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 a_i matches and loses b_i matches, show that ∑ a_i² = ∑ b_i².

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.  Your opponent has chosen a 1 x 4 rectangle on a 7 x 7 board. At each move you are allowed to ask whether a particular square of the board belongs to his rectangle. How many questions do you need to be certain of identifying the rectangle. How many questions are needed for a 2 x 2 rectangle?

5.  Prove that 1¹⁹⁸⁷ + 2¹⁹⁸⁷ + ... + n¹⁹⁸⁷ is divisible by n+2.

6.  An L is an arrangement of 3 adjacent unit squares formed by deleting one unit square from a 2 x 2 square. How many Ls can be placed on an 8 x 8 board (with no interior points overlapping)? Show that if any one square is deleted from a 1987 x 1987 board, then the remaining squares can be covered with Ls (with no interior points overlapping).

7.  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.

8.  A₁, A₂, ... , A_2m+1 and B₁, B₂, ... , B_2n+1 are points in the plane such that the 2m+2n+2 lines A₁A₂, A₂A₃, ... , A_2mA_2m+1, A_2m+1A₁, B₁B₂, B₂B₃, ... , B_2nB_2n+1, B_2n+1B₁ are all different and no three of them are concurrent. Show that we can find i and j such that A_iA_i+1, B_jB_j+1 are opposite sides of a convex quadrilateral (if i = 2m+1, then we take A_i+1 to be A₁. Similarly for j = 2n+1).

9.  Find 5 different relatively prime numbers, so that the sum of any subset of them is composite.

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

11.  Show that there is a real number x such that all of cos x, cos 2x, cos 4x, ... cos(2ⁿx) are negative.

12.  The positive reals a, b, c, x, y, z satisfy a + x = b + y = c + z = k. Show that ax + by + cz ≤ k².

13.  A real number with absolute value at most 1 is put in each square of a 1987 x 1987 board. The sum of the numbers in each 2 x 2 square is 0. Show that the sum of all the numbers does not exceed 1987.

14.  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.

15.  Players take turns in choosing numbers from the set {1, 2, 3, ... , n}. Once m has been chosen, no divisor of m may be chosen. The first player unable to choose a number loses. Who has a winning strategy for n = 10? For n = 1000?

16.  What is the smallest number of subsets of S = {1, 2, ... , 33}, such that each subset has size 9 or 10 and each member of S belongs to the same number of subsets?

17.  Some lattice points in the plane are marked. S is a set of non-zero vectors. If you take any one of the marked points P and add place each vector in S with its beginning at P, then more vectors will have their ends on marked points than not. Show that there are an infinite number of points.

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

19.  The set S₀ = {1, 2!, 4!, 8!, 16!, ... }. The set S_n+1 consists of all finite sums of distinct elements of S_n. Show that there is a positive integer not in S₁₉₈₇.

20.  If the graph of the function f = f(x) is rotated through 90 degrees about the origin, then it is not changed. Show that there is a unique solution to f(b) = b. Give an example of such a function.

21.  A convex polyhedron has all its faces triangles. Show that it is possible to color some edges red and the others blue so that given any two vertices one can always find a path between them along the red edges and another path between them along the blue edges.

22.  Show that (2n+1)ⁿ ≥ (2n)ⁿ + (2n-1)ⁿ for every positive integer n.