Let denote the Euler’s function. Prove that
The key idea is to rewrite the floor as a sum involving divisors. Hence,
Let and let be a line that intersects at the points , and the axis at the points and .
Prove that .
Let with . Then,
Let be a differentiable and convex function such that and . Prove that .
Let be continuous functions. If is symmetric around the line then prove that:
Evaluate the integral