Olympiad problem algebra inequality I'm having trouble solving the following inequality problem:
If $n$ is positive integer greater than $1$, and $x>y>1$, then show that:
$\frac{x^{n+1}-1}{x(x^{n-1}-1)} > \frac{y^{n+1}-1}{y(y^{n-1}-1)}$
Any hints? Thanks.

Since $x^{n-1} > x^{n-2} > \dots > 1$, and $y^{n-1} > y^{n-2} > \dots > 1$,
we have from the rearrangement inequality
$$
x^{n-1} y^{n-1} + x^{n-2} y^{n-2} + \dots + 1
>
x^{n-1} + x^{n-2} y + \dots + y^{n-1}.
$$
Multiplying both side by $(xy - 1)(x - y)$, we get
$$
(x^n y^n - 1)(x - y) > (x^n - y^n)(xy - 1).
$$
Or
$$
(x^{n+1} - 1) (y^n - y)
>
(y^{n+1} - 1) (x^n - x).
$$
which is the required result after dividing both sides by $(y^n -y)(x^n -x)$.
