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# Tag Archives: General

## Square of a number

Let such that . Prove that

is rational.

**Solution**

Setting , and we note that . Hence,

The result follows.

## Proof of “Fermat’s last theorem”

Let and . Prove that the equation

has no solution.

**Solution**

Without loss of generality , assume that . If held , then it would be thus . It follows from Bernoulli’s inequality that,

which is an obscurity. The result follows.

## Double binomial sum

Evaluate the double sum

**Solution**

We sum diagonally , hence:

(1): For it holds .

(2): .

**Conjecture:** Does the following equality

hold?

## Kinda Pythagorean Theorem

Let be a triangle such that , and . Find the area of the triangle.

**Solution**

Since it follows from the law of sines that

Hence . Thus,

To completely justify the title of the post we give another solution based on the following proposition:

**Proposition: **Let be a given triangle such that . Prove that

*Proof: *We are working on the following figure.

Let be the bisector of . Then:

and the result follows.

Now, the area of the initial triangle is given by Heron’s formula.

## An inequality

Let . Prove that

**Solution**

**Lemma: **If or then it holds

*Proof: *Straightforward!

Then,