## summation

Let denote the Euler’s function. Prove that

Solution

The key idea is to rewrite the floor as a sum involving divisors. Hence,

## Equal segments

Let and let be a line that intersects at the points , and the axis at the points and .

Prove that .

Solution

Let with . Then,

Hence,

## Integral inequality (I)

Let be a differentiable and convex function such that and . Prove that .

## Symmetry

Let be continuous functions. If is symmetric around the line then prove that:

## An integral from a geometric view

Evaluate the integral

Solution