geometry problems from old Moscow Mathematical Olympiads (<1997)

with aops links in the names

with aops links in the names

translated by D. Leites (ed.), without figures

written by G. Galperin, A. Tolpygo, P. Grozman, A. Shapovalov, V. Prasolov, A. Fomenko

forum inside aops with MMO problems

forum inside aops with MMO problems

1935 -1956 complete

1957 -1997 under construction

1957 -1997 under construction

In years 1942 -44 it did not take place

1935

Given the lengths of two sides of a triangle and that of the bisector of the angle between these sides, construct the triangle.

The base of a pyramid is an isosceles triangle with the vertex angle $\alpha$. The pyramid’s lateral edges are at angle $\phi$ to the base. Find the dihedral angle $\theta$ at the edge connecting the pyramid’s vertex to that of angle $\alpha$.

Given three parallel straight lines. Construct a square three of whose vertices belong to these lines.

In $\vartriangle ABC$, two straight lines drawn from an arbitrary point $D$ on $AB$ are parallel to $AC$ and BC and intersect $BC$ and $AC$ at $F$ and $G$, respectively. Prove that the sum of the circumferences of the circles circumscribed around $\vartriangle ADG$ and $\vartriangle BDF$ is equal to the circumference of the circle circumscribed around $\vartriangle ABC$.

The unfolding of the lateral surface of a cone is a sector of angle $120^o$. The angles at the base of a pyramid constitute an arithmetic progression with a difference of $15^o$. The pyramid is inscribed in the cone. Consider a lateral face of the pyramid with the smallest area. Find the angle $\alpha$ between the plane of this face and the base.

The median, bisector, and height, all originate at the same vertex of a triangle. Given the intersection points of the median, bisector, and height with the circumscribed circle, construct the triangle.

Find the locus of points on the surface of a cube that serve as the vertex of the smallest angle that subtends the diagonal.

Triangles $\vartriangle ABC$ and $\vartriangle A_1B_1C_1$ lie on different planes. Line $AB$ intersects line $A_1B_1$, line $BC$ intersects line $B_1C_1$ and line $CA$ intersects line $C_1A_1$. Prove that either the three lines $AA_1, BB_1, CC_1$ meet at one point or that they are all parallel.

Prove that it is impossible to divide a scalene triangle into two equal triangles.

What is the largest number of acute angles that a convex polygon can have?

Given an isosceles trapezoid $ABCD$ and a point $P$. Prove that a quadrilateral can be constructed from segments $PA, PB, PC, PD$.

The base of a right pyramid is a quadrilateral whose sides are each of length $a$. The planar angles at the vertex of the pyramid are equal to the angles between the lateral edges and the base. Find the volume of the pyramid.

Prove that if the lengths of the sides of a triangle form an arithmetic progression, then the radius of the inscribed circle is one third of one of the heights of the triangle.

The height of a truncated cone is equal to the radius of its base. The perimeter of a regular hexagon circumscribing its top is equal to the perimeter of an equilateral triangle inscribed in its base. Find the angle $\phi$ between the cone’s generating line and its base.

The unfolding of the lateral surface of a cone is a sector of angle $120^o$. The angles at the base of a pyramid constitute an arithmetic progression with a difference of $15^o$. The pyramid is inscribed in the cone. Consider a lateral face of the pyramid with the smallest area. Find the angle $\alpha$ between the plane of this face and the base.

1936

All rectangles that can be inscribed in an isosceles triangle with two of their vertices on the triangle’s base have the same perimeter. Construct the triangle.

Consider a circle and a point $P$ outside the circle. The angle of given measure with vertex at $P$ subtends a diameter of the circle. Construct the circle’s diameter with ruler and compass.

Given an angle less than $180^o$, and a point $M$ outside the angle. Draw a line through $M$ so that the triangle, whose vertices are the vertex of the angle and the intersection points of its legs with the line drawn, has a given perimeter.

The lengths of a rectangle’s sides and of its diagonal are integers. Prove that the area of the rectangle is an integer multiple of $12$.

Given three planes and a ball in space. In space, find the number of different ways of placing another ball so that it would be tangent the three given planes and the given ball.

Two segments slide along two skew lines. Consider the tetrahedron with vertices at the endpoints of the segments. Prove that the volume of the tetrahedron does not depend on the position of the segments.

1937

* On a plane two points $A$ and $B$ are on the same side of a line. Find point $M$ on the line such that $MA +MB$ is equal to a given length.

Given three points that are not on the same straight line. Three circles pass through each pair of the points so that the tangents to the circles at their intersection points are perpendicular to each other. Construct the circles.

1938

In space $4$ points are given. How many planes equidistant from these points are there? Consider separately

(a) the generic case (the points given do not lie on a single plane) and

(b) the degenerate cases.

Given the base, height and the difference between the angles at the base of a triangle, construct the triangle.

Consider points $A, B, C$. Draw a line through $A$ so that the sum of distances from $B$ and $C$ to this line is equal to the length of a given segment.

(a) the generic case (the points given do not lie on a single plane) and

(b) the degenerate cases.

The following operation is performed over points $O_1, O_2, O_3$ and $A$ in space. The point $A$ is reflected with respect to $O_1$, the resultant point $A_1$ is reflected through $O_2$, and the resultant point $A_2$ through $O_3$. We get some point $A_3$ that we will also consecutively reflect through $O_1, O_2, O_3$. Prove that the point obtained last coincides with $A$.

Given the base, height and the difference between the angles at the base of a triangle, construct the triangle.

1939

Prove that for any triangle the bisector lies between the median and the height drawn from the same vertex.

Given two points $A$ and $B$ and a circle, find a point $X$ on the circle so that points $C$ and $D$ at which lines $AX$ and $BX$ intersect the circle are the endpoints of the chord $CD$ parallel to a given line $MN$.

Consider a regular pyramid and a perpendicular to its base at an arbitrary point $P$. Prove that the sum of the lengths of the segments connecting $P$ to the intersection points of the perpendicular with the planes of the pyramid’s faces does not depend on the location of $P$.

1940

Draw a circle that has a given radius $R$ and is tangent to a given line and a given circle. How many solutions does this problem have?

Construct a circle equidistant from four points on a plane. How many solutions are there?

Given two lines on a plane, find the locus of all points with the difference between the distance to one line and the distance to the other equal to the length of a given segment.

Points $A, B, C$ are vertices of an equilateral triangle inscribed in a circle. Point $D$ lies on the shorter arc $AB$ (not arc $ACB$) . Prove that $AD + BD = DC$.

The center of the circle circumscribing $\vartriangle ABC$ is mirrored through each side of the triangle and three points are obtained: $O_1, O_2, O_3$. Reconstruct $\vartriangle ABC$ from $O_1, O_2, O_3$ if everything else is erased.

1941

Construct a triangle given its height and median — both from the same vertex — and the radius of the circumscribed circle.

Given a quadrilateral, the midpoints $A, B, C, D$ of its consecutive sides, and the midpoints of its diagonals, $P$ and $Q$. Prove that $\vartriangle BCP = \vartriangle ADQ$.

A point $P$ lies outside a circle. Consider all possible lines drawn through $P$ so that they intersect the circle. Find the locus of the midpoints of the chords — segments the circle intercepts on these lines.

On the sides of a parallelogram, squares are constructed outwards. Prove that the centers of these squares are vertices of a square.

Given points $M$ and $N$, the bases of heights $AM$ and $BN$ of $\vartriangle ABC$ and the line to which the side $AB$ belongs. Construct $\vartriangle ABC$.

* Given $\vartriangle ABC$, divide it into the minimal number of parts so that after being flipped over these parts can constitute the same $\vartriangle ABC$.

Consider $\vartriangle ABC$ and a point $M$ inside it. We move $M$ parallel to $BC$ until $M$ meets $CA$, then parallel to $AB$ until it meets $BC$, then parallel to $CA$, and so on. Prove that $M$ traverses a self-intersecting closed broken line and find the number of its straight segments.

Given three points $H_1, H_2, H_3$ on a plane. The points are the reflections of the intersection point of the heights of the triangle $\vartriangle ABC$ through its sides. Construct $\vartriangle ABC$.

Given two skew perpendicular lines in space, find the set of the midpoints of all segments of given length with the endpoints on these lines.

1945

Two circles are tangent externally at one point. Common external tangents are drawn to them and the tangent points are connected. Prove that the sum of the lengths of the opposite sides of the quadrilateral obtained are equal.

A right triangle $ABC$ moves along the plane so that the vertices $B$ and $C$ of the triangle’s acute angles slide along the sides of a given right angle. Prove that point $A$ fills in a line segment and find its length.

The side $AD$ of a parallelogram $ABCD$ is divided into $n$ equal segments. The nearest to $A$ division point $P$ is connected with $B$. Prove that line $BP$ intersects the diagonal $AC$ at point $Q$ such that $AQ = \frac{AC}{n + 1}$

Segments connect vertices $A, B, C$ of $\vartriangle ABC$ with respective points $A_1, B_1, C_1$ on the opposite sides of the triangle. Prove that the midpoints of segments $AA_1, BB_1, CC_1$ do not belong to one straight line.

105 1945 Moscow MO

A circle rolls along a side of an equilateral triangle. The radius of the circle is equal to the height of the triangle. Prove that the measure of the arc intercepted by the sides of the triangle on this circle is equal to $60^o$ at all times.

A circle rolls along a side of an equilateral triangle. The radius of the circle is equal to the height of the triangle. Prove that the measure of the arc intercepted by the sides of the triangle on this circle is equal to $60^o$ at all times.

1946

Given points $A, B, C$ on a line, equilateral triangles $ABC_1$ and $BCA_1$ constructed on segments $AB$ and $BC$, and midpoints $M$ and $N$ of $AA_1$ and $CC_1$, respectively. Prove that $\vartriangle BMN$ is equilateral. (We assume that $B$ lies between $A$ and $C$, and points $A_1$ and $C_1$ lie on the same side of line $AB$)

Given two intersecting planes $\alpha$ and $\beta$ and a point $A$ on the line of their intersection. Prove that of all lines belonging to $\alpha$ and passing through $A$ the line which is perpendicular to the intersection line of $\alpha$ and $\beta$ forms the greatest angle with $\beta$.

Through a point $M$ inside an angle $a$ line is drawn. It cuts off this angle a triangle of the least possible area. Prove that $M$ is the midpoint of the segment on this line that the angle intercepts.

On the legs of $\angle AOB$, the segments $OA$ and $OB$ lie, $OA > OB$. Points $M$ and $N$ on lines $OA$ and $OB$, respectively, are such that $AM = BN = x$. Find $x$ for which the length of $MN$ is minimal.

On the sides $PQ, QR, RP$ of $\vartriangle PQR$ segments $AB, CD, EF$ are drawn.

Given a point $S_0$ inside triangle $\vartriangle PQR$, find the locus of points $S$ for which the sum of the areas of triangles $\vartriangle SAB, \vartriangle SCD$ and $\vartriangle SEF$ is equal to the sum of the areas of triangles $\vartriangle S_0AB, \vartriangle S_0CD, \vartriangle S0EF$.

Consider separately the case $\frac{AB}{PQ }= \frac{CD}{QR} = \frac{EF}{RP}$ .

Given a convex pentagon $ABCDE$, prove that if an arbitrary point $M$ inside the pentagon is connected by lines with all the pentagon’s vertices, then either one or three or five of these lines cross the sides of the pentagon opposite the vertices they pass.

Given a point $S_0$ inside triangle $\vartriangle PQR$, find the locus of points $S$ for which the sum of the areas of triangles $\vartriangle SAB, \vartriangle SCD$ and $\vartriangle SEF$ is equal to the sum of the areas of triangles $\vartriangle S_0AB, \vartriangle S_0CD, \vartriangle S0EF$.

Consider separately the case $\frac{AB}{PQ }= \frac{CD}{QR} = \frac{EF}{RP}$ .

1947

Point $O$ is the intersection point of the heights of an acute triangle $\vartriangle ABC$.

Prove that the three circles which pass:

a) through $O, A, B$,

b) through $O, B, C$, and

c) through $O, C, A$, are equal

Prove that the three circles which pass:

a) through $O, A, B$,

b) through $O, B, C$, and

c) through $O, C, A$, are equal

Given line $AB$ and point $M$. Find all lines in space passing through $M$ at distance $d$.

Consider two triangular pyramids $ABCD$ and $A'BCD$, with a common base $BCD$, and such that $A'$ is inside $ABCD$. Prove that the sum of planar angles at vertex $A'$ of pyramid $A'BCD$ is greater than the sum of planar angles at vertex $A$ of pyramid $ABCD$.

1948

Let $R$ and $r$ be the radii of the circles circumscribed and inscribed, respectively, in a triangle.

Prove that $R \ge 2r$, and that $R = 2r$ only for an equilateral triangle.

Prove that $R \ge 2r$, and that $R = 2r$ only for an equilateral triangle.

Can a figure have a greater than $1$ and finite number of centers of symmetry?

The distance between the midpoints of the opposite sides of a convex quadrilateral is equal to a half sum of lengths of the other two sides. Prove that the first pair of sides is parallel.

a) Two legs of an angle $\alpha$ on a plane are mirrors. Prove that after several reflections in the mirrors any ray leaves in the direction opposite the one from which it came if and only if $\alpha = \frac{90^o}{n}$ for an integer $n$. Find the number of reflections.

b) Given three planar mirrors in space forming an octant (trihedral angle with right planar angles), prove that any ray of light coming into this mirrored octant leaves it, after several reflections in the mirrors, in the direction opposite to the one from which it came. Find the number of reflections.

a) Prove that if a planar polygon has several axes of symmetry, then all of them intersect at one point.

b) A finite solid body is symmetric about two distinct axes. Describe the position of the symmetry planes of the body.

b) Given three planar mirrors in space forming an octant (trihedral angle with right planar angles), prove that any ray of light coming into this mirrored octant leaves it, after several reflections in the mirrors, in the direction opposite to the one from which it came. Find the number of reflections.

* What is the radius of the largest possible circle inscribed into a cube with side $a$?

1949

b) A finite solid body is symmetric about two distinct axes. Describe the position of the symmetry planes of the body.

Prove that for any triangle the circumscribed circle divides the line segment connecting the center of its inscribed circle with the center of one of the exscribed circles in halves.

Prove that if opposite sides of a hexagon are parallel and the diagonals connecting opposite vertices have equal lengths, a circle can be circumscribed around the hexagon.

Consider two triangles, $ABC$ and $DEF$, and any point $O$. We take any point $X$ in $\vartriangle ABC$ and any point $Y$ in $\vartriangle DEF$ and draw a parallelogram $OXY Z$. Prove that the locus of all possible points $Z$ form a polygon. How many sides can it have? Prove that its perimeter is equal to the sum of perimeters of the original triangles.

The midpoints of alternative sides of a hexagon are connected by line segments. Prove that the intersection points of the medians of the two triangles obtained coincide.

What is a centrally symmetric polygon of greatest area one can inscribe in a given triangle?

a) We are given $n$ circles $O_1, O_2, . . . , O_n$, passing through one point $O$. Let $A_1, . . . , A_n$ denote the second intersection points of $O_1$ with $O_2, O_2$ with $O_3$, etc., $O_n$ with $O_1$, respectively. We choose an arbitrary point $B_1$ on $O_1$ and draw a line segment through $A_1$ and $B_1$ to the second intersection with $O_2$ at $B_2$, then draw a line segment through $A_2$ and $B_2$ to the second intersection with $O_3$ at $B_3$, etc., until we get a point $B_n$ on $O_n$. We draw the line segment through $B_n$ and $A_n$ to the second intersection with $O_1$ at $B_{n+1}$. If $B_k$ and $A_k$ coincide for some $k$, we draw the tangent to $O_k$ through $A_k$ until this tangent intersects $O_{k+1}$ at $B_{k+1}$. Prove that $B_{n+1}$ coincides with $B_1$.

b) for $n=3$ the same problem

1950

b) for $n=3$ the same problem

Let $a, b, c$ be the lengths of the sides of a triangle and $A, B, C$, the opposite angles.

Prove that $Aa + Bb + Cc >\frac{Ab + Ac + Ba + Bc + Ca + Cb}{2}$ .

Prove that $Aa + Bb + Cc >\frac{Ab + Ac + Ba + Bc + Ca + Cb}{2}$ .

Two triangular pyramids have common base. One pyramid contains the other. Can the sum of the lengths of the edges of the inner pyramid be longer than that of the outer one?

A circle is inscribed in a triangle and a square is circumscribed around this circle so that no side of the square is parallel to any side of the triangle. Prove that less than half of the square’s perimeter lies outside the triangle.

A spatial quadrilateral is circumscribed around a sphere. Prove that all the tangent points lie in one plane.

1951

Let $ABCD$ and $A'B'C'D'$ be two convex quadrilaterals whose corresponding sides are equal, i.e., $AB = A'B', BC = B'C'$, etc. Prove that if $\angle A > \angle A'$, then $\angle B < \angle B', \angle C > \angle C', \angle D < \angle D'$.

One side of a convex polygon is equal to $a$, the sum of exterior angles at the vertices not adjacent to this side are equal to $120^o$. Among such polygons, find the polygon of the largest area.

We have two concentric circles. A polygon is circumscribed around the smaller circle and is contained entirely inside the greater circle. Perpendiculars from the common center of the circles to the sides of the polygon are extended till they intersect the greater circle. Each of the points obtained is connected with the endpoints of the corresponding side of the polygon . When is the resulting star-shaped polygon the unfolding of a pyramid?

* On a plane, given points $A, B, C$ and angles $\angle D, \angle E, \angle F$ each less than $180^o$ and the sum equal to $360^o$, construct with the help of ruler and protractor a point $O$ such that $\angle AOB = \angle D, \angle BOC = \angle E$ and $\angle COA = \angle F.$

What figure can the central projection of a triangle be?

(The center of the projection does not lie on the plane of the triangle.)

(The center of the projection does not lie on the plane of the triangle.)

A sphere is inscribed in an $n$-angled pyramid. Prove that if we align all side faces of the pyramid with the base plane, flipping them around the corresponding edges of the base, then

(1) all tangent points of these faces to the sphere would coincide with one point, $H$, and

(2) the vertices of the faces would lie on a circle centered at $H$.

(1) all tangent points of these faces to the sphere would coincide with one point, $H$, and

(2) the vertices of the faces would lie on a circle centered at $H$.

Among all orthogonal projections of a regular tetrahedron to all possible planes, find the projection of the greatest area.

The circle is inscribed in $\vartriangle ABC$. Let $L, M, N$ be the tangent points of the circle with sides $AB, AC, BC$, respectively. Prove that $\angle MLN$ is always an acute angle.

1952

Prove that if all faces of a parallelepiped are equal parallelograms, they are rhombuses.

Prove that if the orthocenter divides all heights of a triangle in the same proportion, the triangle is equilateral.

$\vartriangle ABC$ is divided by a straight line $BD$ into two triangles. Prove that the sum of the radii of circles inscribed in triangles $ABD$ and $DBC$ is greater than the radius of the circle inscribed in $\vartriangle ABC$.

Given three skew lines. Prove that they are pair-wise perpendicular to their pair-wise perpendiculars.

A sphere with center at $O$ is inscribed in a trihedral angle with vertex $S$. Prove that the plane passing through the three tangent points is perpendicular to $OS$.

In a convex quadrilateral $ABCD$, let $AB + CD = BC + AD$. Prove that the circle inscribed in $ABC$ is tangent to the circle inscribed in $ACD$.

From a point $C$, tangents $CA$ and $CB$ are drawn to a circle $O$. From an arbitrary point $N$ on the circle, perpendiculars $ND, NE, NF$ are dropped to $AB, CA$ and $CB$, respectively. Prove that the length of $ND$ is the mean proportional of the lengths of $NE$ and $NF$.

How to arrange three right circular cylinders of diameter $a/2$ and height $a$ into an empty cube with side $a$ so that the cylinders could not change position inside the cube? Each cylinder can, however, rotate about its axis of symmetry.

In an isosceles triangle $\vartriangle ABC, \angle ABC = 20^o$ and $BC = AB$. Points $P$ and $Q$ are chosen on sides $BC$ and $AB$, respectively, so that $\angle PAC = 50^o$ and $\angle QCA = 60^o$ . Prove that $\angle PQC = 30^o$ .

Prove that the sum of angles at the longer base of a trapezoid is less than the sum of angles at the shorter base.

A regular star-shaped hexagon is split into $4$ parts. Construct from them a convex polygon.

In an isosceles triangle $\vartriangle ABC, \angle ABC = 20^o$ and $BC = AB$. Points $P$ and $Q$ are chosen on sides $BC$ and $AB$, respectively, so that $\angle PAC = 50^o$ and $\angle QCA = 60^o$ . Prove that $\angle PQC = 30^o$ .

1953

Divide a segment in halves using a right triangle.

(With a right triangle one can draw straight lines and erect perpendiculars but cannot drop perpendiculars.)

(With a right triangle one can draw straight lines and erect perpendiculars but cannot drop perpendiculars.)

Three circles are pair-wise tangent to each other.

Prove that the circle passing through the three tangent points is perpendicular to each of the initial three circles.

Prove that the circle passing through the three tangent points is perpendicular to each of the initial three circles.

Let $AB$ and $A_1B_1$ be two skew segments, $O$ and $O_1$ their respective midpoints. Prove that $OO_1$ is shorter than a half sum of $AA_1$ and $BB_1$.

Let $A$ be a vertex of a regular star-shaped pentagon, the angle at $A$ being less than $180^o$ and the broken line $AA_1BB_1CC_1DD_1EE_1$ being its contour. Lines $AB$ and $DE$ meet at $F$. Prove that polygon $ABB_1CC_1DED_1$ has the same area as the quadrilateral $AD_1EF$.

Given a right circular cone and a point $A$. Find the set of vertices of cones equal to the given one, with axes parallel to that of the given one, and with $A$ inside them.

A quadrilateral is circumscribed around a circle. Its diagonals intersect at the center of the circle. Prove that the quadrilateral is a rhombus.

Let $a, b, c, d$ be the lengths of consecutive sides of a quadrilateral, and $S$ its area. Prove that $S \le \frac{ (a + c)(b + d)}{4}$

Given triangle $\vartriangle A_1A_2A_3$ and a straight line $\ell$ outside it. The angles between the lines $A_1A_2$ and $A_2A_3, A_1A_2$ and $A_2A_3, A_2A_3$ and $A_3A_1$ are equal to $a_3, a_1$ and $a_2$, respectively. The straight lines are drawn through points $A_1, A_2, A_3$ forming with $\ell$ angles of $\pi -a_1, \pi -a_2, \pi -a_3$, respectively. All angles are counted in the same direction from $\ell$ . Prove that these new lines meet at one point.

Given two convex polygons, $A_1A_2...A_n$ and $B_1B_2...B_n$ such that $A_1A_2 = B_1B_2$, $A_2A_3 =B_2B_3$,$ ...$, $A_nA_1 = B_nB_1$ and $n - 3$ angles of one polygon are equal to the respective angles of the other. Find whether these polygons are equal.

From an arbitrary point $O$ inside a convex n-gon, perpendiculars are dropped to the (extensions of the) sides of the $n$-gon. Along each perpendicular a vector is constructed, starting from $O$, directed towards the side onto which the perpendicular is dropped, and of length equal to half the length of the corresponding side. Find the sum of these vectors.

Consider $\vartriangle ABC$ and a point $S$ inside it. Let $A_1, B_1, C_1$ be the intersection points of $AS, BS, CS$ with $BC, AC, AB$, respectively. Prove that at least in one of the resulting quadrilaterals $AB_1SC_1, C_1SA_1B, A_1SB_1C$ both angles at either $C_1$ and $B_1$, or $C_1$ and $A_1$, or $A_1$ and $B_1$ are not acute.

Do there exist points $A, B, C, D$ in space, such that $AB = CD = 8, AC = BD = 10$, and $AD = BC = 13$?

How many axes of symmetry can a heptagon have?

Given four straight lines, $m_1, m_2, m_3, m_4$, intersecting at $O$ and numbered clockwise with $O$ as the center of the clock, we draw a line through an arbitrary point $A_1$ on $m_1$ parallel to $m_4$ until the line meets $m_2$ at $A_2$. We draw a line through $A_2$ parallel to $m_1$ until it meets $m_3$ at $A_3$. We also draw a line through $A_3$ parallel to $m_2$ until it meets $m_4$ at $A_4$. Now, we draw a line through$ A_4$ parallel to $m_3$ until it meets $m_1$ at $B$. Prove that

a) $OB \le \frac{OA_1}{2}$ .

b) $OB \le \frac{OA_1}{4}$ .

a) $OB \le \frac{OA_1}{2}$ .

b) $OB \le \frac{OA_1}{4}$ .

Rays $l_1$ and $l_2$ pass through a point $O$. Segments $OA_1$ and $OB_1$ on $l_1$, and $OA_2$ and $OB_2$ on $l_2$, are drawn so that $\frac{OA_1}{OA_2} \ne \frac{OB_1}{OB_2}$ . Find the set of all intersection points of lines $A_1A_2$ and $B_1B_2$ as $l_2$ rotates around $O$ while $l_1$ is fixed.

How many planes of symmetry can a triangular pyramid have?

1955

We are given a right triangle $ABC$ and the median $BD$ drawn from the vertex $B$ of the right angle. Let the circle inscribed in $\vartriangle ABD$ be tangent to side $AD$ at $K$. Find the angles of $\vartriangle ABC$ if $K$ divides $AD$ in halves.

Consider an equilateral triangle $\vartriangle ABC$ and points $D$ and $E$ on the sides $AB$ and $BC$csuch that $AE = CD$. Find the locus of intersection points of $AE$ with $CD$ as points $D$ and $E$ vary.

Consider a quadrilateral $ABCD$ and points $K, L, M, N$ on sides $AB, BC, CD$ and $AD$, respectively, such that $KB = BL = a, MD = DN = b$ and $KL \nparallel MN$. Find the set of all the intersection points of $KL$ with $MN$ as $a$ and $b$ vary.

There are four points $A, B, C, D$ on a circle. Circles are drawn through each pair of neighboring points. Denote the intersection points of neighboring circles by $A_1, B_1, C_1, D_1$. (Some of these points may coincide with previously given ones.) Prove that points $A_1, B_1, C_1, D_1$ lie on one circle.

Given two distinct nonintersecting circles none of which is inside the other. Find the locus of the midpoints of all segments whose endpoints lie on the circles.

Inside $\vartriangle ABC$, there is fixed a point $D$ such that $AC - DA > 1$ and $BC - BD > 1$. Prove that $EC - ED > 1$ for any point $E$ on segment $AB$.

Given a trihedral angle with vertex $O$. Find whether there is a planar section $ABC$ such that the angles $\angle OAB, \angle OBA, \angle OBC, \angle OCB, \angle OAC, \angle OCA$ are acute?

The centers $O_1, O_2$ and $O_3$ of circles exscribed about $\vartriangle ABC$ are connected.

Prove that $O_1O_2O_3$ is an acute-angled one.

Prove that $O_1O_2O_3$ is an acute-angled one.

* Two circles are tangent to each other externally, and to a third one from the inside. Two common tangents to the first two circles are drawn, one outer and one inner. Prove that the inner tangent divides in halves the arc intercepted by the outer tangent on the third circle.

Let the inequality $Aa(Bb + Cc) + Bb(Aa + Cc) + Cc(Aa + Bb) > \frac{ABc^2 + BCa^2 + CAb^2}{2}$ with given $a > 0, b > 0, c > 0$ hold for all $A > 0, B > 0, C > 0$. Is it possible to construct a triangle with sides of lengths $a, b, c$?

Given $\vartriangle ABC$, points $C_1, A_1, B_1$ on sides $AB, BC, CA$, respectively, such that $\frac{AC_1}{C_1B}= \frac{BA_1}{A_1C}= \frac{CB_1}{B_1A}=\frac{1}{n}$ and points $C_2, A_2, B_2$ on sides $A_1B_1, B_1C_1, C_1A_1$ of $\vartriangle A_1B_1C_1$, respectively, such that $\frac{A_1C_2}{C_2B_1}= \frac{B_1A_2}{A_2C_1}= \frac{C_1B_2}{B_2A_1}= n$. Prove that $A_2C_2 //AC, C_2B_2 // CB, B_2A_2 // BA$.

Given $\vartriangle ABC$, points $C_1, A_1, B_1$ on sides $AB, BC, CA$, respectively, such that $\frac{AC_1}{C_1B}= \frac{BA_1}{A_1C}= \frac{CB_1}{B_1A}=\frac{1}{n}$ and points $C_2, A_2, B_2$ on sides $A_1B_1, B_1C_1, C_1A_1$ of $\vartriangle A_1B_1C_1$, respectively, such that $\frac{A_1C_2}{C_2B_1}= \frac{B_1A_2}{A_2C_1}= \frac{C_1B_2}{B_2A_1}= n$. Prove that $A_2C_2 //AC, C_2B_2 // CB, B_2A_2 // BA$.

A right circular cone stands on plane $P$. The radius of the cone’s base is $r$, its height is $h$. A source of light is placed at distance $H$ from the plane, and distance $1$ from the axis of the cone. What is the illuminated part of the disc of radius $R$, that belongs to $P$ and is concentric with the disc forming the base of the cone?

Consider $\vartriangle A_0B_0C_0$ and points $C_1, A_1, B_1$ on its sides $A_0B_0, B_0C_0, C_0A_0$, points $C_2, A_2,B_2$ on the sides $A_1B_1, B_1C_1, C_1A_1$ of $\vartriangle A_1B_1C_1$, respectively, etc., so that

$\frac{A_0B_1}{B_1C_0}= \frac{B_0C_1}{C_1A_0}= \frac{C_0A_1}{A_1B_0}= k$, $\frac{A_1B_2}{B_2C_1}= \frac{B_1C_2}{C_2A_1}= \frac{C_1A_2}{A_2B_1}= \frac{1}{k^2}$

and, in general,

$\frac{A_nB_{n+1}}{B_{n+1}C_n}= \frac{B_nC_{n+1}}{C_{n+1}A_n}= \frac{C_nA_{n+1}}{A_{n+1}B_n}

=$ $k^{2n}$ for $n$ even , $\frac{1}{k^{2n}}$ for $n$ odd.

Prove that 4ABC formed by lines $A_0A_1, B_0B_1, C_0C_1$ is contained in $\vartriangle A_nB_nC_n$ for any $n$.

under construction

$\frac{A_0B_1}{B_1C_0}= \frac{B_0C_1}{C_1A_0}= \frac{C_0A_1}{A_1B_0}= k$, $\frac{A_1B_2}{B_2C_1}= \frac{B_1C_2}{C_2A_1}= \frac{C_1A_2}{A_2B_1}= \frac{1}{k^2}$

and, in general,

$\frac{A_nB_{n+1}}{B_{n+1}C_n}= \frac{B_nC_{n+1}}{C_{n+1}A_n}= \frac{C_nA_{n+1}}{A_{n+1}B_n}

=$ $k^{2n}$ for $n$ even , $\frac{1}{k^{2n}}$ for $n$ odd.

Prove that 4ABC formed by lines $A_0A_1, B_0B_1, C_0C_1$ is contained in $\vartriangle A_nB_nC_n$ for any $n$.

1956

Prove that there are no four points $A, B, C, D$ on a plane such that all triangles $\vartriangle ABC,\vartriangle BCD, \vartriangle CDA, \vartriangle DAB$ are acute ones.

On sides $AB$ and $CB$ of $\vartriangle ABC$ there are drawn equal segments, $AD$ and $CE$, respectively, of arbitrary length (but shorter than min($AB,BC$)). Find the locus of midpoints of all possible segments $DE$.

1957
In a convex quadrilateral $ABCD$, consider quadrilateral $KLMN$ formed by the centers of mass of triangles $ABC, BCD, DBA, CDA$. Prove that the straight lines connecting the midpoints of the opposite sides of quadrilateral $ABCD$ meet at the same point as the straight lines connecting the midpoints of the opposite sides of $KLMN$.

Consider positive numbers $h, s_1, s_2$, and a spatial triangle $\vartriangle ABC$. How many ways are there to select a point $D$ so that the height of tetrahedron $ABCD$ dropped from $D$ equals $h$, and the areas of faces $ACD$ and $BCD$ equal $s_1$ and $s_2$, respectively?

A square of side $a$ is inscribed in a triangle so that two of the square’s vertices lie on the base, and the other two lie on the sides of the triangle. Prove that if $r$ is the radius of the circle inscribed in the triangle, then $r\sqrt2 < a < 2r$.

Given a closed broken line $A_1A_2A_3...A_n$ in space and a plane intersecting all its segments, $A_1A_2$ at $B_1, A_2A_3$ at $B_2,... , A_nA_1$ at $Bn$, prove that $$\frac{A_1B_1}{B_1A_2}\cdot \frac{A_2B_2}{B_2A_3}\cdot \frac{A_3B_3}{B_3A_4}\cdot ...\cdot \frac{A_nB_n}{B_nA_1}= 1$$

Let $O$ be the center of the circle circumscribed around $\vartriangle ABC$, let $A_1, B_1, C_1$ be symmetric to $O$ through respective sides of $\vartriangle ABC$. Prove that all heights of $\vartriangle A_1B_1C_1$ pass through $O$, and all heights of $\vartriangle ABC$ pass through the center of the circle circumscribed around $\vartriangle A_1B_1C_1$.

a) Points $A_1, A_2, A_3, A_4, A_5, A_6$ divide a circle of radius $1$ into six equal arcs. Ray $\ell_1$ from $A_1$ connects $A_1$ with $A_2$, ray $\ell_2$ from $A_2$ connects $A_2$ with $A_3$, and so on, ray $\ell_6$ from $A_6$ connects $A_6$ with $A_1$. From a point $B_1$ on $\ell_1$ the perpendicular is dropped to $\ell_6$, from the foot of this perpendicular another perpendicular is dropped to $\ell_5$, and so on. Let the foot of the $6$-th perpendicular coincide with $B_1$. Find the length of segment $A_1B_1$.

b) Find points $B_1, B_2,... , B_n$ on the extensions of sides $A_1A_2, A_2A_3,... , A_nA_1$ of a regular $n$-gon $A_1A_2...A_n$ such that $B_1B_2 \perp A_1A_2, B_2B_3 \perp A_2A_3, . . . , B_nB_1 \perp A_nA_1$.

b) Find points $B_1, B_2,... , B_n$ on the extensions of sides $A_1A_2, A_2A_3,... , A_nA_1$ of a regular $n$-gon $A_1A_2...A_n$ such that $B_1B_2 \perp A_1A_2, B_2B_3 \perp A_2A_3, . . . , B_nB_1 \perp A_nA_1$.

Find the union of all projections of a given line segment $AB$ to all lines passing through a given point $O$.

A quadrilateral is circumscribed around a circle. Prove that the straight lines connecting neighboring tangent points either meet on the extension of a diagonal of the quadrilateral or are parallel to it.

* Prove that if the trihedral angles at each of thihe vertices of a triangular pyramid are formed by the identical planar angles, then all faces of this pyramid are equal.

under construction

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