Third Heuristic Method In Solving Problem: Compare $M$ and $N$.

Given that $p,\,q,\,r,\,s,\,a,\,b$ are positive real numbers with $M=\sqrt{ap+br}\cdot \sqrt{\dfrac{q}{a}+\dfrac{s}{b}}$ and $N=\sqrt{pq}+\sqrt{rs}$. Compare $M$ and $N$.

The third inspiring methods of solving the above question will be discussed in this post.

It's solved by one Indian genius, a math friend of mine who compared the quantities between $M^4$ and $N^4$:

He first noticed

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: