Thursday, April 30, 2015

Asian Pacific Mathematics Olympiad Mock Problem

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

At first glance, this looks like a very demanding hard challenge, and you don't even know where to start working on the problem.

But this is necessarily a delicious problem in terms of how well we can milk it for all its worth to benefit students. In case  you might not already know, I am the advocate for teaching students and equip them with analytical thinking and critical problem solving skills, so, this problem can help me to achieve my aim.

According to Wikipedia (Analytical skill):

Analytical skill is the ability to visualize, articulate, and solve both complex and uncomplicated problems and concepts and make decisions that are sensible and based on available information. Such skills include demonstration of the ability to apply logical thinking to gathering and analyzing information, designing and testing solutions to problems, and formulating plans.

Analytical skills are highly important in every aspect of every single task, regardless if you are a student, or a worker, or even a boss out there. If you fully understand the basic principles of the "task", then you will be able to solve the task easily. That is why analytical skills are so important.

This is what this problem offers us, the opportunity to attack it analytically:

If you digest the quotient being summed properly, you will see that the given sum is zero because the numerator and denominator are symmetric about the same value of $x$, i.e. $x=\dfrac{101}{2}$, with the numerator being odd and the denominator even with respect to this axis of symmetry. This is analogous to the odd function rule from integral calculus.

Graphical calculator helps us greatly in seeing how the numerator and denominator are symmetric about  $x=\dfrac{101}{2}$, and if you used that in constructing your argument, you're improving your analytical skill at the same time.

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

Well, learning to think analytically and critically is not easy, and it might be daunting at first, but the good news is, it can be learned!

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