5
Part of 2007 All-Russian Olympiad
Problems(3)
Maykop and Belorechensk
Source: All-Russian 2007
5/4/2007
The distance between Maykop and Belorechensk is km. Two of three friends need to reach Belorechensk from Maykop and another friend wants to reach Maykop from Belorechensk. They have only one bike, which is initially in Maykop. Each guy may go on foot (with velocity at most kmph) or on a bike (with velocity at most kmph). It is forbidden to leave a bike on a road. Prove that all of them may achieve their goals after hours minutes. (Only one guy may seat on the bike simultaneously).Folclore
combinatorics unsolvedcombinatorics
two numbers in each vertice of a convex $100$-gon
Source: All-Russian 2007
5/4/2007
Two numbers are written on each vertex of a convex -gon. Prove that it is possible to remove a number from each vertex so that the remaining numbers on any two adjacent vertices are different.
F. Petrov
inductionalgebrapolynomialfunctioncombinatorics proposedcombinatorics
a set of $n>2$ planar vectors
Source: All-Russian 2007
5/4/2007
Given a set of planar vectors. A vector from this set is called long, if its length is not less than the length of the sum of other vectors in this set. Prove that if each vector is long, then the sum of all vectors equals to zero.
N. Agakhanov
vectorinequalitiesgeometry unsolvedgeometry