On the regular nonagon

Given the regular nonagon below

prove that

Solution

Currently unavailable!

An equivalence relation in a triangle

Prove that in any triangle the following equivalence relation holds:

Solution

We are working on the following shape.

Let be the incenter of the triangle. Thus,

meaning that is tangent to the circumcircle of the triangle . Thus,

(1)

Hence,

On the heptagon

Given a heptagon of side and diagonals such that ,

prove that:

Solution

Let   be the side of the heptagon and be its diagonals respectively. It holds that

(1)

(2)

(3)

(4)

Equation comes naturally from Vieta’s formulae since are the roots of the equation . Thus,

Limit of a sequence

Let be a function such that and is differentiable at . Let us set

Evaluate the limit .

Solution

Since is differentiable at , there is some such that

and is of course continuous.

Thus,

Let . There exists such that which in return means that . Hence , for larger than it holds that

On the other hand , the sum is a Riemann sum and converges to .

In conclusion,

Integral of Jacobi Theta function

Let denote one of the Jacobi Theta functions. Prove that

Solution

We have successively,

The sum is evaluated as follows. Consider the function

and integrate it around a square with vertices . The function has poles at every integer with residue as well as at with residues . We also note that as the contour integral of tends to . Thus,

Hence,

and the exercise is complete.