MathDB

3

Part of 2016 International Zhautykov Olympiad

Problems(2)

izho 2016 p3

Source: IZhO2016 Day1 P3

1/15/2016
There are 6060 towns in GraphlandGraphland every two countries of which are connected by only a directed way. Prove that we can color four towns to red and four towns to green such that every way between green and red towns are directed from red to green
graph theorycombinatoricsProbabilistic Method
Izho 2016 p6

Source: IZhO 2016 Day 2 P6

1/16/2016
We call a positive integer qq a convenient   denominator for a real number α\alpha if ∣α−pq∣<110q\displaystyle |\alpha - \dfrac{p}{q}|<\dfrac{1}{10q} for some integer pp. Prove that if two irrational numbers α\alpha and β\beta have the same set of convenient denominators then either α+β\alpha+\beta or α−β\alpha- \beta is an integer.
number theoryirrational number