## Inequality in acute triangle

Let be an acute triangle. Prove the following inequality

Solution

The solution can be found at cut -the – knot

## Inequality in a triangle

Given a triangle let denote the median points of the sides respectively. Prove that

Solution [Soumava Chakraborty]

We have successively

In all the above denotes the semiperimeter of the triangle. More on Gerretsen’s inequality can be found at this link .

## Inequality

Let satisfying . Prove that

Solution

Since then the numbers

could be sides of a triangle. The area of this triangle is

However , in any triangle is holds that [Weitzenböck]

(1)

where is the area of the triangle. Thus

## Exponential matrix

Let . We define

It is known that this series converges. Prove that

Solution

We triangulise the matrix , that is where is an invertible matrix and is an upper triangular. This is possible since our matrix is over and thus its characteristic polynomial splits. Let be its eigenvalues. Then we note that is upper triangular with in its diagonal. Hence is also upper triangular with in its diagonal. Hence

However and forall . Thus and finally

## An arccot sum

Evaluate the sum

Solution

Well, first of all we note that

We also recall the Fourier series expansion of which is no other than

(1)

Hence