24th Russian 1998

24th Russian 1998 problems


1. a and b are such that there are two arcs of the parabola y = x² + ax + b lying between the ray y = x, x > 0 and y = 2x, x > 0. Show that the projection of the left-hand arc onto the x-axis is smaller than the projection of the right-hand arc by 1.
2. A convex polygon is partitioned into parallelograms, show that at least three vertices of the polygon belong to only one parallelogram.
3. Can you find positive integers a, b, c, so that the sum of any two has digit sum less than 5, but the sum of all three has digit sum more than 50?
4. A maze is a chessboard with walls between some squares. A piece responds to the commands left, right, up, down by moving one square in the indicated direction (parallel to the sides of the board), unless it meets a wall or the edge of the board, in which case it does not move. Is there a universal sequence of moves so that however the maze is constructed and whatever the initial position of the piece, by following the sequence it will visit every square of the board. You should assume that a maze must be constructed, so that some sequence of commands would allow the piece to visit every square.
5. Five watches each have the conventional 12 hour faces. None of them work. You wish to move forward the time on some of the watches so that they all show the same time and so that the sum of the times (in minutes) by which each watch is moved forward is as small as possible. How should the watches be set to maximise this minimum sum?
6. In the triangle ABC, AB > BC, M is the midpoint of AC and BL is the angle bisector of B. The line through L parallel to BC meets BM at E and the line through M parallel to AB meets BL at D. Show that ED is perpendicular to BL.
7. A chain has n > 3 numbered links. A customer asks for the order of the links to be changed to a new order. The jeweller opens the smallest possible number of links, but the customer chooses the new order in order to maximise this number. How many links have to be opened?
8. There are two unequal rational numbers r < s on a blackboard. A move is to replace r by rs/(s - r). The numbers on the board are initially positive integers and a sequence of moves is made, at the end of which the two numbers are equal. Show that the final numbers are positive integers.
9. A, B, C, D, E, F are points on the graph of y = ax³ + bx² + cx + d such that ABC and DEF are both straight lines parallel to the x-axis (with the points in that order from left to right). Show that the length of the projection of BE onto the x-axis equals the sum of the lengths of the projections of AB and CF onto the x-axis.
10. Two polygons are such that the distance between any two vertices of the same polygon is at most 1 and the distance between any vertex of one polygon and any vertex of the other is more than 1/√2. Show that the interiors of the two polygons are disjoint.
11. The point A' on the incircle of ABC is chosen so that the tangent at A' passes through the foot of the bisector of angle A, but A' does not lie on BC. The line L_A is the line through A' and the midpint of BC. The lines L_B and L_C are defined similarly. Show that L_A, L_B and L_C all pass through a single point on the incircle.
12. X is a set. P is a collection of subsets of X, each of which have exactly 2k elements. Any subset of X with at most (k+1)² elements either has no subsets in P or is such that all its subsets which are in P have a common element. Show that every subset in P has a common element.
13. The numbers 19 and 98 are written on a blackboard. A move is to take a number n on the blackboard and replace it by n+1 or by n². Is it possible to obtain two equal numbers by a series of moves?
14. A binary operation * is defined on the real numbers such that (a * b) * c = a + b + c for all a, b, c. Show that * is the same as +.
15. Given a convex n-gon with no 4 vertices lying on a circle, show that the number of circles through three adjacent vertices of the n-gon such that all the other vertices lie inside the circle exceeds by two the number of circles through three vertices, no two of which are adjacent, such that all other vertices lie inside the circle.
16. Find the number of ways of placing a 1 or -1 into each cell of a (2ⁿ - 1) by (2ⁿ - 1) board, so that the number in each cell is the product the numbers in its neighbours (a neighbour is a cell which shares a side).
17. The incircle of the triangle ABC touches the sides BC, CA, AB at D, E, F respectively. D' is the midpoint of the arc BC that contains A, E' is the midpoint of the arc CA that contains B, and F' is the midpoint of the arc AB that contains C. Show that DD', EE', FF' are concurrent.
18. Given a collection of solid equilateral triangles in the plane, each of which is a translate of the others, such that every two have a common point. Show that there are three points, so that every triangle contains at least one of the points.
19. A connected graph has 1998 points and each point has degree 3. If 200 points, no two of them joined by an edge, are deleted, show that the result is a connected graph.
20. C₁ is the circle center (0, 1/2), diameter 1 which touches the parabola y = x² at the point (0, 0). The circle C_n+1 has its center above C_n on the y axis, touches C_n and touches the parabola at two symmetrically placed points. Find the diameter of C₁₉₉₈.
21. Do there exist 1998 different positive integers such that the product of any two is divisible by the square of their difference?
22. The tetrahedron ABCD has all edges less than 100 and contains two disjoint spheres of diameter 1. Show that it contains a sphere of diameter 1.01.
23. A figure is made out of unit squares joined along their sides. It has the property that if the squares of an m x n rectangle are filled with real numbers with positive sum, then the figure can be placed over the rectangle (possibly after being rotated, but with each square of the figure coinciding with a square of the rectangle) so that the sum of the numbers under each square is positive. Prove that a number of copies of the figure can be placed over an m x n rectangle so that each square of the rectangle is covered by the same number of figures.