geometry problems from All - Soviet Union Math Olympiads

with aops links in the names

1961-66 All Russian, 1967-91 All Soviet Union

All Soviet Union Mathematical Olympiad 1961-92 EN with solutions,

Russian Mathematical Olympiad 1995-2002 with partial solutions
both by John Scholes (Kalva)

014 1962 All Russian (also here)

Given the circumference $s$ and the straight line $l$, passing through the centre $O$ of $s$. Another circumference $s'$ passes through the point $O$ and has its centre on the $l$. Describe the set of the points $M$, where the common tangent of $s$ and $s'$ touches $s'$.

018 1962 All Russian (also here)

Given two sides of the triangle. Build that triangle, if medians to those sides are orthogonal.

020 1962 All Russian (also here)

Given right pentagon $ABCDE$. $M$ is an arbitrary point inside $ABCDE$ or on its side. Let the distances $|MA|, |MB|, ... , |ME|$ be renumerated and denoted with $r_1\le r_2\le r_3\le r_4\le r_5$. a) Find all the positions of the $M$, giving $r_3$ the minimal possible value.

b) Find all the positions of the $M$, giving $r_3$ the maximal possible value.

023 1962 All Russian (also here)

What maximal area can have a triangle if its sides $a,b,c$ satisfy inequality $0\le a\le 1\le b\le 2\le c\le 3$ ?

027 1963 All Russian

Given $5$ circumferences, every four of them have a common point. Prove that there exists a point that belongs to all five circumferences.

094 1968 All Soviet Union

with aops links in the names

__named as:__

1961-66 All Russian, 1967-91 All Soviet Union

1992 Commonwealth of Independent States

translated by S/W engineer Vladimir Pertsel

All Soviet Union Mathematical Olympiad 1961-92 EN with solutions,

Russian Mathematical Olympiad 1995-2002 with partial solutions

002 1961 All Russian (also here)

Given a rectangle $A_1A_2A_3A_4$. Four circles with $A_i$ as their centres have their radiuses $r_1, r_2, r_3, r_4$; and $r_1+r_3=r_2+r_4<d$, where d is a diagonal of the rectangle. Two pairs of the outer common tangents to {the first and the third} and {the second and the fourth} circumferences make a quadrangle. Prove that You can inscribe a circle into that quadrangle.

006 1961 All Russian (part a also here) (part b also here)

Given a rectangle $A_1A_2A_3A_4$. Four circles with $A_i$ as their centres have their radiuses $r_1, r_2, r_3, r_4$; and $r_1+r_3=r_2+r_4<d$, where d is a diagonal of the rectangle. Two pairs of the outer common tangents to {the first and the third} and {the second and the fourth} circumferences make a quadrangle. Prove that You can inscribe a circle into that quadrangle.

006 1961 All Russian (part a also here) (part b also here)

a) Points $A$ and $B$ move uniformly and with equal angle speed along the circumferences with $O_a$ and $O_b$ centres (both clockwise). Prove that a vertex $C$ of the equilateral triangle $ABC$ also moves along a certain circumference uniformly.

b) The distance from the point $P$ to the vertices of the equilateral triangle $ABC$ equal $|AP|=2, |BP|=3$. Find the maximal value of $CP$.

1962

013 1962 All Russian (also here)

Given points $A' ,B' ,C' ,D',$ on the continuation of the $[AB], [BC], [CD], [DA]$ sides of the convex quadrangle $ABCD$, such, that the following pairs of vectors are equal: $[BB']=[AB], [CC']=[BC], [DD']=[CD], [AA']=[DA].$ Prove that the quadrangle $A'B'C'D'$ area is five times more than the quadrangle $ABCD$ area.

b) The distance from the point $P$ to the vertices of the equilateral triangle $ABC$ equal $|AP|=2, |BP|=3$. Find the maximal value of $CP$.

1962

Given points $A' ,B' ,C' ,D',$ on the continuation of the $[AB], [BC], [CD], [DA]$ sides of the convex quadrangle $ABCD$, such, that the following pairs of vectors are equal: $[BB']=[AB], [CC']=[BC], [DD']=[CD], [AA']=[DA].$ Prove that the quadrangle $A'B'C'D'$ area is five times more than the quadrangle $ABCD$ area.

014 1962 All Russian (also here)

Given the circumference $s$ and the straight line $l$, passing through the centre $O$ of $s$. Another circumference $s'$ passes through the point $O$ and has its centre on the $l$. Describe the set of the points $M$, where the common tangent of $s$ and $s'$ touches $s'$.

018 1962 All Russian (also here)

Given two sides of the triangle. Build that triangle, if medians to those sides are orthogonal.

020 1962 All Russian (also here)

Given right pentagon $ABCDE$. $M$ is an arbitrary point inside $ABCDE$ or on its side. Let the distances $|MA|, |MB|, ... , |ME|$ be renumerated and denoted with $r_1\le r_2\le r_3\le r_4\le r_5$. a) Find all the positions of the $M$, giving $r_3$ the minimal possible value.

b) Find all the positions of the $M$, giving $r_3$ the maximal possible value.

022 1962 All Russian (also here)

The $M$ point is a middle of a isosceles triangle base $[AC]$. $[MH]$ is orthogonal to $[BC]$ side. Point $P$ is the middle of the segment $[MH]$. Prove that $[AH]$ is orthogonal to $[BP]$.

The $M$ point is a middle of a isosceles triangle base $[AC]$. $[MH]$ is orthogonal to $[BC]$ side. Point $P$ is the middle of the segment $[MH]$. Prove that $[AH]$ is orthogonal to $[BP]$.

What maximal area can have a triangle if its sides $a,b,c$ satisfy inequality $0\le a\le 1\le b\le 2\le c\le 3$ ?

027 1963 All Russian

Given $5$ circumferences, every four of them have a common point. Prove that there exists a point that belongs to all five circumferences.

1963

a) Each diagonal of the quadrangle halves its area. Prove that it is a parallelogram.

b) Three main diagonals of the hexagon halve its area. Prove that they intersect in one point.

[If $ABCDEF$ is a hexagon, then the main diagonals are $AD$, $BE$ and $CF$.]

031 1963 All Russian (also here)

Given two fixed points $A$ and $B$ .The point $M$ runs along the circumference containing $A$ and $B$. $K$ is the middle of the segment $[MB]$. $[KP]$ is a perpendicular to the line $(MA)$.

a) Prove that all the possible lines $(KP)$ pass through one point.

b) Find the set of all the possible points $P$.

b) Three main diagonals of the hexagon halve its area. Prove that they intersect in one point.

[If $ABCDEF$ is a hexagon, then the main diagonals are $AD$, $BE$ and $CF$.]

031 1963 All Russian (also here)

Given two fixed points $A$ and $B$ .The point $M$ runs along the circumference containing $A$ and $B$. $K$ is the middle of the segment $[MB]$. $[KP]$ is a perpendicular to the line $(MA)$.

a) Prove that all the possible lines $(KP)$ pass through one point.

b) Find the set of all the possible points $P$.

032 1963 All Russian (also here)

Given equilateral triangle with the side $l$. What is the minimal length $d$ of a brush (segment), that will paint all the triangle, if its ends are moving along the sides of the triangle.

Given equilateral triangle with the side $l$. What is the minimal length $d$ of a brush (segment), that will paint all the triangle, if its ends are moving along the sides of the triangle.

035 1963 All Russian (also here)

Given a triangle $ABC$. We build two angle bisectors in the corners $A$ and $B$. Than we build two lines parallel to those ones through the point $C$. $D$ and $E$ are intersections of those lines with the bisectors. It happens, that $(DE)$ line is parallel to $(AB)$. Prove that the triangle is isosceles.

Given a triangle $ABC$. We build two angle bisectors in the corners $A$ and $B$. Than we build two lines parallel to those ones through the point $C$. $D$ and $E$ are intersections of those lines with the bisectors. It happens, that $(DE)$ line is parallel to $(AB)$. Prove that the triangle is isosceles.

040 1963 All Russian

Given an isosceles triangle. Find the set of the points inside the triangle such, that the distance from that point to the base equals to the geometric mean of the distances to the sides.

1964

041 1964 All Russian

The two heights in the triangle are not less than the respective sides. Find the angles.

The two heights in the triangle are not less than the respective sides. Find the angles.

045 1964 All Russian

a) Given a convex hexagon $ABCDEF$ with all the equal angles.

Prove that $|AB|-|DE| = |EF|-|BC| = |CD|-|FA|$.

b) The opposite problem:

Prove that it is possible to build a convex hexagon with equal angles of six segments $a_1,a_2,...,a_6$, whose lengths satisfy the condition $a_1-a_4 = a_5-a_2 = a_3-a_6$

047 1964 All Russian

a) Given a convex hexagon $ABCDEF$ with all the equal angles.

Prove that $|AB|-|DE| = |EF|-|BC| = |CD|-|FA|$.

b) The opposite problem:

Prove that it is possible to build a convex hexagon with equal angles of six segments $a_1,a_2,...,a_6$, whose lengths satisfy the condition $a_1-a_4 = a_5-a_2 = a_3-a_6$

047 1964 All Russian

Four perpendiculars are drawn from the vertices of a convex quadrangle to its diagonals.

Prove that their bases make a quadrangle similar to the given one.

Prove that their bases make a quadrangle similar to the given one.

049 1964 All Russian

A honeybug crawls along the honeycombs with the unite length of their hexagons. He has moved from the node $A$ to the node $B$ along the shortest possible trajectory. Prove that the half of his way he moved in one direction.

Given a lattice of regular hexagons. A bug crawls from vertex A to vertex B along the edges of the hexagons, taking the shortest possible path (or one of them). Prove that it travels a distance at least AB/2 in one direction. If it travels exactly AB/2 in one direction, how many edges does it traverse?

A honeybug crawls along the honeycombs with the unite length of their hexagons. He has moved from the node $A$ to the node $B$ along the shortest possible trajectory. Prove that the half of his way he moved in one direction.

*other formulation:*Given a lattice of regular hexagons. A bug crawls from vertex A to vertex B along the edges of the hexagons, taking the shortest possible path (or one of them). Prove that it travels a distance at least AB/2 in one direction. If it travels exactly AB/2 in one direction, how many edges does it traverse?

050 1964 All Russian

The quadrangle $ABCD$ is outscribed around the circle with the centre $O$. Prove that the sum of $AOB$ and $COD$ angles equals $180$ degrees.

The quadrangle $ABCD$ is outscribed around the circle with the centre $O$. Prove that the sum of $AOB$ and $COD$ angles equals $180$ degrees.

055 1964 All Russian

Let $ABCD$ be an outscribed trapezoid, $E$ is a point of its diagonals intersection, $r_1,r_2,r_3,r_4$ -- the radiuses of the circles inscribed in the triangles $ABE, BCE, CDE, DAE$ respectively. Prove that $1/(r_1)+1/(r_3) = 1/(r_2)+1/(r_4)$.

Let $ABCD$ be an outscribed trapezoid, $E$ is a point of its diagonals intersection, $r_1,r_2,r_3,r_4$ -- the radiuses of the circles inscribed in the triangles $ABE, BCE, CDE, DAE$ respectively. Prove that $1/(r_1)+1/(r_3) = 1/(r_2)+1/(r_4)$.

1965

058 1965 All Russian

A circle is outscribed around the triangle $ABC$. Chords, from the middle of the arc $AC$ to the middles of the arcs $AB$ and $BC$, intersect sides $[AB]$ and $[BC]$ in the points $D$ and $E$. Prove that $(DE)$ is parallel to $(AC)$ and passes through the centre of the inscribed circle.

A circle is outscribed around the triangle $ABC$. Chords, from the middle of the arc $AC$ to the middles of the arcs $AB$ and $BC$, intersect sides $[AB]$ and $[BC]$ in the points $D$ and $E$. Prove that $(DE)$ is parallel to $(AC)$ and passes through the centre of the inscribed circle.

062 1965 All Russian

What is the maximal possible length of the segment, being cut out by the sides of the triangle on the tangent to the inscribed circle, being drawn parallel to the base, if the triangle's perimeter equals $2p$?

What is the maximal possible length of the segment, being cut out by the sides of the triangle on the tangent to the inscribed circle, being drawn parallel to the base, if the triangle's perimeter equals $2p$?

070 1965 All Russian

Prove that the sum of the lengths of the polyhedron edges exceeds its tripled diameter (distance between two farest vertices).

Prove that the sum of the lengths of the polyhedron edges exceeds its tripled diameter (distance between two farest vertices).

1966

a) Points $B$ and $C$ are inside the segment $[AD]$. $|AB|=|CD|$. Prove that for all of the points P on the plane holds inequality $|PA|+|PD|>|PB|+|PC|$.

b) Given four points $A,B,C,D$ on the plane. For all of the points $P$ on the plane holds inequality $|PA|+|PD| > |PB|+|PC|$. Prove that points $B$ and C are inside the segment $[AD]$ and$ |AB|=|CD|$..

076 1966 All Russian

A rectangle $ABCD$ is drawn on the cross-lined paper with its sides laying on the lines, and $|AD|$ is $k$ times more than $|AB|$ ($k$ is an integer). All the shortest paths from $A$ to $C$ coming along the lines are considered. Prove that the number of those with the first link on $[AD]$ is $k$ times more then of those with the first link on $[AB]$.

078 1966 All Russian (also here)

Prove that you can always pose a circle of radius $S/P$ inside a convex polygon with the perimeter $P$ and area $S$.

b) Given four points $A,B,C,D$ on the plane. For all of the points $P$ on the plane holds inequality $|PA|+|PD| > |PB|+|PC|$. Prove that points $B$ and C are inside the segment $[AD]$ and$ |AB|=|CD|$..

076 1966 All Russian

A rectangle $ABCD$ is drawn on the cross-lined paper with its sides laying on the lines, and $|AD|$ is $k$ times more than $|AB|$ ($k$ is an integer). All the shortest paths from $A$ to $C$ coming along the lines are considered. Prove that the number of those with the first link on $[AD]$ is $k$ times more then of those with the first link on $[AB]$.

078 1966 All Russian (also here)

Prove that you can always pose a circle of radius $S/P$ inside a convex polygon with the perimeter $P$ and area $S$.

080 1966 All Russian

Given a triangle $ABC$. Consider all the tetrahedrons $PABC$ with $PH$ -- the smallest of all tetrahedron's heights. Describe the set of all possible points $H$.

1967

Given a triangle $ABC$. Consider all the tetrahedrons $PABC$ with $PH$ -- the smallest of all tetrahedron's heights. Describe the set of all possible points $H$.

1967

084 1967 All Soviet Union

092 1967 All Soviet Union

Three vertices $KLM$ of the rhombus (diamond) $KLMN$ lays on the sides $[AB], [BC]$ and $[CD]$ of the given unit square. Find the area of the set of all the possible vertices $N$.

**(1st ASU)**
The maximal height $|AH|$ of the acute-angled triangle $ABC$ equals the median $|BM|$.

Prove that the angle $ABC$ isn't greater than $60$ degrees.

b) The height $|AH|$ of the acute-angled triangle ABC equals the median $|BM|$ and bisectrix $|CD|$. Prove that the angle $ABC$ is equilateral.

Three vertices $KLM$ of the rhombus (diamond) $KLMN$ lays on the sides $[AB], [BC]$ and $[CD]$ of the given unit square. Find the area of the set of all the possible vertices $N$.

1968

Given an octagon with the equal angles. The lengths of all the sides are integers.

Prove that the opposite sides are equal in pairs.

099 1968 All Soviet Union

101 1968 All Soviet Union

103 1968 All Soviet Union

The difference between the maximal and the minimal diagonals of the right $n$-angle equals to its side ( $n > 5$ ). Find $n$.

Given two acute-angled triangles $ABC$ and $A'B'C'$ with the points $O$ and $O'$ inside. Three pairs of the perpendiculars are drawn: $[OA_1]$ to the side $[BC]$, $[O'A'_1]$ to the side $[B'C']$, $[OB_1]$ to the side $[AC]$, $[O'B'_1]$ to the side $[A'C']$, $[OC_1] $ to the side $[AB]$, $[O'C'_1]$ to the side $[A'B']$;

Given a triangle $ABC$, point $D$ on $[AB], E$ on $[AC]$, $|AD| = |DE| = |AC| , |BD| = |AE| , DE$ is parallel to $BC$. Prove that the length $|BD|$ equals to the side of a right decagon (ten-angle) inscribed in a circle with the radius $R=|AC|$.

104 1968 All Soviet Union

106 1968 All Soviet Union

112 1968 All Soviet Union

114 1968 All Soviet Union

Three spheres are built so that the edges $[AB], [BC], [AD]$ of the tetrahedron $ABCD$ are their respective diameters. Prove that the spheres cover all the tetrahedron.

Medians divide the triangle onto $6$ smaller ones. $4$ of the circles inscribed in those small ones are equal. Prove that the triangle is equilateral.

The circle inscribed in the triangle $ABC$ touches the side $[AC]$ in the point $K$. Prove that the line connecting the middle of the $[AC]$ side with the centre of the circle halves the $[BK]$ segment.

Given a quadrangle $ABCD$. The lengths of all its sides and diagonals are the rational numbers. Let $O$ be the point of its diagonals intersection. Prove that $|AO|$ - the length of the $[AO]$ segment is also rational.

1969

115 1969 All Soviet Union

The point $E$ lies on the base $[AD]$ of the trapezoid $ABCD$. The triangles' $ABE, BCE$ and $CDE$ perimeters are equal. Prove that $|BC| = |AD|/2$

1969

The point $E$ lies on the base $[AD]$ of the trapezoid $ABCD$. The triangles' $ABE, BCE$ and $CDE$ perimeters are equal. Prove that $|BC| = |AD|/2$

124 1969 All Soviet Union

129 1970 All Soviet Union
Given a pentagon with all equal sides.

a) Prove that there exist such a point on the maximal diagonal, that every side is seen from it inside a right angle.

(the side $AB$ is seen from the point $C$ inside an arbitrary angle that is greater or equal than angle $ACB$)

b) Prove that the circles build on its sides as on the diameters cannot cover the pentagon entirely

Let $h_k$ be an apothem of the right $k$-angle inscribed into a circle with radius $R$.

Prove that $(n + 1)h_{n+1} - nh_n > R$.

1970

Given a circle, its diameter $[AB]$ and a point $C$ on it. Build (with the help of compasses and ruler) two points $X$ and $Y$, that are symmetric with respect to $(AB)$ line, such that $(YC)$ is orthogonal to $(XA)$.

131 1970 All Soviet Union

135 1970 All Soviet Union

138 1970 All Soviet Union

How many sides of the convex polygon can equal its longest diagonal?

The bisector $[AD]$, the median $[BM]$ and the height $[CH]$ of the acute-angled triangle $ABC$ intersect in one point.Prove that the angle $BAC$ is greater than $45$ degrees.

Given triangle $ABC$, middle $M$ of the side $[BC]$, the centre $O$ of the inscribed circle. The line $(MO)$ crosses the height $AH$ in the point $E$. Prove that the distance $|AE|$ equals the inscribed circle radius.

140 1970 All Soviet Union

145 1971 All Soviet Union

Two equal rectangles are intersecting in $8$ points.

Prove that the common part area is greater than the half of the rectangle's area.

1971

a) Given a triangle $A_1A_2A_3$ and the points $B_1$ and $D_2$ on the side $[A_1A_2], B_2$ and $D_3$ on the side $[A_2A3], B_3$ and $D_1$ on the side $[A_3A_1]$. If you build parallelograms $A_1B_1C_1D_1, A_2B_2C_2D_2$ and $A_3B_3C_3D_3$, the lines $(A_1C_1), (A_2C_2)$ and $(A_3C_3)$, will cross in one point $O$. Prove that if $|A_1B_1| = |A_2D_2|$ and $|A_2B_2| = |A_3D_3|$, than $|A_3B_3| = |A_1D_1|$.

b) Given a convex polygon $A_1A_2 ... A_n$ and the points $B_1$ and $D_2$ on the side $[A_1A_2], B_2$ and $D_3$ on the side $[A_2A_3], ... B_n$ and $D_1$ on the side $[A_nA_1]$. Ifyou build parallelograms $A_1B_1C_1D_1, A_2B_2C_2D_2 ... , A_nB_nC_nD_n$, the lines $(A_1C_1), (A_2C_2), ..., (A_nC_n)$, will cross in one point $O$. Prove that $|A_1B_1| \cdot |A_2B_2|\cdot ... \cdot |A_nB_n| = |A_1D_1|\cdot |A_2D_2|\cdot ...\cdot |A_nD_n|$.

150 1971 All Soviet Union

152 1971 All Soviet Union

159 1972 All Soviet Union

167 1972 All Soviet Union

253 1978 All Soviet Union

The projections of the body on two planes are circles. Prove that they have the same radius.

a) Prove that the line dividing the triangle onto two polygons with equal perimeters and equal areas passes through the centre of the inscribed circle.

b) Prove the same statement for the arbitrary polygon outscribed around the circle.

c) Prove that all the lines halving its perimeter and area simultaneously, intersect in one point.

1972

Given a rectangle $ABCD$, points $M$ -- the middle of $[AD]$ side, $N$ -- the middle of $[BC]$ side. Let us take a point $P$ on the continuation of the $[DC]$ segment over the point $D$. Let us denote the point of intersection of lines $(PM)$ and $(AC)$ as $Q$. Prove that the angles $QNM$ and $MNP$ are equal.

Let $O$ be the intersection point of the of the convex quadrangle $ABCD$ diagonals.Prove that the line drawn through the points of intersection of the medians of $AOB$ and $COD$ triangles is orthogonal to the line drawn through the points of intersection of the heights of $BOC$ and $AOD$ triangles.

The $7$-angle $A_1A_2A_3A_4A_5A_6A_7$ is inscribed in a circle.

Prove that if the centre of the circle is inside the $7$-angle, than the sum of $A_1,A_2$ and $A_3$ angles is less than $450$ degrees.

170 1972 All Soviet Union

The point $O$ inside the convex polygon makes isosceles triangle with all the pairs of its vertices.

Prove that $O$ is the centre of the outscribed circle.

*other formulation:*

P is a convex polygon and X is an interior point such that for every pair of vertices A, B, the triangle XAB is isosceles. Prove that all the vertices of P lie on some circle center X.

1973

Given an angle with the vertex $O$ and a circle touching its sides in the points $A$ and $B$. A ray is drawn from the point $A$ parallel to $[OB)$. It intersects with the circumference in the point $C$. The segment $[OC]$ intersects the circumference in the point $E$. The straight lines $(AE)$ and $(OB)$ intersect in the point $K$. Prove that $|OK| = |KB|$.

182 1973 All Soviet Union

Three similar acute-angled triangles $AC_1B, BA_1C$ and $CB_1A$ are built on the outer side of the acute-angled triangle $ABC$. (Equal triples of the angles are $AB1_C, ABC_1, A_1BC$ and $BA_1C, BAC_1, B_1AC$.)

a) Prove that the circumferences outscribed around the outer triangles intersect in one point.

b) Prove that the straight lines $AA_1, BB_1$ and $CC_1$ intersect in the same point

185 1973 All Soviet Union

Given a triangle with $a,b,c$ sides and with the area $1$ ($a \ge b \ge c$). Prove that $b^2 \ge 2$.

191 1974 All Soviet Union

Three similar acute-angled triangles $AC_1B, BA_1C$ and $CB_1A$ are built on the outer side of the acute-angled triangle $ABC$. (Equal triples of the angles are $AB1_C, ABC_1, A_1BC$ and $BA_1C, BAC_1, B_1AC$.)

a) Prove that the circumferences outscribed around the outer triangles intersect in one point.

b) Prove that the straight lines $AA_1, BB_1$ and $CC_1$ intersect in the same point

185 1973 All Soviet Union

Given a triangle with $a,b,c$ sides and with the area $1$ ($a \ge b \ge c$). Prove that $b^2 \ge 2$.

1974

a) Each of the side of the convex hexagon ($6$-angle) is longer than $1$. Does it necessary have a diagonal longer than $2$?

b) Each of the main diagonals of the convex hexagon is longer than $2$. Does it necessary have a side longer than $1$?

192 1974 All Soviet Union

Given two circles with the radiuses $R$ and $r$, touching each other from the outer side. Consider all the trapezoids, such that its lateral sides touch both circles, and its bases touch different circles. Find the shortest possible lateral side.

Circles radius r and R touch externally. AD is parallel to BC. AB and CD touch both circles. AD touches the circle radius r, but not the circle radius R, and BC touches the circle radius R, but not the circle radius r. What is the smallest possible length for AB?

195 1974 All Soviet Union (also here)

198 1974 All Soviet Union

Given two circles with the radiuses $R$ and $r$, touching each other from the outer side. Consider all the trapezoids, such that its lateral sides touch both circles, and its bases touch different circles. Find the shortest possible lateral side.

*other formulation:*

195 1974 All Soviet Union (also here)

Given a square $ABCD$. Points $P$ and $Q$ are in the sides $[AB]$ and $[BC]$ respectively. $|BP|=|BQ|$. Let $H$ be the base of the perpendicular from the point $B$ to the segment $[PC]$. Prove that the angle $DHQ$ is a right one.

Given points $D$ and $E$ on the legs $[CA]$ and $[CB]$, respectively, of the isosceles right triangle. $|CD| = |CE|$. The extensions of the perpendiculars from $D$ and $C$ to the line $AE$ cross the hypotenuse $AB$ in the points $K$ and $L$. Prove that $|KL| = |LB|$

204 1974 All Soviet Union

Given a triangle $ABC$ with the are $1$. Let $A',B'$ and $C' $ are the middles of the sides $[BC], [CA]$ and $[AB]$ respectively. What is the minimal possible area of the common part of two triangles $A'B'C'$ and $KLM$, if the points $K,L$ and $M$ are lying on the segments $[AB'], [CA']$ and $[BC']$ respectively?

205 1975 All Soviet UnionGiven a triangle $ABC$ with the are $1$. Let $A',B'$ and $C' $ are the middles of the sides $[BC], [CA]$ and $[AB]$ respectively. What is the minimal possible area of the common part of two triangles $A'B'C'$ and $KLM$, if the points $K,L$ and $M$ are lying on the segments $[AB'], [CA']$ and $[BC']$ respectively?

1975

a) The triangle $ABC$ was turned around the centre of the outscribed circle by the angle less than $180$ degrees and thus was obtained the triangle $A_1B_1C_1$. The corresponding segments $[AB]$ and $[A_1B_1]$ intersect in the point $C_2, [BC]$ and $[B_1C_1]$ -- $A_2, [AC]$ and $[A_1C_1]$ -- $B_2$. Prove that the triangle $A_2B_2C_2$ is similar to the triangle $ABC$.

b) The quadrangle $ABCD$ was turned around the centre of the outscribed circle by the angle less than $180$ degrees and thus was obtained the quadrangle $A_1B1C_1D_1$. Prove that the points of intersection of the corresponding lines ( $(AB$) and $(A_1B_1), (BC)$ and $(B_1C_1), (CD)$ and $(C_1D_1), (DA)$ and $(D_1A_1)$ ) are the vertices of the parallelogram..

206 1975 All Soviet Union

209 1975 All Soviet Union

213 1975 All Soviet Union

222 1976 All Soviet Union

237 1977 All Soviet Union

Given a triangle $ABC$ with the unit area. The first player chooses a point $X$ on the side $[AB]$, than the second -- $Y$ on $[BC]$ side, and, finally, the first chooses a point $Z$ on $[AC]$ side. The first tries to obtain the greatest possible area of the $XYZ$ triangle, the second -- the smallest. What area can obtain the first for sure and how?

Denote the middles of the convex hexagon $A_1A_2A_3A_4A_5A_6$ diagonals $A_6A_2, A_1A_3, A_2A_4, A_3A_5, A_4A_6, A_5A_1$ as $B_1, B_2, B_3, B_4, B_5, B_6$ respectively. Prove that if the hexagon $B_1B_2B_3B_4B_5B_6$ is convex, than its area equals to the quarter of the initial hexagon.

Three flies are crawling along the perimeter of the $ABC$ triangle in such a way, that the centre of their masses is a constant point. One of the flies has already passed along all the perimeter. Prove that the centre of the flies' masses coincides with the centre of masses of the $ABC$ triangle. (The centre of masses for the triangle is the point of medians intersection.

1976

Given three circumferences of the same radius in a plane.

a) All three are crossing in one point $K$. Consider three arcs $AK,CK,EK$ : the $A,C,E$ are the points of the circumferences intersection and the arcs are taken in the clockwise direction. Every arc is inside one circle, outside the second and on the border of the third one. Prove that the sum of the arcs is $180$ degrees.

b) Consider the case, when the three circles give a curvilinear triangle $BDF$ as their intersection (instead of one point $K$). The arcs are taken in the clockwise direction. Every arc is inside one circle, outside the second and on the border of the third one. Prove that the sum of the $AB, CD$ and $EF$ arcs is $180$ degrees.

1977

(a) Given a circle with two inscribed triangles $T_1$ and $T_2$. The vertices of $T_1$ are the middles of the arcs with the ends in the vertices of $T_2$. Consider a hexagon -- the intersection of $T_1$ and $T_2$. Prove that its main diagonals are parallel to $T_1$ sides and are intersecting in one point.

b) The segment, that connects the middles of the arcs $AB$ and $AC$ of the circle outscribed around the $ABC$ triangle, intersects $[AB]$ and $[AC]$ sides in $D$ and $K$ points. Prove that the points $A,D,K$ and $O$ -- the centre of the circle -- are the vertices of a diamond.

241 1977 All Soviet Union

Every vertex of a convex polyhedron belongs to three edges. It is possible to outscribe a circle around all its faces. Prove that the polyhedron can be inscribed in a sphere.

1978

Given a quadrangle $ABCD$ and a point $M$ inside it such that $ABMD$ is a parallelogram. the angle $CBM$ equals to $CDM$. Prove that the angle $ACD$ equals to $BCM$.

261 1978 All Soviet Union

266 1978 All Soviet Union

1979

269 1979 All Soviet Union

282 1979 All Soviet Union

Given a circle with radius $R$ and inscribed $n$-angle with area $S$. We mark one point on every side of the given polygon. Prove that the perimeter of the polygon with the vertices in the marked points is not less than $2S/R$.

Prove that for every tetrahedron there exist two planes such that the projection areas on those planes relation is not less than $\sqrt 2$.

1979

What is the least possible relation of two isosceles triangles areas, if three vertices of the first one belong to three different sides of the second one?

282 1979 All Soviet Union

The convex quadrangle is divided by its diagonals onto four triangles. The circles inscribed in those triangles are equal. Prove that the given quadrangle is a diamond.

1980

287 1980 All Soviet Union

The points $M$ and $P$ are the middles of $[BC]$ and $[CD]$ sides of a convex quadrangle $ABCD$. It is known that $|AM| + |AP| = a$. Prove that the $ABCD$ area is less than $\frac{a^2}{2}$.

289 1980 All Soviet Union

298 1980 All Soviet Union

302 1980 All Soviet Union

305 1981 All Soviet Union

Given a point $E$ on the diameter $AC$ of the certain circle. Draw a chord $BD$ to maximise the area of the quadrangle $ABCD$.

Given equilateral triangle $ABC$. Some line, parallel to $[AC]$ crosses $[AB]$ and $[BC]$ in $M$ and $P$ points respectively. Let $D$ be the centre of $PMB$ triangle, $E$ - the middle of the $[AP]$ segment. Find the angles of $DEC$ triangle.

The edge $[AC]$ of the tetrahedron $ABCD$ is orthogonal to $[BC]$, and $[AD]$ is orthogonal to $[BD]$. Prove that the cosine of the angle between $(AC)$ and $(BD)$ lines is less than $|CD|/|AB|$.

1981

Given points $A,B,M,N$ on the circumference. Two chords $[MA_1]$ and $[MA_2]$ are orthogonal to $(NA)$ and $(NB)$ lines respectively. Prove that $(AA_1)$ and $(BB_1)$ lines are parallel.

309 1981 All Soviet Union

312 1981 All Soviet Union

315 1981 All Soviet Union

Given two points $M$ and $K$ on the circumference with radius $r_1$ and centre $O_1$. The circumference with radius $r_2$ and centre $O_2$ is inscribed in $MO_1K$ angle. Find the $MO_1KO_2$ quadrangle area.

Three equilateral triangles $ABC, CDE, EHK$ (the vertices are mentioned counterclockwise) are lying in the plane so, that the vectors $\overrightarrow{AD}$ and $\overrightarrow{DK}$ are equal. Prove that the triangle $BHD$ is also equilateral

The points $K$ and $M$ are the centres of the $AB$ and $CD$ sides of the convex quadrangle $ABCD$. The points $L$ and $M$ belong to two other sides and $KLMN$ is a rectangle. Prove that $KLMN$ area is a half of $ABCD$ area.

The quadrangles $AMBE, AHBT, BKXM$, and $CKXP$ are parallelograms. Prove that the quadrangle $ABTE$ is also parallelogram. (the vertices are mentioned counterclockwise)

318 1981 All Soviet Union

326 1981 All Soviet Union

327 1982 All Soviet Union
The points $C_1, A_1, B_1$ belong to $[AB], [BC], [CA]$ sides, respectively, of the $ABC$ triangle. $\frac{|AC_1|}{|C_1B| }=\frac{ |BA_1|}{|A_1C| }= \frac{|CB_1|}{|B_1A| }= \frac{1}{3}$.

Prove that the perimeter $P$ of the $ABC$ triangle and the perimeter $p$ of the $A_1B_1C_1$ triangle, satisfy inequality $\frac{P}{2} < p < \frac{3P}{4}$.

The segments $[AD], [BE]$ and $[CF]$ are the side edges of the right triangle prism. (the equilateral triangle is a base) Find all the points in its base $ABC$, situated on the equal distances from the $(AE), (BF)$ and $(CD)$ lines.

1982

332 1982 All Soviet Union

The parallelogram $ABCD$ isn't a diamond. The relation of the diagonal lengths $|AC|/|BD|$ equals to $k$. The $[AM)$ ray is symmetric to the $[AD)$ ray with respect to the $(AC)$ line. The $[BM)$ ray is symmetric to the $[BC)$ ray with respect to the $(BD)$ line. ($M$ point is those rays intersection.) Find the $|AM|/|BM|$ relation

334 1982 All Soviet Union

Given a point $M$ inside a right tetrahedron.

336 1982 All Soviet Union

339 1982 All Soviet Union

363 1983 All Soviet Union (also)

The points $A_1,B_1,C_1$ belong to $[BC],[CA],[AB]$ sides of the $ABC$ triangle respectively. The $[AA_1], [BB_1], [CC_1]$ segments split the $ABC$ onto $4$ smaller triangles and $3$ quadrangles. It is known, that the smaller triangles have the same area. Prove that the quadrangles have equal areas. What is the quadrangle area, it the small triangle has the unit area?

As shown in the figure below, the four triangles colored in yellow have equal areas. Show that the three uncolored quadrilaterals also have equal areas.

A point is chosen on each of the three sides of a triangle and joined to the opposite vertex. The resulting lines divide the triangle into four triangles and three quadrilaterals. The four triangles all have area A. Show that the three quadrilaterals have equal area. What is it (in terms of A)?

The parallelogram $ABCD$ isn't a diamond. The relation of the diagonal lengths $|AC|/|BD|$ equals to $k$. The $[AM)$ ray is symmetric to the $[AD)$ ray with respect to the $(AC)$ line. The $[BM)$ ray is symmetric to the $[BC)$ ray with respect to the $(BD)$ line. ($M$ point is those rays intersection.) Find the $|AM|/|BM|$ relation

334 1982 All Soviet Union

Given a point $M$ inside a right tetrahedron.

Prove that at least one tetrahedron edge is seen from the $M$ in an angle, that has a cosine not greater than $-1/3$. (e.g. if $A$ and $B$ are the vertices, corresponding to that edge, $cos(\widehat{AMB}) \le -1/3$)

336 1982 All Soviet Union

The closed broken line $M$ has odd number of vertices -- $A_1,A_2,..., A_{2n+1}$ in sequence.

Let us denote with $S(M)$ a new closed broken line with vertices $B_1,B_2,...,B_{2n+1}$ -- the middles of the first line links: $B_1$ is the middle of $[A_1A_2], ... , B_{2n+1}$ -- of $[A_{2n+1}A_1]$. Prove that in a sequence $M_1=S(M), ... , M_k = S(M_{k-1}), ...$ there is a broken line, homothetic to the $M$.

339 1982 All Soviet Union

There is a parabola $y = x^2$ drawn on the coordinate plane. The axes are deleted. Can you restore them with the help of compass and ruler?

The $KLMN$ tetrahedron (triangle pyramid) vertices are situated inside or on the faces or on the edges of the $ABCD$ tetrahedron. Prove that $KLMN$ perimeter is less than $4/3$ $ABCD$ perimeter.

1983

Three disks touch pairwise from outside in the points $X,Y,Z$. Then the radiuses of the disks were expanded by $2/\sqrt3$ times, and the centres were reserved. Prove that the $XYZ$ triangle is completely covered by the expanded disks.

355 1983 All Soviet Union

358 1983 All Soviet Union

The point $D$ is the middle of the $[AB]$ side of the $ABC$ triangle. The points $E$ and F belong to $[AC]$ and $[BC]$ respectively. Prove that the $DEF$ triangle area does not exceed the sum of the $ADE$ and $BDF$ triangles areas.

The points $A1,B1,C1,D1$ and $A2,B2,C2,D2$ are orthogonal projections of the $ABCD$ tetrahedron vertices on two planes. Prove that it is possible to move one of the planes to provide the parallelness of $(A_1A_2), (B_1B_2), (C_1C_2)$ and $(D_1D_2)$ lines.

The points $A_1,B_1,C_1$ belong to $[BC],[CA],[AB]$ sides of the $ABC$ triangle respectively. The $[AA_1], [BB_1], [CC_1]$ segments split the $ABC$ onto $4$ smaller triangles and $3$ quadrangles. It is known, that the smaller triangles have the same area. Prove that the quadrangles have equal areas. What is the quadrangle area, it the small triangle has the unit area?

__another formulation__

As shown in the figure below, the four triangles colored in yellow have equal areas. Show that the three uncolored quadrilaterals also have equal areas.

__different formulation__(without figure)

A point is chosen on each of the three sides of a triangle and joined to the opposite vertex. The resulting lines divide the triangle into four triangles and three quadrilaterals. The four triangles all have area A. Show that the three quadrilaterals have equal area. What is it (in terms of A)?

366 1983 All Soviet Union

Given a point $O$ inside $ABC$ triangle. Prove that $S_A * \overrightarrow{OA} + S_B * \overrightarrow{OB} + S_C * \overrightarrow{OC} = \overrightarrow{0}$,where $S_A, S_B, S_C$ denote $BOC, COA, AOB$ triangles areas respectively..

368 1983 All Soviet Union

373 1984 All Soviet Union

The points $D,E,F$ belong to the sides $(AB), (BC)$ and $(CA)$ of the $ABC$ triangle respectively (but they are not vertices). Let us denote with $d_0, d_1, d_2$, and $d_3$ the maximal side length of the $DEF, DEA, DBF, CEF$, triangles respectively. Prove that $d_0 \ge \frac{\sqrt3}{2} min\{d_1, d_2, d_3\}$. When the equality takes place?

1984

Given two equilateral triangles $A_1B_1C_1$ and $A_2B_2C_2$ in the plane. (The vertices are mentioned counterclockwise.) We draw vectors $\overrightarrow{OA}, \overrightarrow{OB}, \overrightarrow{OC}$, from the arbitrary point $O$, equal to $\overrightarrow{A_1A_2}, \overrightarrow{B_1B_2}, \overrightarrow{C_1C_2}$ respectively. Prove that the triangle $ABC$ is equilateral.

378 1984 All Soviet Union

381 1984 All Soviet Union

384 1984 All Soviet Union

395 1985 All Soviet Union

The circle with the centre $O$ is inscribed in the $ABC$ triangle. The circumference touches its sides $[BC], [CA], [AB]$ in $A_1, B_1, C_1$ points respectively. The $[AO], [BO], [CO]$ segments cross the circumference in $A_2, B_2, C_2$ points respectively. Prove that $(A_1A_2),(B_1B_2)$ and $(C_1C_2)$ lines intersect in one point.

Given $ABC$ triangle. From the $P$ point three lines $(PA),(PB),(PC)$ are drawn. They cross the outscribed circumference in $A_1, B_1,C_1$ points respectively. It comes out that the $A_1B_1C_1$ triangle equals to the initial one. Prove that there are not more than eight such a points $P$ in a plane.

The centre of the coin with radius $r$ is moved along some polygon with the perimeter $P$, that is outscribed around the circle with radius $R$ ($R>r$). Find the coin trace area (a sort of polygon ring).

388 1984 All Soviet Union

393 1984 All Soviet Union

394 1984 All Soviet Union
The $A,B,C$ and $D$ points (from left to right) belong to the straight line. Prove that every point $E$, that doesn't belong to the line satisfy: $|AE| + |ED| + | |AB| - |CD| | > |BE| + |CE|$.

Given three circles $c_1,c_2,c_3$ with $r_1,r_2,r_3$ radiuses, $r_1 > r2, r_1 > r_3$. Each lies outside of two others. The A point -- an intersection of the outer common tangents to $c_1$ and $c_2$ -- is outside $c_3$. The $B$ point -- an intersection of the outer common tangents to $c_1$ and $c_3$ -- is outside $c_2$. Two pairs of tangents -- from $A$ to $c_3$ and from $B$ to $c_2$ -- are drawn. Prove that the quadrangle, they make, is outscribed around some circle and find its radius.

Prove that every cube's cross-section, containing its centre, has the area not less then its face's area.

1985

Two perpendiculars are drawn from the middles of each side of the acute-angle triangle to two other sides. Those six segments make hexagon. Prove that the hexagon area is a half of the triangle area.

399 1985 All Soviet Union

404 1985 All Soviet Union

408 1985 All Soviet Union

Given a straight line $\ell$ and the point $O$ out of the line. Prove that it is possible to move an arbitrary point $A$ in the same plane to the $O$ point, using only rotations around $O$ and symmetry with respect to the $\ell$.

The convex pentagon $ABCDE$ was drawn in the plane.

$A_1$ was symmetric to $A$ with respect to $B$, $B_1$ was symmetric to $B$ with respect to $C$,

$C_1$ was symmetric to $C$ with respect to $D$, $D_1$ was symmetric to $D$ with respect to $E$,

$E_1$ was symmetric to $E$ with respect to $A$.

How is it possible to restore the initial pentagon with the compasses and ruler, knowing $A_1,B_1,C_1,D_1,E_1$ points?

The $[A_0A_5]$ diameter divides a circumference with the $O$ centre onto two hemicircumferences. One of them is divided onto five equal arcs $A_0A_1, A_1A_2, A_2A_3, A_3A_4, A_4A_5$. The $(A_1A_4)$ line crosses $(OA_2)$ and $(OA_3)$ lines in $M$ and $N$ points. Prove that $(|A_2A_3| + |MN|)$ equals to the circumference radius.

412 1985 All Soviet Union

415 1985 All Soviet Union

417 1985 All Soviet Union

419 1986 All Soviet Union
One of two circumferences of radius $R$ comes through $A$ and $B$ vertices of the $ABCD$ parallelogram. Another comes through $B$ and $D$. Let $M$ be another point of circumferences intersection. Prove that the circle outscribed around $AMD$ triangle has radius $R$.

All the points situated more close than $1$ cm to ALL the vertices of the right pentagon with $1$ cm side, are deleted from that pentagon. Find the area of the remained figure.

The $ABCDA_1B_1C_1D_1$ cube has unit length edges.Find the distance between two circumferences, one of those is inscribed into the $ABCD$ base, and another comes through $A,C$ and $B_1$ points.

1986

Two equal squares, one with red sides, another with blue ones, give an octagon in intersection. Prove that the sum of red octagon sides lengths is equal to the sum of blue octagon sides lengths.

420 1986 All Soviet Union

422 1986 All Soviet Union

424 1986 All Soviet Union

428 1986 All Soviet Union

447 1987 All Soviet Union

The point $M$ belongs to the $[AC]$ side of the acute-angle triangle $ABC$. Two circles are outscribed around $ABM$ and $BCM$ triangles. What $M$ position corresponds to the minimal area of those circles intersection?

Prove that it is impossible to draw a convex quadrangle, with one diagonal equal to doubled another, the angle between them $45$ degrees, on the coordinate plane, so, that all the vertices' coordinates would be integers.

Two circumferences, with the distance $d$ between centres, intersect in $P$ and $Q$ points. Two lines are drawn through the $A$ point on the first circumference ($Q\ne A\ne P$) and $P$ and $Q$ points. They intersect the second circumference in the $B$ and $C$ points.

a) Prove that the radius of the circle, outscribed around the $ABC$ triangle, equals $d$.

b) Describe the set of the new circle's centres, if the $A$ point moves along all the first circumference.

A line is drawn through the $A$ vertex of $ABC$ triangle with $|AB|\ne|AC|$. Prove that the line can not contain more than one point $M$ such, that $M$ is not a triangle vertex, and the angles $ABM$ and $ACM$ are equal. What lines do not contain such a point $M$ at all?

431 1986 All Soviet Union

434 1986 All Soviet Union

$\pm \overrightarrow{MA_1} \pm \overrightarrow{MA_2} \pm ... \pm \overrightarrow{MA_n}$to make it equal to the zero vector .

b) $n$ is odd, than the abovementioned expression equals to the zero vector for the finite set of $M$ points only

438 1986 All Soviet Union

440 1986 All Soviet Union

Given two points inside a convex dodecagon (twelve sides) situated $10$ cm far from each other. Prove that the difference between the sum of distances, from the point to all the vertices, is less than $1$ m for those points.

Given right $n$-angle $A_1A_2...A_n$. Prove that if

a) $n$ is even number, than for the arbitrary point $M$ in the plane, it is possible to choose signs in an expression$\pm \overrightarrow{MA_1} \pm \overrightarrow{MA_2} \pm ... \pm \overrightarrow{MA_n}$to make it equal to the zero vector .

b) $n$ is odd, than the abovementioned expression equals to the zero vector for the finite set of $M$ points only

438 1986 All Soviet Union

A triangle and a square are outscribed around the unit circle. Prove that the intersection area is more than $3.4$. Is it possible to assert that it is more than $3.5$?

Consider all the tetrahedrons $AXBY$, outscribed around the sphere. Let $A$ and $B$ points be fixed. Prove that the sum of angles in the non-plane quadrangle $AXBY$ doesn't depend on $X$ and $Y$ points.

1987

Given right heptagon ($7$-angle) $A_1...A_7$. Prove that $\frac{1}{|A_1A_5|} + \frac{1}{|A_1A_3| }= \frac{1}{|A_1A_7|}$.

Three lines are drawn parallel to the sides of the triangles in the opposite to the vertex, not belonging to the side, part of the plane. The distance from each side to the corresponding line equals the length of the side. Prove that six intersection points of those lines with the continuations of the sides are situated on one circumference.

450 1987 All Soviet Union

454 1987 All Soviet Union

Given a convex pentagon. The angles $ABC$ and $ADE$ are equal. The angles $AEC$ and $ADB$ are equal too. Prove that the angles $BAC$ and $DAE$ are equal also.

The $B$ vertex of the $ABC$ angle lies out the circle, and the $[BA)$ and $[BC)$ beams intersect it. The $K$ point belongs to the intersection of the $[BA)$ beam and the circumference. The $KP$ chord is orthogonal to the angle $ABC$ bisector. The $(KP)$ line intersects the $BC$ beam in the M point. Prove that the $[PM]$ segment is twice as long as the distance from the circle centre to the angle $ABC$ bisector.

458 1987 All Soviet Union

464 1988 All Soviet Union
The convex $n$-angle ($n\ge 5$) is cut along all its diagonals.Prove that there are at least a pair of parts with the different areas.

1988

$ABCD$ is a convex quadrilateral. The midpoints of the diagonals and the midpoints of $AB$ and $CD$ form another convex quadrilateral $Q$. The midpoints of the diagonals and the midpoints of $BC$ and $CA$ form a third convex quadrilateral $Q'$. The areas of $Q$ and $Q'$ are equal. Show that either $AC$ or $BD$ divides $ABCD$ into two parts of equal area.

467 1988 All Soviet Union

474 1988 All Soviet Union

476 1988 All Soviet Union

The quadrilateral $ABCD$ is inscribed in a fixed circle. It has $AB$ parallel to $CD$ and the length $AC$ is fixed, but it is otherwise allowed to vary. If $h$ is the distance between the midpoints of $AC$ and $BD$ and $k$ is the distance between the midpoints of $AB$ and $CD$, show that the ratio $h/k$ remains constant.

In the triangle $ABC$, the angle $C$ is obtuse and $D$ is a fixed point on the side $BC$, different from $B$ and $C$. For any point $M$ on the side $BC$, different from $D$, the ray $AM$ intersects the circumcircle $S$ of $ABC$ at $N$. The circle through $M, D$ and $N$ meets $S$ again at $P$, different from $N$. Find the location of the point $M$ which minimises $MP$.

$ABC$ is an acute-angled triangle. The tangents to the circumcircle at $A$ and $C$ meet the tangent at $B$ at $M$ and $N$. The altitude from $B$ meets $AC$ at $P$. Show that $BP$ bisects the angle $MPN$

479 1988 All Soviet Union

486 1988 All Soviet Union

Prove that for any tetrahedron the radius of the inscribed sphere $r <\frac{ ab}{ 2(a + b)}$, where $a$ and $b$ are the lengths of any pair of opposite edges.

489 1989 All Soviet Union
In the acute-angled triangle $ABC$, the altitudes $BD$ and $CE$ are drawn. Let $F$ and $G$ be the points of the line $ED$ such that $BF$ and $CG$ are perpendicular to $ED$. Prove that $EF = DG$.

Prove that for any tetrahedron the radius of the inscribed sphere $r <\frac{ ab}{ 2(a + b)}$, where $a$ and $b$ are the lengths of any pair of opposite edges.

1989

The incircle of $ABC$ touches $AB$ at $M$. $N$ is any point on the segment $BC$. Show that the incircles of $AMN, BMN, ACN$ have a common tangent.

492 1989 All Soviet Union

496 1989 All Soviet Union

A triangle with perimeter $1$ has side lengths $a, b, c$. Show that $a^2 + b^2 + c^2 + 4abc <\frac 12$.

497 1989 All Soviet Union

501 1989 All Soviet Union

$S$ and $S'$ are two intersecting spheres. The line $BXB'$ is parallel to the line of centers, where $B$ is a point on $S, B'$ is a point on $S'$ and $X$ lies on both spheres. $A$ is another point on $S$, and $A'$ is another point on S' such that the line $AA'$ has a point on both spheres. Show that the segments $AB$ and $A'B'$ have equal projections on the line $AA'$.

510 1989 All Soviet Union

A convex polygon is such that any segment dividing the polygon into two parts of equal area which has at least one end at a vertex has length $< 1$. Show that the area of the polygon is $< \pi /4$.

$ABC$ is a triangle. $A' , B' , C'$ are points on the segments $BC, CA, AB$ respectively. $\angle B' A' C' = \angle A$ , $\frac{AC'}{C'B} = \frac{BA' }{A' C} = \frac{CB'}{B'A}$. Show that $ABC$ and $A'B'C'$ are similar.

A triangle with perimeter $1$ has side lengths $a, b, c$. Show that $a^2 + b^2 + c^2 + 4abc <\frac 12$.

497 1989 All Soviet Union

$ABCD$ is a convex quadrilateral. $X$ lies on the segment $AB$ with $\frac{AX}{XB} = \frac{m}{n}$. $Y$ lies on the segment $CD$ with $\frac{CY}{YD} = \frac{m}{n}$. $AY$ and $DX$ intersect at $P$, and $BY$ and $CX$ intersect at $Q$. Show that $\frac{S_{XQYP}}{S_{ABCD}} < \frac{mn}{m^2 + mn + n^2}$.

$ABCD$ has $AB = CD$, but $AB$ not parallel to $CD$, and $AD$ parallel to $BC$. The triangle is $ABC$ is rotated about $C$ to $A'B'C$. Show that the midpoints of $BC, B'C$ and $A'D$ are collinear.

504 1989 All Soviet Union

505 1989 All Soviet Union
$ABC$ is a triangle. Points $D, E, F$ are chosen on $BC, CA, AB$ such that $B$ is equidistant from $D$ and $F$, and $C$ is equidistant from $D$ and $E$. Show that the circumcenter of $AEF$ lies on the bisector of $EDF$.

$S$ and $S'$ are two intersecting spheres. The line $BXB'$ is parallel to the line of centers, where $B$ is a point on $S, B'$ is a point on $S'$ and $X$ lies on both spheres. $A$ is another point on $S$, and $A'$ is another point on S' such that the line $AA'$ has a point on both spheres. Show that the segments $AB$ and $A'B'$ have equal projections on the line $AA'$.

A convex polygon is such that any segment dividing the polygon into two parts of equal area which has at least one end at a vertex has length $< 1$. Show that the area of the polygon is $< \pi /4$.

1990

512 1990 All Soviet Union

515 1990 All Soviet Union

521 1990 All Soviet Union

Two unequal circles intersect at $X$ and $Y$. Their common tangents intersect at $Z$. One of the tangents touches the circles at $P$ and $Q$. Show that $ZX$ is tangent to the circumcircle of $PXQ$.

532 1990 All Soviet Union

www.kalva.demon.co.uk

olympiads.win.tue.nl/imo/soviet/RusMath.html

The line joining the midpoints of two opposite sides of a convex quadrilateral makes equal angles with the diagonals. Show that the diagonals are equal.

The point $P$ lies inside the triangle $ABC$. A line is drawn through $P$ parallel to each side of the triangle. The lines divide $AB$ into three parts length $c, c', c"$ (in that order), and $BC$ into three parts length $a, a', a"$ (in that order), and $CA$ into three parts length $b, b', b"$ (in that order). Show that $abc = a'b'c' = a"b"c"$.

$ABCD$ is a convex quadrilateral. $X$ is a point on the side $AB. AC$ and $DX$ intersect at $Y$. Show that the circumcircles of $ABC, CDY$ and $BDX$ have a common point.

524 1990 All Soviet Union

526 1990 All Soviet Union

527 1990 All Soviet Union
$A, B, C$ are adjacent vertices of a regular $2n$-gon and $D$ is the vertex opposite to $B$ (so that $BD$ passes through the center of the $2n$-gon). $X$ is a point on the side $AB$ and $Y$ is a point on the side $BC$ so that angle $XDY = \frac{\pi}{2n}$. Show that DY bisects angle $\angle XYC$.

Given a point $X$ and $n$ vectors $\overrightarrow{x_i}$ with sum zero in the plane. For each permutation of the vectors we form a set of $n$ points, by starting at $X$ and adding the vectors in order. For example, with the original ordering we get $X_1$ such that $XX_1 = \overrightarrow{x_1}, X_2$ such that $X_1X_2 = \overrightarrow{x_2}$ and so on. Show that for some permutation we can find two points $Y, Z$ with angle $\angle YXZ = 60^o $, so that all the points lie inside or on the triangle $XYZ$.

Two unequal circles intersect at $X$ and $Y$. Their common tangents intersect at $Z$. One of the tangents touches the circles at $P$ and $Q$. Show that $ZX$ is tangent to the circumcircle of $PXQ$.

If every altitude of a tetrahedron is at least $1$, show that the shortest distance between each pair of opposite edges is more than $2$.

$ABCD$ is a rectangle. Points $K, L, M, N$ are chosen on $AB, BC, CD, DA$ respectively so that $KL$ is parallel to $MN$, and $KM$ is perpendicular to $LN$. Show that the intersection of $KM$ and $LN$ lies on $BD$.

544 1991 All Soviet Union

547 1991 All Soviet Union

548 1991 All Soviet Union

A polygon can be transformed into a new polygon by making a straight cut, which creates two new pieces each with a new edge. One piece is then turned over and the two new edges are reattached. Can repeated transformations of this type turn a square into a triangle?

553 1991 All Soviet Union

sources:
Does there exist a triangle in which two sides are integer multiples of the median to that side?Does there exist a triangle in which every side is an integer multiple of the median to that side?

$ABC$ is an acute-angled triangle with circumcenter $O$. The circumcircle of $ABO$ intersects$ AC$ and $BC$ at $M$ and $N$. Show that the circumradii of $ABO$ and $MNC$ are the same.

A polygon can be transformed into a new polygon by making a straight cut, which creates two new pieces each with a new edge. One piece is then turned over and the two new edges are reattached. Can repeated transformations of this type turn a square into a triangle?

The chords $AB$ and $CD$ of a sphere intersect at $X. A, C$ and $X$ are equidistant from a point $Y$ on the sphere. Show that $BD$ and $XY$ are perpendicular.

555 1991 All Soviet Union

599 1992 Commonwealth of Independent States (ASU)

$ABCD$ is a square. The points $X$ on the side $AB$ and $Y$ on the side $AD$ are such that $AX\cdot AY = 2 BX\cdot DY$. The lines $CX$ and $CY$ meet the diagonal $BD$ in two points. Show that these points lie on the circumcircle of $AXY$.

1992

$E$ is a point on the diagonal $BD$ of the square $ABCD$. Show that the points $A, E$ and the circumcenters of $ABE$ and $ADE$ form a square.

563 1992 Commonwealth of Independent States (ASU)

569 1992 Commonwealth of Independent States (ASU)

571 1992 Commonwealth of Independent States (ASU)
$A$ and $B$ lie on a circle. $P$ lies on the minor arc $AB$. $Q$ and $R$ (distinct from $P$) also lie on the circle, so that $P$ and $Q$ are equidistant from $A$, and $P$ and $R$ are equidistant from $B$. Show that the intersection of $AR$ and $BQ$ is the reflection of $P$ in $AB$.

Circles $C$ and $C'$ intersect at $O$ and $X$. A circle center $O$ meets $C$ at $Q$ and $R$ and meets $C'$ at $P$ and $S$. $PR$ and $Q$ meet at $Y$ distinct from $X$. Show that $\angle YXO = 90^o$.

$ABCD$ is a parallelogram. The excircle of $ABC$ opposite $A$ has center $E$ and touches the line $AB$ at $X$. The excircle of $ADC$ opposite $A$ has center $F$ and touches the line $AD$ at $Y$. The line $FC$ meets the line$ AB$ at $W$, and the line $EC$ meets the line $AD$ at $Z$. Show that $WX = YZ$.

575 1992 Commonwealth of Independent States (ASU)

A plane intersects a sphere in a circle $C$. The points $A$ and $B$ lie on the sphere on opposite sides of the plane. The line joining $A$ to the center of the sphere is normal to the plane. Another plane $p$ intersects the segment $AB$ and meets $C$ at $P$ and $Q$. Show that $BP\cdot BQ$ is independent of the choice of $p$.

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