Πέμπτη 12 Ιανουαρίου 2012

▪ Turkey National Olympiad 2011

1. and are subsets of , such that for all Exactly one of is empty set. What is the maximum possible ?
2. Let be a triangle (different than and ). is the midpoint of . such that and Circumcircle of intersect at different than .Prove that tangent to circumcircle of at is touch circumcircle of too.
3. positive real numbers such that
Prove that:
 
 
4. and for all . is a prime such that and . Show that .
5. Let and be two regular polygonic area.Define as the midpoints of segments such that belong to and belong to . Find all situations of and such that is a regualr polygonic area too.
6. Let and two countries which inlude exactly cities.There is exactly one flight from a city of to a city of and there is no domestic flights (flights are bi-directional).For every city (doesn't matter from or from ), there exist at most different airline such that airline have a flight from to the another city.For an integer , (it doesn't matter how flights arranged ) we can say that there exists at least cities such that it is possible to trip from one of these cities to another with same airline.So find the maximum value of .

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