Mathematical Problem Solving Skill

In this blog post, I will show one really insightful method given by one very intelligent U.K. mathematician on how he used his own elegant way to prove that $\tan^2 20^{\circ}+\tan^2 40^{\circ}+\tan^2 80^{\circ}=33$.

He first noticed that

$\tan 3(20^\circ)=\tan 60^\circ=\sqrt{3}$

$\tan 3(40^\circ)=\tan 120^\circ=-\sqrt{3}$

$\tan 3(80^\circ)=\tan 240^\circ=\sqrt{3}$

Algebraic Method to Tackle the Mock APMO Problem

 There exists another way to tackle the previously discussed AMPO mock problem (Asian Pacific Mathematics Olympiad Mock Problem ).

Find [MATH]\sum_{x=0}^{101}\dfrac{\dfrac{2x}{101}-1}{\dfrac{3x^2}{10201}-\dfrac{3x}{101}+1}[/MATH].

In case you are not well prepared to attack the problem analytically, you could still tackle it algebraically, that is purely allowable and no one will ever say algebraic method is not awesome!

For simplicity's sake, we let $x_i=\dfrac{i}{101}$ and $f(x)=\dfrac{\dfrac{2x}{101}-1}{\dfrac{3x^2}{10201}-\dfrac{3x}{101}+1}$, we then have:

A Trigonometric Sum

It can be shown that the following sum:

[MATH]S=\sum_{k=1}^{89}\left(\sin^6\left(k^{\circ}\right)\right)[/MATH]

is rational. Find the value of $S$.