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# Tag Archives: General

## Constant area

Let be a positive real number. The parabolas defined by and intersect at the points and .

Prove that the area enclosed by the two curves is constant. Explain why.

Solution

First of all we note that

Hence,

## Polynomial equation

Let denote the golden ratio. Solve the equation

Solution

First of all we note that

We easily note that is one root of the equation, hence using Horner we get that

Hence is a double root and the other root is .

## Binomial sum

Let be positive numbers with . Prove that

Solution

Using the exercise here we have that

Hence,

## Logarithmic inequality

Let . Prove that

Solution

Let and . Thus,

Thus,

The result follows.

## Nested binomial sum

Prove that

Solution

We may begin with the beta function identity for non negative integer values of .

Hence, for non-negative integers

As a result we may compute the nested summation as,