Let be positive real numbers satisfying the following equality
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.
Let be positive numbers other than such that . Prove that
The Engels form of the Cauchy – Schwartz inequality gives us:
and the inequality is proven.
Let be a sequence of real numbers. Compute:
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,
Prove that in any triangle it holds that
where denotes the circumradius and the inradius.
Using the law of sines we have that
and if we denote the area of the triangle then
Let be three positive real numbers such that . Prove that
By AM – GM we have,
Hence and the exercise is complete.