28th All Russian Mathematical Olympiad Problems 2002

1.  Can the cells of a 2002 x 2002 table be filled with the numbers from 1 to 20022 (one per cell) so that for any cell we can find three numbers a, b, c in the same row or column (or the cell itself) with a = bc? 2.  ABC is a triangle. D is a point on the side BC. A is equidistant from the incenter of ABD and the excenter of ABC which lies on the internal angle

27th All Russian Mathematical Olympiad Problems 2001

1.  Are there more positive integers under a million for which the nearest square is odd or for which it is even? 2.  A monic quartic and a monic quadratic both have real coefficients. The quartic is negative iff the quadratic is negative and the set of values for which they are negative is an interval of length more than 2. Show that at some point the quartic

26th All Russian Mathematical Olympiad Problems 2000

1.  The equations x2 + ax + 1 = 0 and x2 + bx + c = 0 have a common real root, and the equations x2 + x + a = 0 and x2 + cx + b = 0 have a common real root. Find a + b + c. 2.  A chooses a positive integer X ≤ 100. B has to find it. B is allowed to ask 7 questions of the form "What is the greatest common divisor of X + m and n?" for positive integers m,

25th All Russian Mathematical Olympiad Problems 1999

1.  The digits of n strictly increase from left to right. Find the sum of the digits of 9n. 2.  Each edge of a finite connected graph is colored with one of N colors in such a way that there is just one edge of each color at each point. One edge of each color but one is deleted. Show that the graph remains connected. 3.  ABC is a triangle. A' is the

24th All Russian Mathematical Olympiad Problems 1998

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

23rd All Russian Mathematical Olympiad Problems 1997

1.  p(x) is a quadratic polynomial with non-negative coefficients. Show that p(xy)2 ≤ p(x2)p(y2). 2.  A convex polygon is invariant under a 90o rotation. Show that for some R there is a circle radius R contained in the polygon and a circle radius R√2 which contains the polygon. 3.  A rectangular box has integral sides a, b, c, with c odd. Its surface is

22nd All Russian Mathematical Olympiad Problems 1996

1.  Can a majority of the numbers from 1 to a million be represented as the sum of a square and a (non-negative) cube? 2.  Non-intersecting circles of equal radius are drawn centered on each vertex of a triangle. From each vertex a tangent is drawn to the other circles which intersects the opposite side of the triangle. The six resulting lines enclose a

6th All Russian Mathematical Olympiad Problems 1966

6th All Russian Mathematical Olympiad Problems 19661.  There are an odd number of soldiers on an exercise. The distance between every pair of soldiers is different. Each soldier watches his nearest neighbour. Prove that at least one soldier is not being watched.