▪ Korea National Olympiad 2012

Hμέρα 1η

1. Let be positive real numbers. Prove that

 

2. For triangle such that and , let be the incenter and be the foot of perpendicular from to .
Let . Let be the circle with diameter and (incircle) . ( are in the different side from and are in the same side from . ) Define .

 

Now think of an isosceles triangle such that . Let be the point on the side such that .

Let . Prove that . 

3. are given subsets. Let . For any , let

Let be an integer such that . Prove that there exist such that and . 

Hμέρα 2η

1. Let be an acute triangle. Let be the foot of perpendicular from to . are the points on and let be the foot of perpendicular from to . Assume that is on . Let be the foot of perpendicular from to . Prove that . 

2. Let be a given positive integer. Prove that there exist infinitely many integer triples such that

 

3. Let be the set of positive integers which do not have as a prime divisor. For any infinite family of subsets of , say , prove that there exist such that for each there exists some such that .