
A circle is circumscribed around triangle $ABC$ and a circle is inscribed in it, which touches the sides of the triangle $BC,CA,AB$  at points $A_1,B_1,C_1$, respectively. The line $B_1C_1$ intersects the line $BC$ at the point $P$, and $M$ is the midpoint of the segment $PA_1$. Prove that the segments of the tangents drawn from the point $M$ to the inscribed and circumscribed circle are equal.
2006 Sharygin Geometry Olympiad
Regular triangles are built on the sides of the triangle $ABC$. It turned out that their vertices form a regular triangle. Is the original triangle regular also?
2006 Sharygin Geometry Olympiad
In two circles intersecting at points $A$ and $B$, parallel chords $A_1B_1$ and $A_2B_2$ are drawn.

The lines $AA_1$ and $BB_2$ intersect at the point $X, AA_2$ and $BB_1$ intersect at the point $Y$.

Prove that $XY // A_1B_1$.
2006 Sharygin Geometry Olympiad
Two perpendicular lines are drawn through the orthocenter $H$ of triangle $ABC$, one of which intersects $BC$ at point $X$, and the other intersects $AC$ at point $Y$. Lines $AZ, BZ$ are parallel, respectively with $HX$ and $HY$. Prove that the points $X, Y, Z$ lie on the same line.
2006 Sharygin Geometry Olympiad
Through the midpoints of the sides of the triangle $T$, straight lines are drawn perpendicular to the bisectors of the opposite angles of the triangle. These lines formed a triangle $T_1$. Prove that the center of the circle circumscribed about $T_1$ is in the midpoint of the segment formed by the center of the inscribed circle and the intersection point of the heights of triangle $T$.
2006 Sharygin Geometry Olympiad
Four points are given $A, B, C, D$. Points $A_1, B_1, C_1,D_1$ are orthocenters of the triangles $BCD, CDA, DAB, ABC$ and  $A_2, B_2, C_2,D_2$ are orthocenters of the triangles $B_1C_1D_1, C_1D_1A_1, D_1A_1B_1,A_1B_1C_1$  etc. Prove that the circles passing through the midpoints of the sides of all the triangles intersect at one point.
2006 Sharygin Geometry Olympiad
On the sides $AB, BC, CA$ of triangle $ABC$, points $C', A', B'$ are taken.

Prove that for the areas of the corresponding triangles, the inequality holds:

$$S_{ABC}S^2_{A'B'C'}\ge  4S_{AB'C'}S_{BC'A'}S_{CA'B'}$$and equality is achieved  if and only if the lines $AA', BB', CC'$ intersect at one point.
2006 Sharygin Geometry Olympiad
Given points $A, B$ on a circle and a point $P$ not lying on the circle. $X$ is an arbitrary point of the circle, $Y$ is the intersection point of lines $AX$ and $BP$. Find the locus of the centers of the circles circumscribed around the triangles $PXY$.
2006 Sharygin Geometry Olympiad
$ABCD$ is a convex quadrangle, $G$ is its center of gravity as a homogeneous plate (i.e., the intersection point of two lines, each of which connects the centroids of triangles having a common diagonal).

a) Suppose that around $ABCD$ we can circumscribe a circle centered on $O$. We define $H$ similarly to $G$, taking orthocenters instead of centroids. Then the points of $H, G, O$ lie on the same line and $HG: GO = 2: 1$.

b) Suppose that in $ABCD$ we can inscribe a circle centered on $I$. The Nagel point N of the circumscribed quadrangle is the intersection point of two lines, each of which passes through points on opposite sides of the quadrangle that are symmetric to the tangent points of the inscribed circle relative to the midpoints of the sides. (These lines divide the perimeter of the quadrangle in half). Then $N, G, I$ lie on one straight line, with $NG: GI = 2: 1$.
2006 Sharygin Geometry Olympiad
a) Two perpendicular rays are drawn through a fixed point $P$ inside a given circle, intersecting the circle at points $A$ and $B$. Find the geometric locus of the projections of $P$ on the lines $AB$.

b) Three pairwise perpendicular rays passing through the fixed point $P$ inside a given sphere intersect the sphere at points $A, B, C$. Find the geometrical locus of the projections $P$ on the $ABC$ plane
2006 Sharygin Geometry Olympiad
In the tetrahedron $ABCD$ , the dihedral angles at the $BC, CD$, and $DA$ edges are equal to $\alpha$, and for the remaining edges equal to $\beta$. Find the ratio $AB / CD$.
2006 Sharygin Geometry Olympiad
Four cones are given with a common vertex and the same generatrix, but with, generally speaking, different radii of the bases. Each of them is tangent to two others. Prove that the four tangent points of the circles of the bases of the cones lie on the same circle.
2006 Sharygin Geometry Olympiad
Inscribe the equilateral triangle of the largest perimeter in a given semicircle.
2006 Sharygin Geometry Olympiad
What $n$ is the smallest such that “there is a $n$-gon that can be cut into a triangle, a quadrilateral, ..., a $2006$-gon''?
2006 Sharygin Geometry Olympiad
A parallelogram $ABCD$ is given. Two circles with centers at the vertices $A$ and $C$ pass through $B$. The straight line $\ell$ that passes through $B$ and crosses the circles at second time at points $X, Y$ respectively. Prove that $DX = DY$.
2006 Sharygin Geometry Olympiad
Two equal circles intersect at points $A$ and $B$. $P$ is the point of one of the circles that is different from $A$ and $B, X$ and $Y$ are the second intersection points of the lines of $PA, PB$ with the other circle. Prove that the line passing through $P$ and perpendicular to $AB$ divides one of the arcs $XY$ in half.
2006 Sharygin Geometry Olympiad
Is there a convex polygon with each side equal to some diagonal, and each diagonal equal to some side?
2006 Sharygin Geometry Olympiad
A triangle $ABC$ and a point $P$ inside it are given. $A', B', C'$ are the projections of $P$ onto the straight lines ot the sides $BC,CA,AB$. Prove that the center of the circle circumscribed around the triangle $A'B'C'$  lies inside the triangle $ABC$.
2006 Sharygin Geometry Olympiad
Given a circle of radius $K$. Two other circles, the sum of the radii of which are also equal to $K$, tangent to the circle from the inside. Prove that the line connecting the points of tangency passes through one of the common points of these circles.
2006 Sharygin Geometry Olympiad
Given a circle, point $A$ on it and point $M$ inside it. We consider the chords $BC$ passing through $M$. Prove that the circles passing through the midpoints of the sides of all the triangles $ABC$ are tangent to a fixed circle.
2006 Sharygin Geometry Olympiad
Triangles $ABC$ and $A_1B_1C_1$ are similar and differently oriented. On the segment $AA_1$, a point $A'$ is taken such that $AA' / A_1A'= BC / B_1C_1$. We similarly construct $B'$ and $C'$. Prove that $A', B',C'$ lie on one straight line.
2006 Sharygin Geometry Olympiad
In a non-convex hexagon, each angle is either $90$ or $270$ degrees. Is it true that for some lengths of the sides it can be cut into two hexagons similar to it and unequal to each other?
2006 Sharygin Geometry Olympiad
A straight line passing through the center of the circumscribed circle and the intersection point of the heights of the non-equilateral triangle $ABC$ divides its perimeter and area in the same ratio.Find this ratio.
2006 Sharygin Geometry Olympiad
A convex quadrilateral $ABC$ is given. $A',B',C',D'$ are the  orthocenters of triangles $BCD, CDA, DAB, ABC$ respectively. Prove that in the quadrilaterals $ABCP$ and $A'B'C'D'$, the corresponding diagonals share the intersection points in the same ratio.
2006 Sharygin Geometry Olympiad
Five lines go through one point. Prove that there exists a closed five-segment polygonal line, the vertices and the middle of the segments of which lie on these lines, and each line has exactly one vertex.
2006 Sharygin Geometry Olympiad
The projections of the point $X$ onto the sides of the $ABCD$ quadrangle lie on the same circle. $Y$ is a point symmetric to $X$ with respect to the center of this circle. Prove that the projections of the point $B$ onto the lines $AX,XC, CY, YA$ also lie on the same circle.
2006 Sharygin Geometry Olympiad
Given a circle and a point $P$ inside it, different from the center. We consider pairs of circles tangent to the given internally and to each other at point $P$. Find the locus of the points of intersection of the common external tangents to these circles.
2006 Sharygin Geometry Olympiad
Lines containing the medians of the triangle $ABC$ intersect its circumscribed circle for a second time at the points $A_1, B_1, C_1$. The straight lines passing through $A,B,C$ parallel to opposite sides intersect it at points $A_2, B_2, C_2$. Prove that lines $A_1A_2,B_1B_2,C_1C_2$ intersect at one point.
2006 Sharygin Geometry Olympiad
Can a tetrahedron scan turn out to be a triangle with sides $3, 4$ and $5$ (a tetrahedron can be cut only along the edges)?
2006 Sharygin Geometry Olympiad
A quadrangle was drawn on the board, that you can inscribe and circumscribe  a circle. Marked are the centers of these circles and the intersection point of the lines connecting the midpoints of the opposite sides, after which the quadrangle itself was erased. Restore it with a compass and ruler.
2005 Sharygin Geometry Olympiad
The chords $AC$ and $BD$ of the circle intersect at point $P$. The perpendiculars to $AC$ and $BD$ at points $C$ and $D$, respectively, intersect at point $Q$. Prove that the lines $AB$ and $PQ$ are perpendicular.
2005 Sharygin Geometry Olympiad
Cut a cross made up of five identical squares into three polygons, equal in area and perimeter.
2005 Sharygin Geometry Olympiad
Given a circle and a point $K$ inside it. An arbitrary circle equal to the given one and passing through the point $K$ has a common chord with the given circle. Find the geometric locus of the midpoints of these chords.
2005 Sharygin Geometry Olympiad
At what smallest $n$ is there a convex $n$-gon for which the sines of all angles are equal and the lengths of all sides are different?
2005 Sharygin Geometry Olympiad
There are two parallel lines $p_1$ and $p_2$. Points $A$ and $B$ lie on $p_1$, and $C$ on $p_2$. We will move the segment $BC$ parallel to itself and consider all the triangles $AB'C '$ thus obtained. Find the locus of the points in these triangles:

a) points of intersection of heights,

b) the intersection points of the medians,

c) the centers of the circumscribed circles.
2005 Sharygin Geometry Olympiad
Side $AB$ of triangle $ABC$ was divided into $n$ equal parts (dividing points $B_0 = A, B_1, B_2, ..., B_n = B$), and side $AC$ of this triangle was divided into $(n + 1)$ equal parts (dividing points $C_0 = A, C_1, C_2, ..., C_{n+1} = C$). Colored are the triangles $C_iB_iC_{i+1}$ (where $i = 1,2, ..., n$). What part of the area of the triangle is painted over?
2005 Sharygin Geometry Olympiad
Two circles with radii $1$ and $2$ have a common center at the point $O$. The vertex $A$ of the regular triangle $ABC$ lies on the larger circle, and the middpoint of the base $CD$ lies on the smaller one. What can  the angle $BOC$ be equal to?
2005 Sharygin Geometry Olympiad
Around the convex quadrilateral $ABCD$, three rectangles are circumscribed .

It is known that two of these rectangles are squares. Is it true that the third one is necessarily a square?

(A rectangle is circumscribed around the quadrilateral $ABCD$ if there is one vertex $ABCD$ on each side of the rectangle).
2005 Sharygin Geometry Olympiad
Let $O$ be the center of a regular triangle $ABC$. From an arbitrary point $P$ of the plane, the perpendiculars were drawn on the sides of the triangle. Let $M$ denote the intersection point of the medians of the triangle , having vertices the bases of the perpendiculars. Prove that $M$ is the midpoint of the segment $PO$.
2005 Sharygin Geometry Olympiad
Cut the non-equilateral triangle into four similar triangles, among which not all are the same.
2005 Sharygin Geometry Olympiad
The square was cut into $n^2$ rectangles with sides $a_i \times b_j, i , j= 1,..., n$.

For what is the smallest $n$ in the set $\{a_1, b_1, ..., a_n, b_n\}$ all the numbers can be different?
2005 Sharygin Geometry Olympiad
Construct a quadrangle along the given sides $a, b, c$, and $d$ and the distance $I$ between the midpoints of its diagonals.
2005 Sharygin Geometry Olympiad
A triangle $ABC$ and two lines $\ell_1, \ell_2$ are given. Through an arbitrary point $D$ on the side $AB$, a line parallel to $\ell_1$ intersects the $AC$ at point $E$ and a line parallel to $\ell_2$ intersects the $BC$ at point $F$. Construct a point $D$ for which the segment $EF$ has the smallest length.