USA Mathematical Olympiad 1989 Problem (Second Method)

USA Mathematical Olympiad 1989 Problem

Which is larger, the real root of $x + x^2 + ... + x^8 = 8 - 10x^9$, or the real root of $x + x^2 + ... + x^{10} = 8 - 10x^{11}$?

Previously we checked for the maximum and minimum points to ascertain if the first derivative of $f(x)=x+x^2+\cdots +x^8 +10x^9 - 8$ and $g(x)=x+x^2+\cdots +x^{10}+10x^{11}-8$ are always greater than or equal to zero.

IMO (Hong Kong) Trigonometric Problem (Modified)

Let $A,\,B$ be acute angles such that $\tan B=2015\sin A \cos A-2015\sin^2 A \tan B$.

Find the greatest possible value of $\tan B$.

This is a fun IMO problem, since it has many ways (all are nothing less than remarkable) to approach it and without any further ado, I will post with the first approach here:

$\tan B=2015\sin A \cos A-2015\sin^2 A \tan B$

$\tan B(1+2015\sin^2 A )=2015\sin A \cos A$

Slideshow 1: Teaching Mathematics Methodology For Success