6th All Soviet Union Mathematical Olympiad 1972 Problems & Solutions

1.  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. 2.  Given 50 segments on a line show that you can always find either 8 segments which are disjoint or 8 segments with a common point. 3.  Find the largest integer

5th All Soviet Union Mathematical Olympiad 1971 Problems & Solutions

1.  Prove that we can find a number divisible by 2n whose decimal representation uses only the digits 1 and 2. 2.  (1) A1A2A3 is a triangle. Points B1, B2, B3 are chosen on A1A2, A2A3, A3A1 respectively and points D1, D2 D3 on A3A1, A1A2, A2A3 respectively, so that if parallelograms AiBiCiDi are formed, then the lines AiCi

4th All Soviet Union Mathematical Olympiad 1970 Problems & Solutions

1.  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. 2.  The product of three positive numbers is 1, their sum is greater than the sum of their inverses. Prove that just one of the numbers is greater than 1. 3.  What

3rd All Soviet Union Mathematical Olympiad 1969 Problems & Solutions

1.  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. 2.  A wolf is in the center of a square field and there is a dog at each corner. The wolf can run anywhere in the field, but the dogs can only run along the sides. The dogs' speed is 3/2 times the wolf's

2nd All Soviet Union Mathematical Olympiad 1968 Problems & Solutions

1.  An octagon has equal angles. The lengths of the sides are all integers. Prove that the opposite sides are equal in pairs. 2.  Which is greater: 3111 or 1714? [No calculators allowed!] 3.  A circle radius 100 is drawn on squared paper with unit squares. It does not touch any of the grid lines or pass through any of the lattice points. What is the maximum number of

1st All Soviet Union Mathematical Olympiad 1967 Problems & Solutions

1.  In the acute-angled triangle ABC, AH is the longest altitude (H lies on BC), M is the midpoint of AC, and CD is an angle bisector (with D on AB). (a)  If AH ≤ BM, prove that the angle ABC ≤ 60. (b)  If AH = BM = CD, prove that ABC is equilateral. 2. (a)  The digits of a natural number are rearranged and the resultant number is added to the original

25th All Soviet Union Mathematical Olympiad Problems 1991

1.  Find all integers a, b, c, d such that ab - 2cd = 3, ac + bd = 1. 2.  n numbers are written on a blackboard. Someone then repeatedly erases two numbers and writes half their arithmetic mean instead, until only a single number remains. If all the original numbers were 1, show that the final number is not less than 1/n. 3.  Four lines in the plane

24th All Soviet Union Mathematical Olympiad Problems 1990

1.  Show that x4 > x - 1/2 for all real x. 2.  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. 3.  A graph has 30 points and each point has 6 edges. Find the total number of triples such that each pair of points is joined or each pair of points is

23rd All Soviet Union Mathematical Olympiad Problems 1989

1.  7 boys each went to a shop 3 times. Each pair met at the shop. Show that 3 must have been in the shop at the same time. 2.  Can 77 blocks each 3 x 3 x 1 be assembled to form a 7 x 9 x 11 block? 3.  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. 4.  A positive integer n has

21st All Soviet Union Mathematical Olympiad Problems 1987

1.  Ten players play in a tournament. Each pair plays one match, which results in a win or loss. If the ith player wins ai matches and loses bi matches, show that ∑ ai2 = ∑ bi2. 2.  Find all sets of 6 weights such that for each of n = 1, 2, 3, ... , 63, there is a subset of weights weighing n. 3.  ABCDEFG is a regular 7-gon. Prove that 1/AB = 1/AC + 1/AD. 4. 

19th All Soviet Union Mathematical Olympiad Problems 1985

1.  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. 2.  Is there an integer n such that the sum of the (decimal) digits of n is 1000 and the sum of

20th All Soviet Union Mathematical Olympiad Problems 1986

1.  The quadratic x2 + ax + b + 1 has roots which are positive integers. Show that (a2 + b2) is composite. 2.  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. 3.  ABC is acute-angled.

18th All Soviet Union Mathematical Olympiad Problems 1984

1.  Show that we can find n integers whose sum is 0 and whose product is n iff n is divisible by 4. 2.  Show that (a + b)2/2 + (a + b)/4 ≥ a√b + b √a for all positive a and b. 3.  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

16th All Soviet Union Mathematical Olympiad Problems 1982

1.  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. 2.  The sequence an is defined by a1 = 1, a2 = 2, an+2 = an+1 + an. The sequence bn is defined by b1 = 2, b2 = 1, bn+2 = bn+1 + bn. How many integers belong to both

17th All Soviet Union Mathematical Olympiad Problems 1983

1.  A 4 x 4 array of unit cells is made up of a grid of total length 40. Can we divide the grid into 8 paths of length 5? Into 5 paths of length 8? 2.  Three positive integers are written on a blackboard. A move consists of replacing one of the numbers by the sum of the other two less one. For example, if the numbers are 3, 4, 5, then one move could lead to 4, 5

15th All Soviet Union Mathematical Olympiad Problems 1981

1.  A chess board is placed on top of an identical board and rotated through 45 degrees about its center. What is the area which is black in both boards? 2.  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

14th All Soviet Union Mathematical Olympiad Problems 1980

1.  All two digit numbers from 19 to 80 inclusive are written down one after the other as a single number N = 192021...7980. Is N divisible by 1980? 2.  A square is divided into n parallel strips (parallel to

13th All Soviet Union Mathematical Olympiad Problems 1979

1.  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? 2.  A grasshopper hops about in the first quadrant (x, y >= 0).

12th All Soviet Union Mathematical Olympiad Problems 1978

1.  an is the nearest integer to √n. Find 1/a1 + 1/a2 + ... + 1/a1980. 2.  ABCD is a quadrilateral. M is a point inside it such that ABMD is a parallelogram. ∠CBM = ∠CDM. Show that ∠ACD = ∠BCM. 3.  Show that there is no

10th All Soviet Union Mathematical Olympiad Problems 1976

1.  50 watches, all keeping perfect time, lie on a table. Show that there is a moment when the sum of the distances from the center of the table to the center of each dial equals the sum of the distances from the center of the table to the tip of each minute hand.