▪ Kosovo National Mathematical Olympiad 2013

1. Prove that:

.

2. Find all integer such that divide . 

3. For all real numbers prove that 

.

4. Find all value of parameter such that equations 

and  

have at least one same solution. For this value find same solution of this equations(real or imaginary). 

5. Let be an equilateral triangle, with sidelength equal to . Let be a point in the interior of triangle , and let and be the feet of the altitudes from on and , respectively. Prove that 

.

Πηγή: artofproblemsolving