▪ Turkey National Olympiad Second Round 2012

 Ημέρα 2η 

1. Find all polynomials with integer coefficients such that for all positive integers satisfies  

2. Let be a isosceles triangle with an be the foot of perpendicular of . be an interior point of triangle such that

and .

and intersects at , and intersects at . Let be a point on and be a point on and not belongs to satisfying

  and . 

Show that .

3. Find all non-decreasing functions from real numbers to itself such that for all real numbers

holds. 

 Ημέρα 2η 

1 For all positive real numbers , show that

   

is true.

2. Let be set of all -tuples such that for every

  , .

For each and for each ,

if a subset satisfies the condition

, 

we call to an decreasing set, if satisfies

, 

we call it to an increasing set.

Let be a non-empty decreasing set, and be a non-empty increasing set. Find the maximum possible value of . 

3. Let and be points on segments and respectively. Excircles of triangles and touching sides and is the same, and its center is . and intersects at . Let be the circumcenters of triangles

  

respectively.

a) Show that points concylic and points concylic.

b) Denote centers of theese circles as and . Prove that and are collinear.

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Turkey National Olympiad First Round 2012