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 the squares of the digits is 1000²?

3.  An 8 x 8 chess-board is colored in the usual way. What is the largest number of pieces can be placed on the black squares (at most one per square), so that each piece can be taken by at least one other? A piece A can take another piece B if they are (diagonally) adjacent and the square adjacent to B and opposite to A is empty.

4.  Call a side or diagonal of a regular n-gon a segment. How many colors are required to paint all the segments of a regular n-gon, so that each segment has a single color and every two segments with a vertex in common have different colors.

5.  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?

6.  The quadratic p(x) = ax² + bx + c has a > 100. What is the maximum possible number of integer values x such that |p(x)| < 50?

7.  In the diagram below a, b, c, d, e, f, g, h, i, j are distinct positive integers and each (except a, e, h and j) is the sum of the two numbers to the left and above. For example, b = a + e, f = e + h, i = h + j. What is the smallest possible value of d?

j
      h  i
   e  f  g
a  b  c  d

8.  a₁ < a₂ < ... < a_n < ... is an unbounded sequence of positive reals. Show that there exists k such that a₁/a₂ + a₂/a₃ + ... + a_h/a_h+1 < h-1 for all h > k. Show that we can also find a k such that a₁/a₂ + a₂/a₃ + ... + a_h/a_h+1 < h-1985 for all h > k.

9.  Find all pairs (x, y) such that |sin x - sin y| + sin x sin y <= 0.

10.  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?

11.  The sequence a₁, a₂, a₃, ... satisfies a_4n+1 = 1, a_4n+3 = 0, a_2n = a_n. Show that it is not periodic.

12.  n lines are drawn in the plane. Some of the resulting regions are colored black, no pair of painted regions have a boundary line in common (but they may have a common vertex). Show that at most (n² + n)/3 regions are black.

13.  Each face of a cube is painted a different color. The same colors are used to paint every face of a cubical box a different color. Show that the cube can always be placed in the box, so that every face is a different color from the box face it is in contact with.

14.  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.

15.  A move replaces the real numbers a, b, c, d by a-b, b-c, c-d, d-a. If a, b, c, d are not all equal, show that at least one of the numbers can exceed 1985 after a finite number of moves.

16.  a₁ < a₂ < ... < a_n and b₁ > b₂ > ... > b_n. Taken together the a_i and b_i constitute the numbers 1, 2, ... , 2n. Show that |a₁ - b₁| + |a₂ - b₂| + ... + |a_n - b_n| = n².

17.  An r x s x t cuboid is divided into rst unit cubes. Three faces of the cuboid, having a common vertex, are colored. As a result exactly half the unit cubes have at least one face colored. What is the total number of unit cubes?

18.  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.

19.  A regular hexagon is divided into 24 equilateral triangles by lines parallel to its sides. 19 different numbers are assigned to the 19 vertices. Show that at least 7 of the 24 triangles have the property that the numbers assigned to its vertices increase counterclockwise.

20.  x is a real number. Define x₀ = 1 + √(1 + x), x₁ = 2 + x/x₀, x₂ = 2 + x/x₁, ... , x₁₉₈₅ = 2 + x/x₁₉₈₄. Find all solutions to x₁₉₈₅ = x.

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

22.  Given a large sheet of squared paper, show that for n > 12 you can cut along the grid lines to get a rectangle of more than n unit squares such that it is impossible to cut it along the grid lines to get a rectangle of n unit squares from it.

23.  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. 

Solutions

Problem 20

x is a real number. Define x₀ = 1 + √(1 + x), x₁ = 2 + x/x₀, x₂ = 2 + x/x₁, ... , x₁₉₈₅ = 2 + x/x₁₉₈₄. Find all solutions to x₁₉₈₅ = x.

Answer

3

Solution

If x = 0, then x¹⁹⁸⁵ = 2 ≠ x. Otherwise we find x₁ = 2 + x/(1+√(1+x)) = 2 + (√(1+x) - 1) = 1 + √(1+x). Hence x₁₉₈₅ = 1 + √(1+x). So x - 1 = √(1+x). Squaring, x = 0 or 3. We have already ruled out x = 0. It is easy to check that x = 3 is a solution.