3

Part of 2005 Cono Sur Olympiad

Problems(2)

Cono Sur Olympiad 2005, Problem 3

Source:

The monetary unit of a certain country is called Reo, and all the coins circulating are integers values of Reos. In a group of three people, each one has 60 Reos in coins (but we don't know what kind of coins each one has). Each of the three people can pay each other any integer value between 1 and 15 Reos, including, perhaps with change. Show that the three persons together can pay exactly (without change) any integer value between 45 and 135 Reos, inclusive.

combinatorics proposedcombinatorics

Cono Sur Olympiad 2005, Problem 6

Source:

On the cartesian plane we draw circunferences of radii 1/20 centred in each lattice point. Show that any circunference of radii 100 in the cartesian plane intersect at least one of the small circunferences.

symmetryanalytic geometrycombinatorics unsolvedcombinatorics