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 $CP$.

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 $CP$.

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.

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.

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.

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.

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?

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 $1/(r_1)+1/(r_3) = 1/(r_2)+1/(r_4)$.

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).

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 $H$.

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 $H$.

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 $N$.

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

In the quadrilateral ABCD, BC is parallel to AD. The point E lies on the segment AD and the perimeters of ABE, BCE and CDE are equal. Prove that BC = AD/2.

Given a circle, diameter AB and a point C on AB, show how to construct two points X and Y on the circle such that (1) Y is the reflection of X in the line AB, (2) YC is perpendicular to XA.

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

1969 All Soviet Union P1
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.

1969-1992 under construction

In the quadrilateral ABCD, BC is parallel to AD. The point E lies on the segment AD and the perimeters of ABE, BCE and CDE are equal. Prove that BC = AD/2.

1969 All Soviet Union P10

Given a pentagon with equal sides.

(a) Prove that there is a point X on the longest diagonal such that every side subtends an angle at most 90 degrees at X.

(b) Prove that the five circles with diameter one of the pentagon's sides do not cover the pentagon.

1969 All Soviet Union P13

A regular n-gon is inscribed in a circle radius R. The distance from the center of the circle to the center of a side is h

1970 All Soviet Union P1Given a pentagon with equal sides.

(a) Prove that there is a point X on the longest diagonal such that every side subtends an angle at most 90 degrees at X.

(b) Prove that the five circles with diameter one of the pentagon's sides do not cover the pentagon.

1969 All Soviet Union P13

A regular n-gon is inscribed in a circle radius R. The distance from the center of the circle to the center of a side is h

_{n}. Prove that (n+1)h_{n+1}- nh_{n}> R.Given a circle, diameter AB and a point C on AB, show how to construct two points X and Y on the circle such that (1) Y is the reflection of X in the line AB, (2) YC is perpendicular to XA.

1970 All Soviet Union P3

What is the greatest number of sides of a convex polygon that can equal its longest diagonal?

1970 All Soviet Union P7

ABC is an acute-angled triangle. The angle bisector AD, the median BM and the altitude CH are concurrent. Prove that angle A is more than 45 degrees.

1970 All Soviet Union P10

ABC is a triangle with incenter I. M is the midpoint of BC. IM meets the altitude AH at E. Show that AE = r, the radius of the inscribed circle.

What is the greatest number of sides of a convex polygon that can equal its longest diagonal?

1970 All Soviet Union P7

ABC is an acute-angled triangle. The angle bisector AD, the median BM and the altitude CH are concurrent. Prove that angle A is more than 45 degrees.

1970 All Soviet Union P10

ABC is a triangle with incenter I. M is the midpoint of BC. IM meets the altitude AH at E. Show that AE = r, the radius of the inscribed circle.

1970 All Soviet Union P12

Two congruent rectangles of area A intersect in eight points. Show that the area of the intersection is more than A/2.

1971 All Soviet Union P2

(a) A

(b) A

Two congruent rectangles of area A intersect in eight points. Show that the area of the intersection is more than A/2.

1971 All Soviet Union P2

(a) A

_{1}A_{2}A_{3}is a triangle. Points B_{1}, B_{2}, B_{3}are chosen on A_{1}A_{2}, A_{2}A_{3}, A_{3}A_{1}respectively and points D_{1}, D_{2}D_{3}on A_{3}A_{1}, A_{1}A_{2}, A_{2}A_{3}respectively, so that if parallelograms A_{i}B_{i}C_{i}D_{i}are formed, then the lines A_{i}C_{i}concur. Show that A_{1}B_{1}·A_{2}B_{2}·A_{3}B_{3}= A_{1}D_{1}·A_{2}D_{2}·A_{3}D_{3}.(b) A

_{1}A_{2}... A_{n}is a convex polygon. Points B_{i}are chosen on A_{i}A_{i+1}(where we take A_{n+1}to mean A_{1}), and points D_{i}on A_{i-1}A_{i}(where we take A_{0}to mean A_{n}) such that if parallelograms A_{i}B_{i}C_{i}D_{i}are formed, then the n lines A_{i}C_{i}concur. Show that ∏ A_{i}B_{i}= ∏ A_{i}D_{i}.
1971 All Soviet Union P7

The projections of a body on two planes are circles. Show that the circles have the same radius.

1971 All Soviet Union P9

A polygon P has an inscribed circle center O. If a line divides P into two polygons with equal areas and equal perimeters, show that it must pass through O.

1972 All Soviet Union P1

ABCD is a rectangle. M is the midpoint of AD and N is the midpoint of BC. P is a point on the ray CD on the opposite side of D to C. The ray PM intersects AC at Q. Show that MN bisects the angle PNQ.

The projections of a body on two planes are circles. Show that the circles have the same radius.

1971 All Soviet Union P9

A polygon P has an inscribed circle center O. If a line divides P into two polygons with equal areas and equal perimeters, show that it must pass through O.

1972 All Soviet Union P1

ABCD is a rectangle. M is the midpoint of AD and N is the midpoint of BC. P is a point on the ray CD on the opposite side of D to C. The ray PM intersects AC at Q. Show that MN bisects the angle PNQ.

1972 All Soviet Union P7

O is the point of intersection of the diagonals of the convex quadrilateral ABCD. Prove that the line joining the centroids of ABO and CDO is perpendicular to the line joining the orthocenters of BCO and ADO.

1972 All Soviet Union P9

A 7-gon is inscribed in a circle. The center of the circle lies inside the 7-gon. A, B, C are adjacent vertices of the 7-gon show that the sum of the angles at A, B, C is less than 450 degrees.

OA and OB are tangent to a circle at A and B. The line parallel to OB through A meets the circle again at C. The line OC meets the circle again at E. The ray AE meets the line OB at K. Prove that K is the midpoint of OB.

O is the point of intersection of the diagonals of the convex quadrilateral ABCD. Prove that the line joining the centroids of ABO and CDO is perpendicular to the line joining the orthocenters of BCO and ADO.

1972 All Soviet Union P9

A 7-gon is inscribed in a circle. The center of the circle lies inside the 7-gon. A, B, C are adjacent vertices of the 7-gon show that the sum of the angles at A, B, C is less than 450 degrees.

1972 All Soviet Union P12

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 All Soviet Union P4P 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.

OA and OB are tangent to a circle at A and B. The line parallel to OB through A meets the circle again at C. The line OC meets the circle again at E. The ray AE meets the line OB at K. Prove that K is the midpoint of OB.

1973 All Soviet Union P9

ABC is an acute-angled triangle. D is the reflection of A in BC, E is the reflection of B in AC, and F is the reflection of C in AB. Show that the circumcircles of DBC, ECA, FAB meet at a point and that the lines AD, BE, CF meet at a point.

1974 All Soviet Union P3

Each side of a convex hexagon is longer than 1. Is there always a diagonal longer than 2? If each of the main diagonals of a hexagon is longer than 2, is there always a side longer than 1?

ABC is an acute-angled triangle. D is the reflection of A in BC, E is the reflection of B in AC, and F is the reflection of C in AB. Show that the circumcircles of DBC, ECA, FAB meet at a point and that the lines AD, BE, CF meet at a point.

Each side of a convex hexagon is longer than 1. Is there always a diagonal longer than 2? If each of the main diagonals of a hexagon is longer than 2, is there always a side longer than 1?

1974 All Soviet Union P4

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?

1974 All Soviet Union P7

ABCD is a square. P is on the segment AB and Q is on the segment BC such that BP = BQ. H lies on PC such that BHC is a right angle. Show that DHQ is a right angle.

1974 All Soviet Union P10

In the triangle ABC, angle C is 90 deg and AC = BC. Take points D on CA and E on CB such that CD = CE. Let the perpendiculars from D and C to AE meet AB at K and L respectively. Show that KL = LB.

(a) O is the circumcenter of the triangle ABC. The triangle is rotated about O to give a new triangle A'B'C'. The lines AB and A'B' intersect at C'', BC and B'C' intersect at A'', and CA and C'A' intersect at B''. Show that A''B''C'' is similar to ABC.

(b) O is the center of the circle through ABCD. ABCD is rotated about O to give the quadrilateral A'B'C'D'. Prove that the intersection points of corresponding sides form a parallelogram.

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?

1974 All Soviet Union P7

ABCD is a square. P is on the segment AB and Q is on the segment BC such that BP = BQ. H lies on PC such that BHC is a right angle. Show that DHQ is a right angle.

1974 All Soviet Union P10

In the triangle ABC, angle C is 90 deg and AC = BC. Take points D on CA and E on CB such that CD = CE. Let the perpendiculars from D and C to AE meet AB at K and L respectively. Show that KL = LB.

1974 All Soviet Union P16

The triangle ABC has area 1. D, E, F are the midpoints of the sides BC, CA, AB. P lies in the segment BF, Q lies in the segment CD, R lies in the segment AE. What is the smallest possible area for the intersection of triangles DEF and PQR?

1975 All Soviet Union P1The triangle ABC has area 1. D, E, F are the midpoints of the sides BC, CA, AB. P lies in the segment BF, Q lies in the segment CD, R lies in the segment AE. What is the smallest possible area for the intersection of triangles DEF and PQR?

(a) O is the circumcenter of the triangle ABC. The triangle is rotated about O to give a new triangle A'B'C'. The lines AB and A'B' intersect at C'', BC and B'C' intersect at A'', and CA and C'A' intersect at B''. Show that A''B''C'' is similar to ABC.

(b) O is the center of the circle through ABCD. ABCD is rotated about O to give the quadrilateral A'B'C'D'. Prove that the intersection points of corresponding sides form a parallelogram.

1975 All Soviet Union P2

A triangle ABC has unit area. The first player chooses a point X on side AB, then the second player chooses a point Y on side BC, and finally the first player chooses a point Z on side CA. The first player tries to arrange for the area of XYZ to be as large as possible, the second player tries to arrange for the area to be as small as possible. What is the optimum strategy for the first player and what is the best he can do (assuming the second player plays optimally)?

1975 All Soviet Union P5

Given a convex hexagon, take the midpoint of each of the six diagonals joining vertices which are separated by a single vertex (so if the vertices are in order A, B, C, D, E, F, then the diagonals are AC, BD, CE, DF, EA, FB). Show that the midpoints form a convex hexagon with a quarter the area of the original.

1975 All Soviet Union P9

Three flies crawl along the perimeter of a triangle. At least one fly makes a complete circuit of the perimeter. For the entire period the center of mass of the flies remains fixed. Show that it must be at the centroid of the triangle. [You may not assume, without proof, that the flies have the same mass, or that they crawl at the same speed, or that any fly crawls at a constant speed.]

1976 All Soviet Union P3

The circles C

A triangle ABC has unit area. The first player chooses a point X on side AB, then the second player chooses a point Y on side BC, and finally the first player chooses a point Z on side CA. The first player tries to arrange for the area of XYZ to be as large as possible, the second player tries to arrange for the area to be as small as possible. What is the optimum strategy for the first player and what is the best he can do (assuming the second player plays optimally)?

1975 All Soviet Union P5

Given a convex hexagon, take the midpoint of each of the six diagonals joining vertices which are separated by a single vertex (so if the vertices are in order A, B, C, D, E, F, then the diagonals are AC, BD, CE, DF, EA, FB). Show that the midpoints form a convex hexagon with a quarter the area of the original.

1975 All Soviet Union P9

Three flies crawl along the perimeter of a triangle. At least one fly makes a complete circuit of the perimeter. For the entire period the center of mass of the flies remains fixed. Show that it must be at the centroid of the triangle. [You may not assume, without proof, that the flies have the same mass, or that they crawl at the same speed, or that any fly crawls at a constant speed.]

1976 All Soviet Union P3

The circles C

_{1}, C_{2}, C_{3}with equal radius all pass through the point X. C_{i}and C_{j}also intersect at the point Y_{ij}. Show that angle XO_{1}Y_{12}+ angle XO_{2}Y_{23}+ angle XO_{3}Y_{31}= 180 deg, where O_{i}is the center of circle C_{i}.
1977 All Soviet Union P3

(a) The triangle ABC is inscribed in a circle. D is the midpoint of the arc BC (not containing A), similarly E and F. Show that the hexagon formed by the intersection of ABC and DEF has its main diagonals parallel to the sides of ABC and intersecting in a single point.

(b) EF meets AB at X and AC at Y. Prove that AXIY is a rhombus, where I is the center of the circle inscribed in ABC.

(a) The triangle ABC is inscribed in a circle. D is the midpoint of the arc BC (not containing A), similarly E and F. Show that the hexagon formed by the intersection of ABC and DEF has its main diagonals parallel to the sides of ABC and intersecting in a single point.

(b) EF meets AB at X and AC at Y. Prove that AXIY is a rhombus, where I is the center of the circle inscribed in ABC.

1977 All Soviet Union P7

Each vertex of a convex polyhedron has three edges. Each face is a cyclic polygon. Show that its vertices all lie on a sphere.

1978 All Soviet Union P2

ABCD is a quadrilateral. M is a point inside it such that ABMD is a parallelogram. ∠CBM = ∠CDM. Show that ∠ACD = ∠BCM.

Each vertex of a convex polyhedron has three edges. Each face is a cyclic polygon. Show that its vertices all lie on a sphere.

1978 All Soviet Union P2

ABCD is a quadrilateral. M is a point inside it such that ABMD is a parallelogram. ∠CBM = ∠CDM. Show that ∠ACD = ∠BCM.

1978 All Soviet Union P10

An n-gon area A is inscribed in a circle radius R. We take a point on each side of the polygon to form another n-gon. Show that it has perimeter at least 2A/R.

1978 All Soviet Union P15

Given any tetrahedron, show that we can find two planes such that the areas of the projections of the tetrahedron onto the two planes have ratio at least √2.

1979 All Soviet Union P1

T is an isosceles triangle. Another isosceles triangle T' has one vertex on each side of T. What is the smallest possible value of area T'/area T?

An n-gon area A is inscribed in a circle radius R. We take a point on each side of the polygon to form another n-gon. Show that it has perimeter at least 2A/R.

1978 All Soviet Union P15

Given any tetrahedron, show that we can find two planes such that the areas of the projections of the tetrahedron onto the two planes have ratio at least √2.

1979 All Soviet Union P1

T is an isosceles triangle. Another isosceles triangle T' has one vertex on each side of T. What is the smallest possible value of area T'/area T?

1979 All Soviet Union P14

A convex quadrilateral is divided by its diagonals into four triangles. The incircles of each of the four are equal. Show that the quadrilateral has all its sides equal.

A convex quadrilateral is divided by its diagonals into four triangles. The incircles of each of the four are equal. Show that the quadrilateral has all its sides equal.

1980 All Soviet Union P4

ABCD is a convex quadrilateral. M is the midpoint of BC and N is the midpoint of CD. If k = AM + AN show that the area of ABCD is less than k

ABCD is a convex quadrilateral. M is the midpoint of BC and N is the midpoint of CD. If k = AM + AN show that the area of ABCD is less than k

^{2}/2.
1980 All Soviet Union P6

Given a point P on the diameter AC of the circle K, find the chord BD through P which maximises the area of ABCD.

1980 All Soviet Union P15

ABC is equilateral. A line parallel to AC meets AB at M and BC at P. D is the center of the equilateral triangle BMP. E is the midpoint of AP. Find the angles of DEC.

1980 All Soviet Union P19

ABCD is a tetrahedron. Angles ACB and ADB are 90 deg. Let k be the angle between the lines AC and BD. Show that cos k < CD/AB.

1981 All Soviet Union P2

AB is a diameter of the circle C. M and N are any two points on the circle. The chord MA' is perpendicular to the line NA and the chord MB' is perpendicular to the line NB. Show that AA' and BB' are parallel.

Given a point P on the diameter AC of the circle K, find the chord BD through P which maximises the area of ABCD.

1980 All Soviet Union P15

ABC is equilateral. A line parallel to AC meets AB at M and BC at P. D is the center of the equilateral triangle BMP. E is the midpoint of AP. Find the angles of DEC.

1980 All Soviet Union P19

ABCD is a tetrahedron. Angles ACB and ADB are 90 deg. Let k be the angle between the lines AC and BD. Show that cos k < CD/AB.

AB is a diameter of the circle C. M and N are any two points on the circle. The chord MA' is perpendicular to the line NA and the chord MB' is perpendicular to the line NB. Show that AA' and BB' are parallel.

1981 All Soviet Union P6

ABC, CDE, EFG are equilateral triangles (not necessarily the same size). The vertices are counter-clockwise in each case. A, D, G are collinear and AD = DG. Show that BFD is equilateral.

1981 All Soviet Union P9

ABCD is a convex quadrilateral. K is the midpoint of AB and M is the midpoint of CD. L lies on the side BC and N lies on the side AD. KLMN is a rectangle. Show that its area is half that of ABCD.

1981 All Soviet Union P12

ACPH, AMBE, AHBT, BKXM and CKXP are parallelograms. Show that ABTE is also a parallelogram (vertices are labeled anticlockwise).

The circle C has center O and radius r and contains the points A and B. The circle C' touches the rays OA and OB and has center O' and radius r'. Find the area of the quadrilateral OAO'B.

ABC, CDE, EFG are equilateral triangles (not necessarily the same size). The vertices are counter-clockwise in each case. A, D, G are collinear and AD = DG. Show that BFD is equilateral.

1981 All Soviet Union P9

ABCD is a convex quadrilateral. K is the midpoint of AB and M is the midpoint of CD. L lies on the side BC and N lies on the side AD. KLMN is a rectangle. Show that its area is half that of ABCD.

1981 All Soviet Union P12

ACPH, AMBE, AHBT, BKXM and CKXP are parallelograms. Show that ABTE is also a parallelogram (vertices are labeled anticlockwise).

1981 All Soviet Union P15

ABC is a triangle. A' lies on the side BC with BA'/BC = 1/4. Similarly, B' lies on the side CA with CB'/CA = 1/4, and C' lies on the side AB with AC'/AB = 1/4. Show that the perimeter of A'B'C' is between 1/2 and 3/4 of the perimeter of ABC.

1981 All Soviet Union P23

ABCDEF is a prism. Its base ABC and its top DEF are congruent equilateral triangles. The side edges are AD, BE and CF. Find all points on the base wich are equidistant from the three lines AE, BF and CD.

1982 All Soviet Union P1ABC is a triangle. A' lies on the side BC with BA'/BC = 1/4. Similarly, B' lies on the side CA with CB'/CA = 1/4, and C' lies on the side AB with AC'/AB = 1/4. Show that the perimeter of A'B'C' is between 1/2 and 3/4 of the perimeter of ABC.

1981 All Soviet Union P23

ABCDEF is a prism. Its base ABC and its top DEF are congruent equilateral triangles. The side edges are AD, BE and CF. Find all points on the base wich are equidistant from the three lines AE, BF and CD.

The circle C has center O and radius r and contains the points A and B. The circle C' touches the rays OA and OB and has center O' and radius r'. Find the area of the quadrilateral OAO'B.

1982 All Soviet Union P6

ABCD is a parallelogram and AB is not equal to BC. M is chosen so that (1) ∠MAC = ∠DAC and M is on the opposite side of AC to D, and (2) ∠MBD = ∠CBD and M is on the opposite side of BD to C. Find AM/BM in terms of k = AC/BD.

1982 All Soviet Union P8

M is a point inside a regular tetrahedron. Show that we can find two vertices A, B of the tetrahedron such that cos AMB ≤ -1/3.

1982 All Soviet Union P10

P is a polygon with 2n+1 sides. A new polygon is derived by taking as its vertices the midpoints of the sides of P. This process is repeated. Show that we must eventually reach a polygon which is homothetic to P.

1982 All Soviet Union P13

The parabola y = x

ABCD is a parallelogram and AB is not equal to BC. M is chosen so that (1) ∠MAC = ∠DAC and M is on the opposite side of AC to D, and (2) ∠MBD = ∠CBD and M is on the opposite side of BD to C. Find AM/BM in terms of k = AC/BD.

1982 All Soviet Union P8

M is a point inside a regular tetrahedron. Show that we can find two vertices A, B of the tetrahedron such that cos AMB ≤ -1/3.

1982 All Soviet Union P10

P is a polygon with 2n+1 sides. A new polygon is derived by taking as its vertices the midpoints of the sides of P. This process is repeated. Show that we must eventually reach a polygon which is homothetic to P.

1982 All Soviet Union P13

The parabola y = x

^{2}is drawn and then the axes are deleted. Can you restore them using ruler and compasses?
1982 All Soviet Union P22

A tetrahedron $T'$ has all its vertices inside the tetrahedron $T$. Show that the sum of the edge lengths of $T'$ is less than $4/3$ times the corresponding sum for $T$.

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)?

A tetrahedron $T'$ has all its vertices inside the tetrahedron $T$. Show that the sum of the edge lengths of $T'$ is less than $4/3$ times the corresponding sum for $T$.

1983 All Soviet Union P3

C

1983 All Soviet Union grade VII P (Kalva p14)C

_{1}, C_{2}, C_{3}are circles, none of which lie inside either of the others. C_{1}and C_{2}touch at Z, C_{2}and C_{3}touch at X, and C_{3}and C_{1}touch at Y. Prove that if the radius of each circle is increased by a factor 2/√3 without moving their centers, then the enlarged circles cover the triangle XYZ.
1983 All Soviet Union P6

M is the midpoint of BC. E is any point on the side AC and F is any point on the side AB. Show that area MEF ≤ area BMF + area CME.

1983 All Soviet Union P9

The projection of a tetrahedron onto the plane P is ABCD. Can we find a distinct plane P' such that the projection of the tetrahedron onto P' is A'B'C'D' and AA', BB', CC' and DD' are all parallel?

M is the midpoint of BC. E is any point on the side AC and F is any point on the side AB. Show that area MEF ≤ area BMF + area CME.

1983 All Soviet Union P9

The projection of a tetrahedron onto the plane P is ABCD. Can we find a distinct plane P' such that the projection of the tetrahedron onto P' is A'B'C'D' and AA', BB', CC' and DD' are all parallel?

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)?

1983 All Soviet Union P17

O is a point inside the triangle ABC. a = area OBC, b = area OCA, c = area OAB. Show that the vector sum a

O is a point inside the triangle ABC. a = area OBC, b = area OCA, c = area OAB. Show that the vector sum a

**OA**+ b**OB**+ c**OC**is zero.
1983 All Soviet Union P19

Interior points D, E, F are chosen on the sides BC, CA, AB (not at the vertices). Let k be the length of the longest side of DEF. Let a, b, c be the lengths of the longest sides of AFE, BDF, CDE respectively. Show that k ≥ √3 min(a, b, c) /2. When do we have equality?

1984 All Soviet Union P3

ABC and A'B'C' are equilateral triangles and ABC and A'B'C' have the same sense (both clockwise or both counter-clockwise). Take an arbitrary point O and points P, Q, R so that OP is equal and parallel to AA', OQ is equal and parallel to BB', and OR is equal and parallel to CC'. Show that PQR is equilateral.

Interior points D, E, F are chosen on the sides BC, CA, AB (not at the vertices). Let k be the length of the longest side of DEF. Let a, b, c be the lengths of the longest sides of AFE, BDF, CDE respectively. Show that k ≥ √3 min(a, b, c) /2. When do we have equality?

1984 All Soviet Union P3

ABC and A'B'C' are equilateral triangles and ABC and A'B'C' have the same sense (both clockwise or both counter-clockwise). Take an arbitrary point O and points P, Q, R so that OP is equal and parallel to AA', OQ is equal and parallel to BB', and OR is equal and parallel to CC'. Show that PQR is equilateral.

1984 All Soviet Union P8

The incircle of the triangle ABC has center I and touches BC, CA, AB at D, E, F respectively. The segments AI, BI, CI intersect the circle at D', E', F' respectively. Show that DD', EE', FF' are collinear.

1984 All Soviet Union P11

ABC is a triangle and P is any point. The lines PA, PB, PC cut the circumcircle of ABC again at A'B'C' respectively. Show that there are at most eight points P such that A'B'C' is congruent to ABC.

1984 All Soviet Union P14

The center of a coin radius r traces out a polygon with perimeter p which has an incircle radius R > r. What is the area of the figure traced out by the coin?

Show that any cross-section of a cube through its center has area not less than the area of a face.

1985 All Soviet Union P1

ABC is an acute angled triangle. The midpoints of BC, CA and AB are D, E, F respectively. Perpendiculars are drawn from D to AB and CA, from E to BC and AB, and from F to CA and BC. The perpendiculars form a hexagon. Show that its area is half the area of the triangle.

The incircle of the triangle ABC has center I and touches BC, CA, AB at D, E, F respectively. The segments AI, BI, CI intersect the circle at D', E', F' respectively. Show that DD', EE', FF' are collinear.

1984 All Soviet Union P11

ABC is a triangle and P is any point. The lines PA, PB, PC cut the circumcircle of ABC again at A'B'C' respectively. Show that there are at most eight points P such that A'B'C' is congruent to ABC.

1984 All Soviet Union P14

The center of a coin radius r traces out a polygon with perimeter p which has an incircle radius R > r. What is the area of the figure traced out by the coin?

1984 All Soviet Union P18

A, B, C and D lie on a line in that order. Show that if X does not lie on the line then |XA| + |XD| + | |AB| - |CD| | > |XB| + |XC|.

1984 All Soviet Union P23

C

1984 All Soviet Union P24A, B, C and D lie on a line in that order. Show that if X does not lie on the line then |XA| + |XD| + | |AB| - |CD| | > |XB| + |XC|.

1984 All Soviet Union P23

C

_{1}, C_{2}, C_{3}are circles with radii r_{1}, r_{2}, r_{3}respectively. The circles do not intersect and no circle lies inside any other circle. C_{1}is larger than the other two. The two outer common tangents to C_{1}and C_{2}meet at A ("outer" means that the points where the tangent touches the two circles lie on the same side of the line of centers). The two outer common tangents to C_{1}and C_{3}intersect at B. The two tangents from A to C_{3}and the two tangents from B to C_{2}form a quadrangle. Show that it has an inscribed circle and find its radius.Show that any cross-section of a cube through its center has area not less than the area of a face.

1985 All Soviet Union P1

ABC is an acute angled triangle. The midpoints of BC, CA and AB are D, E, F respectively. Perpendiculars are drawn from D to AB and CA, from E to BC and AB, and from F to CA and BC. The perpendiculars form a hexagon. Show that its area is half the area of the triangle.

1985 All Soviet Union P5

Given a line L and a point O not on the line, can we move an arbitrary point X to O using only rotations about O and reflections in L?

1985 All Soviet Union P10

ABCDE is a convex pentagon. A' is chosen so that B is the midpoint of AA', B' is chosen so that C is the midpoint of BB' and so on. Given A', B', C', D', E', how do we construct ABCDE using ruler and compasses?

1985 All Soviet Union P14

The points A, B, C, D, E, F are equally spaced on the circumference of a circle (in that order) and AF is a diameter. The center is O. OC and OD meet BE at M and N respectively. Show that MN + CD = OA.

Two equal squares, one with blue sides and one with red sides, intersect to give an octagon with sides alternately red and blue. Show that the sum of the octagon's red side lengths equals the sum of its blue side lengths.

Given a line L and a point O not on the line, can we move an arbitrary point X to O using only rotations about O and reflections in L?

1985 All Soviet Union P10

ABCDE is a convex pentagon. A' is chosen so that B is the midpoint of AA', B' is chosen so that C is the midpoint of BB' and so on. Given A', B', C', D', E', how do we construct ABCDE using ruler and compasses?

1985 All Soviet Union P14

The points A, B, C, D, E, F are equally spaced on the circumference of a circle (in that order) and AF is a diameter. The center is O. OC and OD meet BE at M and N respectively. Show that MN + CD = OA.

1985 All Soviet Union P18

ABCD is a parallelogram. A circle through A and B has radius R. A circle through B and D has radius R and meets the first circle again at M. Show that the circumradius of AMD is R.

1985 All Soviet Union P21

A regular pentagon has side 1. All points whose distance from every vertex is less than 1 are deleted. Find the area remaining.

1985 All Soviet Union P23

The cube ABCDA'B'C'D' has unit edges. Find the distance between the circle circumscribed about the base ABCD and the circumcircle of AB'C.

1986 All Soviet Union P2ABCD is a parallelogram. A circle through A and B has radius R. A circle through B and D has radius R and meets the first circle again at M. Show that the circumradius of AMD is R.

1985 All Soviet Union P21

A regular pentagon has side 1. All points whose distance from every vertex is less than 1 are deleted. Find the area remaining.

1985 All Soviet Union P23

The cube ABCDA'B'C'D' has unit edges. Find the distance between the circle circumscribed about the base ABCD and the circumcircle of AB'C.

Two equal squares, one with blue sides and one with red sides, intersect to give an octagon with sides alternately red and blue. Show that the sum of the octagon's red side lengths equals the sum of its blue side lengths.

1986 All Soviet Union P3

ABC is acute-angled. What point P on the segment BC gives the minimal area for the intersection of the circumcircles of ABP and ACP?

1986 All Soviet Union P7

Two circles intersect at P and Q. A is a point on one of the circles. The lines AP and AQ meet the other circle at B and C respectively. Show that the circumradius of ABC equals the distance between the centers of the two circles. Find the locus of the circumcircle as A varies.

1986 All Soviet Union P11

ABC is a triangle with AB ≠ AC. Show that for each line through A, there is at most one point X on the line (excluding A, B, C) with ∠ABX = ∠ACX. Which lines contain no such points X?

ABCDEFG is a regular 7-gon. Prove that 1/AB = 1/AC + 1/AD.

ABC is acute-angled. What point P on the segment BC gives the minimal area for the intersection of the circumcircles of ABP and ACP?

1986 All Soviet Union P7

Two circles intersect at P and Q. A is a point on one of the circles. The lines AP and AQ meet the other circle at B and C respectively. Show that the circumradius of ABC equals the distance between the centers of the two circles. Find the locus of the circumcircle as A varies.

1986 All Soviet Union P11

ABC is a triangle with AB ≠ AC. Show that for each line through A, there is at most one point X on the line (excluding A, B, C) with ∠ABX = ∠ACX. Which lines contain no such points X?

1986 All Soviet Union P14

Two points A and B are inside a convex 12-gon. Show that if the sum of the distances from A to each vertex is a and the sum of the distances from B to each vertex is b, then |a - b| < 10 |AB|.

1986 All Soviet Union P21

The incircle of a triangle has radius 1. It also lies inside a square and touches each side of the square. Show that the area inside both the square and the triangle is at least 3.4. Is it at least 3.5?

1986 All Soviet Union P23

A and B are fixed points outside a sphere S. X and Y are chosen so that S is inscribed in the tetrahedron ABXY. Show that the sum of the angles AXB, XBY, BYA and YAX is independent of X and Y.

1987 All Soviet Union P3Two points A and B are inside a convex 12-gon. Show that if the sum of the distances from A to each vertex is a and the sum of the distances from B to each vertex is b, then |a - b| < 10 |AB|.

1986 All Soviet Union P21

The incircle of a triangle has radius 1. It also lies inside a square and touches each side of the square. Show that the area inside both the square and the triangle is at least 3.4. Is it at least 3.5?

1986 All Soviet Union P23

A and B are fixed points outside a sphere S. X and Y are chosen so that S is inscribed in the tetrahedron ABXY. Show that the sum of the angles AXB, XBY, BYA and YAX is independent of X and Y.

ABCDEFG is a regular 7-gon. Prove that 1/AB = 1/AC + 1/AD.

1987 All Soviet Union P7

Squares ABC'C", BCA'A", CAB'B" are constructed on the outside of the sides of the triangle ABC. The line A'A" meets the lines AB and AC at P and P'. Similarly, the line B'B" meets the lines BC and BA at Q and Q', and the line C'C" meets the lines CA and CB at R and R'. Show that P, P', Q, Q', R and R' lie on a circle.

1987 All Soviet Union P10

ABCDE is a convex pentagon with ∠ABC = ∠ADE and ∠AEC = ∠ADB. Show that ∠BAC = ∠DAE.

1987 All Soviet Union P14

AB is a chord of the circle center O. P is a point outside the circle and C is a point on the chord. The angle bisector of APC is perpendicular to AB and a distance d from O. Show that BC = 2d.

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.

Squares ABC'C", BCA'A", CAB'B" are constructed on the outside of the sides of the triangle ABC. The line A'A" meets the lines AB and AC at P and P'. Similarly, the line B'B" meets the lines BC and BA at Q and Q', and the line C'C" meets the lines CA and CB at R and R'. Show that P, P', Q, Q', R and R' lie on a circle.

1987 All Soviet Union P10

ABCDE is a convex pentagon with ∠ABC = ∠ADE and ∠AEC = ∠ADB. Show that ∠BAC = ∠DAE.

1987 All Soviet Union P14

AB is a chord of the circle center O. P is a point outside the circle and C is a point on the chord. The angle bisector of APC is perpendicular to AB and a distance d from O. Show that BC = 2d.

1987 All Soviet Union P18

A convex pentagon is cut along all its diagonals to give 11 pieces. Show that the pieces cannot all have equal areas.

1988 All Soviet Union P2A convex pentagon is cut along all its diagonals to give 11 pieces. Show that the pieces cannot all have equal areas.

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.

1988 All Soviet Union P5

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.

1988 All Soviet Union P12

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.

1988 All Soviet Union P14

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.

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.

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.

1988 All Soviet Union P12

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.

1988 All Soviet Union P14

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.

1988 All Soviet Union P17

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.

1988 All Soviet Union P24

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 All Soviet Union P3In 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.

1988 All Soviet Union P24

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.

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.

1989 All Soviet Union P6

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

1989 All Soviet Union P10

A triangle with perimeter 1 has side lengths a, b, c. Show that a

1989 All Soviet Union P11

ABCD is a convex quadrilateral. X lies on the segment AB with AX/XB = m/n. Y lies on the segment CD with CY/YD = m/n. AY and DX intersect at P, and BY and CX intersect at Q. Show that area XQYP/area ABCD < mn/(m

1989 All Soviet Union P15

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.

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'.

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

1989 All Soviet Union P10

A triangle with perimeter 1 has side lengths a, b, c. Show that a

^{2}+ b^{2}+ c^{2}+ 4abc < 1/2.1989 All Soviet Union P11

ABCD is a convex quadrilateral. X lies on the segment AB with AX/XB = m/n. Y lies on the segment CD with CY/YD = m/n. AY and DX intersect at P, and BY and CX intersect at Q. Show that area XQYP/area ABCD < mn/(m

^{2}+ mn + n^{2}).1989 All Soviet Union P15

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.

1989 All Soviet Union P18

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.

1989 All Soviet Union P19ABC 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'.

1990 All Soviet Union P2

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.

1990 All Soviet Union P5

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".

1990 All Soviet Union P11

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.

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.

1990 All Soviet Union P22

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.

1991 All Soviet Union P6

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.

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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.

1990 All Soviet Union P5

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".

1990 All Soviet Union P11

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.

1990 All Soviet Union P14

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 = π/2n. Show that DY bisects angle XYC.

1990 All Soviet Union P16

Given a point X and n vectors

1990 All Soviet Union P17A, 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 = π/2n. Show that DY bisects angle XYC.

1990 All Soviet Union P16

Given a point X and n vectors

**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}=**x**_{1}, X_{2}such that X_{1}X_{2}=**x**_{2}and so on. Show that for some permutation we can find two points Y, Z with angle YXZ = 60 deg, 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.

1990 All Soviet Union P22

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.

1991 All Soviet Union P6

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.

1991 All Soviet Union P10

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?

1991 All Soviet Union P13

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.

1991 All Soviet Union P19

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.

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.

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?

1991 All Soviet Union P13

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.

1991 All Soviet Union P19

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.

1991 All Soviet Union P21

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

1992 Commonwealth of Independent States P1

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.

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

1992 Commonwealth of Independent States P1

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.

1992 Commonwealth of Independent States P6

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.

1992 Commonwealth of Independent States P12

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 QS meet at Y distinct from X. Show that ∠YXO = 90

1992 Commonwealth of Independent States P14A 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.

1992 Commonwealth of Independent States P12

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 QS meet at Y distinct from X. Show that ∠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.

1992 Commonwealth of Independent States P18

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·BQ is independent of the choice of p.

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·BQ is independent of the choice of p.

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