One Worked Example of Solving Heuristic IMO Math Problem

Olympiad Algebra Problem:

If $a,\,b,\,x,\,y\in R$ such that

$ax +by =7$

$ax^2+by^2=49$

$ax^3+by^3=133$

$ax^4+by^4=406$

Evaluate $ax^5+by^5$.

The solution from the U.K. mathematician:

Challenging Math Contest Problem: Prove that $\tan^2 x+\tan^2 (x+60^{\circ})+\tan^2 (60^{\circ}-x)=9\tan^2 3x+6$

Given $\tan x+\tan (x+60^{\circ})-\tan (60^{\circ}-x)=3\tan 3x$,

prove that $\tan^2 x+\tan^2 (x+60^{\circ})+\tan^2 (60^{\circ}-x)=9\tan^2 3x+6$.

Wow! This is another exquisite problem that one cannot afford to pass it up but to take it as one tough learning example problem so to train students to be the best.  Remember that good teachers will forever encourage learning for understanding and are concerned with developing their students’ critical-thinking skills, problem-solving skills, and problem-approach behaviors. This problem fulfills the these goals of training students to think creatively and that is the reason I bring it to our table.

I won't beat about the bush, so here goes my plan of attack:

Find The Sum Involving The Inverse Tangent Function

We are given to evaluate:

[math]S_n=\sum_{k=0}^n\left[\tan^{-1}\left(\frac{1}{k^2+k+1} \right) \right][/math]

Vietnamese Mathematical Olympiad (Trigonometric) Problem of 1962

Solve the equation $\sin^6 x+\cos^6 x=\dfrac{1}{4}$.

This is one of the brilliant Mathematics Olympiad Contest Problems because we can show to the students how there are plenty of ways to attacking a good problem and how one approach is different from the other and how heuristic skill enable us to find solution quickly that save us time for more challenging problems!