Let be the set of all functions with the following properties:

(i) is defined for all real numbers and takes only real values.

(ii) For all the following equality holds:

(iii)

Determine all functions such that

(a) ,

(b) .

find all such that:
a)
b)

In a convex quadrilateral , the diagonal bisects neither the angle nor the angle . The point lies inside and satisfies Prove that is a cyclic quadrilateral if and only if .

Let be a cyclic quadrilateral in which the lines and meet at a point . Let be the point of the line , different from , such that . We construct the parallelograms and . Prove that the points lie on the same circle.

Determine all functions defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions:
[list]
[*] for at least one ;
[*] for every positive integers and ;
[*] there are infinitely many positive integers such that for all .
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Let be positive real numbers such that . Find the minimum value of the following sum :
knowing that the denominators are positive real numbers.

Inside of convex quadrilateral point was chosen such that and . Prove that if perpendicular from to passes from the intersection of diagonals of , then .

Initially, Aslı distributes balls to boxes as she wishes. After that, Aslı and Zehra make alternated moves which consists of taking a ball in any wanted box starting with Aslı. One who takes the last ball from any box takes that box to herself. What is the maximum number of boxes can Aslı guarantee to take herself regardless of Zehra's moves?

In , . The internal angle bisector of and the external angle bisector of intersect the ray at points and , respectively. Given that , prove that .

Find all functions from the reals to the reals satisfying