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# Tag Archives: General

## 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, ## Trigonometric inequality on an acute triangle

Prove that in any acute triangle the following inequality holds: Solution

Since it holds that

(1) and thus

(2) Using Nesbitt’s inequality we see that Equality holds if-f .

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