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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 .

## 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,