Let be a real numbers such that Prove thatWhere

Let be natural number and be the set of distinct natural numbers. Define function with . Prove that have at least distinct values.

Find all prime numbers and such that
is a prime number.

Proposed by Dorlir Ahmeti, Albania

Find positiv integer numbers s.t. and be positiv integer numbers.

Let be the midpoint of the side of triangle . The bisector of the exterior angle of point intersects the side in . Let the circumcircle of triangle intersect the lines and in and respectively. If the midpoint of is , prove that .

Are there any positive integers and satisfying the equation

?

Let triangle ABC be inscribed in a circle with center O. A line d intersects sides AB and AC at points E and D, respectively. Let M, N, and P be the midpoints of segments BD, CE, and DE, respectively. Let Q be the foot of the perpendicular from O to line DE. Prove that the points M, N, P, and Q lie on a circle.

If is not a perfect square, then there exists an odd prime s.t. is a quadratic non-residue mod .

Let be the orthocenter of the triangle . Let and be the midpoints of the sides and , respectively. Assume that lies inside the quadrilateral and that the circumcircles of triangles and are tangent to each other. The line through parallel to intersects the circumcircles of the triangles and in the points and , respectively. Let be the intersection point of and and let be the incenter of triangle . Prove that .

Find all functions that satisfies for all real

Given triangle with . We take , point on AC, AB respectively such that , . , lines intersect at point . If , find