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. What point P on the segment BC gives the minimal area for the intersection of the circumcircles of ABP and ACP?

4.  Given n points can one build n-1 roads, so that each road joins two points, the shortest distance between any two points along the roads belongs to {1, 2, 3, ... , n(n-1)/2 }, and given any element of {1, 2, 3, ... , n(n-1)/2 } one can find two points such that the shortest distance between them along the roads is that element?

5.  Prove that there is no convex quadrilateral with vertices at lattice points so that one diagonal has twice the length of the other and the angle between them is 45 degrees.

6.  Prove that we can find an m x n array of squares so that the sum of each row and the sum of each column is also a square.

7.  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.

8.  A regular hexagon has side 1000. Each side is divided into 1000 equal parts. Let S be the set of the vertices and all the subdividing points. All possible lines parallel to the sides and with endpoints in S are drawn, so that the hexagon is divided into equilateral triangles with side 1. Let X be the set of all vertices of these triangles. We now paint any three unpainted members of X which form an equilateral triangle (of any size). We then repeat until every member of X except one is painted. Show that the unpainted vertex is not a vertex of the original hexagon.

9.  Let d(n) be the number of (positive integral) divisors of n. For example, d(12) = 6. Find all n such that n = d(n)².

10.  Show that for all positive reals x_i we have 1/x₁ + 1/(x₁ + x₂) + ... + n/(x₁ + ... + x_n) < 4/a₁ + 4/a₂ + ... + 4/a_n.

11.  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?

12.  An n x n x n cube is divided into n³ unit cubes. Show that we can assign a different integer to each unit cube so that the sum of each of the 3n² rows parallel to an edge is zero.

13.  Find all positive integers a, b, c so that a² + b = c and a has n > 1 decimal digits all the same, b has n decimal digits all the same, and c has 2n decimal digits all the same.

14.  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|.

15.  There are 30 cups each containing milk. An elf is able to transfer milk from one cup to another so that the amount of milk in the two cups is equalised. Is there an initial distribution of milk so that the elf cannot equalise the amount in all the cups by a finite number of such transfers?

16.  A 99 x 100 chess board is colored in the usual way with alternate squares black and white. What fraction of the main diagonal is black? What if the board is 99 x 101?

17.  A₁A₂ ... A_n is a regular n-gon and P is an arbitrary point in the plane. Show that if n is even we can choose signs so that the vector sum ± PA₁ ± PA₂ ± ... ± PA_n = 0, but if n is odd, then this is only possible for finitely many points P.

18.  A 1 or a -1 is put into each cell of an n x n array as follows. A -1 is put into each of the cells around the perimeter. An unoccupied cell is then chosen arbitrarily. It is given the product of the four cells which are closest to it in each of the four directions. For example, if the cells below containing a number or letter (except x) are filled and we decide to fill x next, then x gets the product of a, b, c and d.

-1  -1  -1  -1  -1
-1   a   1      -1
 c   x       d  -1
-1
-1   b  -1  -1  -1  

What is the minimum and maximum number of 1s that can be obtained?

19.  Prove that |sin 1| + |sin 2| + ... + |sin 3n| > 8n/5.

20.  Let S be the set of all numbers which can be written as 1/mn, where m and n are positive integers not exceeding 1986. Show that the sum of the elements of S is not an integer.

21.  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?

22.  How many polynomials p(x) have all coefficients 0, 1, 2 or 3 and take the value n at x = 2?

23.  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. 

Solutions

Problem 1

The quadratic x² + ax + b + 1 has roots which are positive integers. Show that (a² + b²) is composite.

Solution

Let the roots be c, d, so c + d = -a, cd = b+1. Hence a² + b² = (c² + 1)(d² + 1).