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, 8 or 3, 5, 7 or 3, 4, 6. After a series of moves the three numbers are 17, 1967 and 1983. Could the initial set have been 2, 2, 2? 3, 3, 3?

3.  C₁, C₂, C₃ are circles, none of which lie inside either of the others. C₁ and C₂ touch at Z, C₂ and C₃ touch at X, and C₃ and C₁ 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.

4.  Find all real solutions x, y to y² = x³ - 3x² + 2x, x² = y³ - 3y² + 2y.

5.  The positive integer k has n digits. It is rounded to the nearest multiple of 10, then to the nearest multiple of 100 and so on (n-1 roundings in all). Numbers midway between are rounded up. For example, 1474 is rounded to 1470, then to 1500, then to 2000. Show that the final number is less than 18k/13.

6.  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.

7.  a_n is the last digit of [10^n/2]. Is the sequence a_n periodic? b_n is the last digit of [2^n/2]. Is the sequence b_n periodic?

8.  A and B are acute angles such that sin²A + sin²B = sin(A + B). Show that A + B = π/2.

9.  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?

10.  Given a quadratic equation ax² + bx + c. If it has two real roots A ≤ B, transform the equation to x² + Ax + B. Show that if we repeat this process we must eventually reach an equation with complex roots. What is the maximum possible number of transformations before we reach such an equation?

11.  a, b, c are positive integers. If a^b divides b^a and c^a divides a^c, show that c^b divides b^c.

12.  A word is a finite string of As and Bs. Can we find a set of three 4-letter words, ten 5-letter words, thirty 6-letter words and five 7-letter words such that no word is the beginning of another word. [For example, if ABA was a word, then ABAAB could not be a word.]

13.  Can you place an integer in every square of an infinite sheet of squared paper so that the sum of the integers in every 4 x 6 (or 6 x 4) rectangle is (1) 10, (2) 1?

14.  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)?

15.  A group of children form two equal lines side-by-side. Each line contains an equal number of boys and girls. The number of mixed pairs (one boy in one line next to one girl in the other line) equals the number of unmixed pairs (two girls side-by-side or two boys side-by-side). Show that the total number of children in the group is a multiple of 8.

16.  A 1 x k rectangle can be divided by two perpendicular lines parallel to the sides into four rectangles, each with area at least 1 and one with area at least 2. What is the smallest possible k?

17.  O is a point inside the triangle ABC. a = area OBC, b = area OCA, c = area OAB. Show that the vector sum aOA + bOB + cOC is zero.

18.  Show that given any 2m+1 different integers lying between -(2m-1) and 2m-1 (inclusive) we can always find three whose sum is zero.

19.  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?

20.  X is a union of k disjoint intervals of the real line. It has the property that for any h < 1 we can find two points of X which are a distance h apart. Show that the sum of the lengths of the intervals in X is at least 1/k.

21.  x is a real. The decimal representation of x includes all the digits at least once. Let f(n) be the number of distinct n-digit segments in the representation. Show that if for some n we have f(n) ≤ n+8, then x is rational.