Olympiad problems

2018 Pan African, 5

Let $a$, $b$, $c$ and $d$ be non-zero pairwise different real numbers such that $$ \frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} = 4 \text{ and } ac = bd. $$ Show that $$ \frac{a}{c} + \frac{b}{d} + \frac{c}{a} + \frac{d}{b} \leq -12 $$ and that $-12$ is the maximum.

2011 Gheorghe Vranceanu, 2

Tags: AMGM , squeezing , AM-GM , algebra , limits , caluclus , calculus

$ a>0,\quad\lim_{n\to\infty }\sum_{i=1}^n \frac{1}{n+a^i} $

2018 Pan-African Shortlist, A4

Tags: inequalities , algebra , AMGM

Let $a$, $b$, $c$ and $d$ be non-zero pairwise different real numbers such that $$ \frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} = 4 \text{ and } ac = bd. $$ Show that $$ \frac{a}{c} + \frac{b}{d} + \frac{c}{a} + \frac{d}{b} \leq -12 $$ and that $-12$ is the maximum.