## Root inequality

Let be positive real numbers satisfying the following equality

Prove that

**Solution**

We begin by stating a lemma:

**Lemma:** Let be positive real numbers, then:

Now, making use of the lemma we have that:

Making use of the Cauchy – Schwartz inequality we have that

The inequality now follows.

## A logarithmic inequality

Let be positive numbers other than such that . Prove that

**Solution**

The Engels form of the Cauchy – Schwartz inequality gives us:

and the inequality is proven.

## On an infinite summation

Let be a sequence of real numbers. Compute:

**Solution**

First and foremost we set and it is obvious that . We are making use of probabilistic methods. Suppose than an infinite number of coins are flipped. Let be the probability that the -th coin toss lands heads and let us consider the first time heads comes up. Then is the probability that the first head appears in the – th flip and is the probability that all flips come up tails. Thus,

## Trigonometric equality

Prove that in any triangle it holds that

where denotes the circumradius and the inradius.

**Solution**

Using the law of sines we have that

and if we denote the area of the triangle then

Thus,

## Root inequality

Let be three positive real numbers such that . Prove that

**Solution**

By AM – GM we have,

However,

Hence and the exercise is complete.