Question 83
a) For non-zero integers $a, b, c$ it holds that: $\frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{a} = \frac{a^2}{c} + \frac{c^2}{b} + \frac{b^2}{a}$.
Does it follow then that: $\frac{a}{b} + \frac{b}{c} + \frac{c}{a} = \frac{a}{c} + \frac{c}{b} + \frac{b}{a}$?
6) Consider all triples of non-zero integers $(a, b, c)$, which belong to the interval $[-2020, 2020]$. Find for how many of them the conditions $a+b+c=0$ and $\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}=\frac{a^2}{c}+\frac{c^2}{b}+\frac{b^2}{a}$ are equivalent? In other words, how many such triples are there, such that either both conditions hold, or none of the two holds?
Does it follow then that: $\frac{a}{b} + \frac{b}{c} + \frac{c}{a} = \frac{a}{c} + \frac{c}{b} + \frac{b}{a}$?
6) Consider all triples of non-zero integers $(a, b, c)$, which belong to the interval $[-2020, 2020]$. Find for how many of them the conditions $a+b+c=0$ and $\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}=\frac{a^2}{c}+\frac{c^2}{b}+\frac{b^2}{a}$ are equivalent? In other words, how many such triples are there, such that either both conditions hold, or none of the two holds?