2
Part of 1996 All-Russian Olympiad
Problems(3)
three circles of equal radius
Source: All-Russian Olympiad 1996, Grade 9, First Day, Problem 2
4/18/2013
The centers ; ; of three nonintersecting circles of equal radius are positioned at the vertices of a triangle. From each of the points ; ; one draws tangents to the other two given circles. It is
known that the intersection of these tangents form a convex hexagon. The sides of the hexagon are alternately colored red and blue. Prove that the sum of the lengths of the red sides equals the sum of the lengths of the blue sides.D. Tereshin
geometry proposedgeometry
Prove that any two counters can be made to coincide
Source: All-Russian Olympiad 1996, Grade 10, First Day, Problem 2
4/18/2013
On a coordinate plane are placed four counters, each of whose centers has integer coordinates. One can displace any counter by the vector joining the centers of two of the other counters. Prove that any two preselected counters can be made to coincide by a finite sequence of moves.Р. Sadykov
analytic geometryvectoralgorithmnumber theoryEuclidean algorithmcombinatorics proposedcombinatorics
sum of pairwise distances is decreasing
Source: All-Russian Olympiad 1996, Grade 11, First Day, Problem 2
4/19/2013
Several hikers travel at fixed speeds along a straight road. It is known that over some period of time, the sum of their pairwise distances is monotonically decreasing. Show that there is a hiker, the sum of whose distances to the other hikers is monotonically decreasing over the same period.A. Shapovalov
geometry proposedgeometry