Some Tripos Problems Practice for RMO

Posted by Nalin Pithwa

Solve the equation: $\frac{(x-a)(x-b)}{x-a-b} = \frac{(x-c)(x-d)}{x-c-d}$
Find the value of a for which the fraction $\frac{x^{3}-ax^{2}+19x-a-4}{x^{3}-(a+1)x^{2}+23x-a-7}$ admits of reduction. Reduce it to lowest terms.
An AP, a GP and an HP have a and b for their first two terms; show that their $(n+2)^{th}$ terms will be in GP if $\frac{b^{2m+2}-a^{2m+1}}{ba(b^{2n}-a^{2n})}=\frac{n+1}{n}$