Moscow City MO 1935-56 (MMO) 136p

geometry problems from old Moscow Mathematical Olympiads (<1997)
with aops links in the names

60 Odd Years of Moscow Mathematical Olympiads (1935-1997) EN with solutions 

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

1935 -1956 complete
1957 -1997 under construction

In years 1942 -44 it did not take place

1935

002 1935 Moscow MO 

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

003 1935 Moscow MO 

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 $\alpha$.

005 1935 Moscow MO 

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

006 1935 Moscow MO 

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.

008 1935 Moscow MO 

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.

009 1935 Moscow MO

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.

011 1935 Moscow MO 

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 $\vartriangle ABC$.

012 1935 Moscow MO 

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.

013 1935 Moscow MO 

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.

014 1935 Moscow MO

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

015 1935 Moscow MO 

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.

1936

023 1936 Moscow MO

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.

025 1936 Moscow MO 

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.

028 1936 Moscow MO

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.

029 1936 Moscow MO 

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 $12$.

031 1936 Moscow MO

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.

1937

033 1937 Moscow MO

* 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.

034 1937 Moscow MO

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.
 

035 1937 Moscow MO

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

038 1938 Moscow MO

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.

039 1938 Moscow MO

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 $A$.

041 1938 Moscow MO

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

1939

045 1939 Moscow MO

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.

047 1939 Moscow MO

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

050 1939 Moscow MO

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 $MN$.

052 1939 Moscow MO

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

057 1940 Moscow MO

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?

060 1940 Moscow MO

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

061 1940 Moscow MO

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.

063 1940 Moscow MO

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 $AD + BD = DC$.

068 1940 Moscow MO

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

071 1941 Moscow MO

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

073 1941 Moscow MO

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

074 1941 Moscow MO

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.

076 1941 Moscow MO

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

078 1941 Moscow MO

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 $\vartriangle ABC$.

082 1941 Moscow MO

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

083 1941 Moscow MO

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.

086 1941 Moscow MO

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 $\vartriangle ABC$.

089 1941 Moscow MO

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.

090 1941 Moscow MO

Construct a right triangle, given two medians drawn to its legs.

1945

094 1945 Moscow MO

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

095 1945 Moscow MO

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.

098 1945 Moscow MO

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.

101 1945 Moscow MO

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 $AQ = \frac{AC}{n + 1}$

102 1945 Moscow MO

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.

1946

106 1946 Moscow MO

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

107 1946 Moscow MO

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 $AB$)

111 1946 Moscow MO

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 $\beta$.

112 1946 Moscow MO

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.

119 1946 Moscow MO

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.

122 1946 Moscow MO

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 $\frac{AB}{PQ }= \frac{CD}{QR} = \frac{EF}{RP}$ .

1947

126 1947 Moscow MO

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.

127 1947 Moscow MO

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

132 1947 Moscow MO

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

135- 1947 Moscow MO

Position the $4$ points on plane so that when measuring of all pairwise distances between them, it turned out only two different numbers. Find all such locations.

1948

146 1948 Moscow MO

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 $ABCD$.

149 1948 Moscow MO

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.

150 1948 Moscow MO

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

151 1948 Moscow MO

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.

152 1948 Moscow MO

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.

153 1948 Moscow MO

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

1949

157 1949 Moscow MO

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.

160 1949 Moscow MO

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.

163 1949 Moscow MO

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.

165 1949 Moscow MO

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.

167 1949 Moscow MO

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.

169 1949 Moscow MO

Construct a convex polyhedron of equal “bricks” shown in Figure. 

170 1949 Moscow MO

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

1950

175 1950 Moscow MO

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

176 1950 Moscow MO

Let $a, b, c$ be the lengths of the sides of a triangle and $A, B, C$, the opposite angles.
Prove that $Aa + Bb + Cc >\frac{Ab + Ac + Ba + Bc + Ca + Cb}{2}$ .

179 1950 Moscow MO

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?

183 1950 Moscow MO

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.

186 1950 Moscow MO

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

1951

189 1951 Moscow MO

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 $\angle B < \angle B', \angle C > \angle C', \angle D < \angle D'$.

191 1951 Moscow MO

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

194 1951 Moscow MO

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.

195 1951 Moscow MO

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?

198 1951 Moscow MO

* 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 $\angle COA = \angle F.$

200 1951 Moscow MO

What figure can the central projection of a triangle be?
(The center of the projection does not lie on the plane of the triangle.)

203 1951 Moscow MO

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 $H$.

205 1951 Moscow MO

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

1952

208 1952 Moscow MO

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.

210 1952 Moscow MO

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

212 1952 Moscow MO

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

215 1952 Moscow MO

$\vartriangle ABC$ 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 $\vartriangle ABC$.

217 1952 Moscow MO

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

220 1952 Moscow MO

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 $OS$.

223 1952 Moscow MO

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

224- 1952 Moscow MO
You are given a segment $AB$. Find the locus of the vertices $C$ of acute-angled triangles $ABC$.

225 1952 Moscow MO

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 $NF$.

228 1952 Moscow MO

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.

229 1952 Moscow MO

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

233 1953 Moscow MO

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

235 1953 Moscow MO

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.)

237 1953 Moscow MO

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.

240 1953 Moscow MO

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 $BB_1$.

242 1953 Moscow MO

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 $AD_1EF$.

243 1953 Moscow MO

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.

245 1953 Moscow MO

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

249 1953 Moscow MO

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

252 1953 Moscow MO

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.

255 1953 Moscow MO

Divide a cube into three equal pyramids.

1954

259 1954 Moscow MO

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

260 1954 Moscow MO

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.

265 1954 Moscow MO

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.

270  1954 Moscow MO

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.

271 1954 Moscow MO

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

275 1954 Moscow MO

How many axes of symmetry can a heptagon have?

279 1954 Moscow MO

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) $OB < \frac{OA_1}{2}$ .
b) $OB \le \frac{OA_1}{4}$ .

280 1954 Moscow MO

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.

284 1954 Moscow MO

How many planes of symmetry can a triangular pyramid have?

1955

288 1955 Moscow MO

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.

289 1955 Moscow MO

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.

293 1955 Moscow MO

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.

296 1955 Moscow MO

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.

297 1955 Moscow MO

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.

300 1955 Moscow MO

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 $AB$.

301 1955 Moscow MO

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?

304 1955 Moscow MO

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.

308 1955 Moscow MO

* 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.

310 1955 Moscow MO

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 $a, b, c$?

312 1955 Moscow MO

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 $A_2C_2 //AC, C_2B_2 // CB, B_2A_2 // BA$.

317 1955 Moscow MO

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?

319 1955 Moscow MO

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 $n$.

1956

320 1956 Moscow MO

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.

325 1956 Moscow MO

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 $DE$.

328 1956 Moscow MO

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 $KLMN$.

329 1956 Moscow MO

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?

330 1956 Moscow MO

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 $r\sqrt2 < a < 2r$.

331 1956 Moscow MO

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  $$\frac{A_1B_1}{B_1A_2}\cdot \frac{A_2B_2}{B_2A_3}\cdot \frac{A_3B_3}{B_3A_4}\cdot ...\cdot \frac{A_nB_n}{B_nA_1}= 1$$

333 1956 Moscow MO

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 $\vartriangle A_1B_1C_1$.

334 1956 Moscow MO

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 $B_1B_2 \perp  A_1A_2, B_2B_3 \perp  A_2A_3, . . . , B_nB_1 \perp A_nA_1$.

339 1956 Moscow MO

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

343 1956 Moscow MO

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.

345 1956 Moscow MO

* 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.

1957


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