Inequalities – Why is Reduce in Mathematica the solution in other symbols that are not present in the original equation?

Here is an example:

To reduce[P/Q[P/Q[P/Q[P/Q< (P - X)/(Q - X) && X > 0 && P> 0 && Q> 0 && P> Q, {X}, integers]

The solution to the above equation is:

(C[1] | C[2] | C[3])[Element] Integers &&
C[1] > = 0 &&[2] > = 0 &&
C[3] > = 0 && P = 3 + C[1] + C[2] + C[3] &&
Q == 2 + C[1] + C[2] &&
X = 1 + C[2]

How do I interpret it? What is C and why are the numbers 1, 2 and 3 in brackets? The Reduction documentation does not use this notation in any of the examples. https://reference.wolfram.com/language/ref/Reduce.html

Inequalities – Find the maximum of $ f (x) $

The following problem is due to a problem that I encountered while studying inequality. I find it hard to prove

To let $ n $ give positive integer,$ (n ge 2) $,and
$$ f (x) = (nx) ln {(n + x + 1)} – (nx) ln {(nx)}, 0 le x le n-1, x in N ^ {+ $$
Find the maximum of $ f (x) $,from where $ x $ be positive integers.

since
$$ f & # 39; (x) = ln { left ( dfrac {n-x} {n + x + 1} right)} + dfrac {2n + 1} {n + x + 1} $$

I suspect when $ x = lfloor dfrac {n + 1} {2} rfloor $ is maximum