MathDB

2

Part of 2000 Tuymaada Olympiad

Problems(3)

When a tangent line meets a rhombus..

Source: Tuymaada 2000, day 1, problem 2.

4/30/2007
A tangent ll to the circle inscribed in a rhombus meets its sides ABAB and BCBC at points EE and FF respectively. Prove that the product AEā‹…CFAE\cdot CF is independent of the choice of ll.
geometryrhombusinradiusgeometry proposed
What if we connect the cities of Graphland?

Source: Tuymaada 2000, day 2, problem 2.

4/30/2007
There are 2000 cities in Graphland; some of them are connected by roads. For every city the number of roads going from it is counted. It is known that there are exactly two equal numbers among all the numbers obtained. What can be these numbers?
inductioncombinatorics proposedcombinatorics
4 colors to paint the plane, inside any circle all 4 colors, is it possible?

Source: Tuymaada 2000 Juniors 2

4/28/2019
Is it possible to paint the plane in 44 colors so that inside any circle are the dots of all four colors?
combinatoricscombinatorial geometryColoringPlaneColor problem