Find all $f:R^+ \rightarrow R^+$ such that 
$f(f(x) + y)f(x) = f(xy + 1)\ \ \forall x, y \in R^+$

@below: [url]https://artofproblemsolving.com/community/c6h2254883_2020_imoc_problems[/url]

[quote]Feel free to start individual threads for the problems as usual[/quote]
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    - We conclude that \( f(x) = 1 \Leftrightarrow x = 1 \).
    - Using \( P(x, \frac{x-1}{x}) \) for \( x > 1 \), we conclude \( f(x) = \frac{1}{x} \) for \( x > 1 \).

The final answer is \( \boxed{ f(x) = \frac{1}{x} } \) for \( x > 1 \) and \( f(x) = 1 \) for \( x = 1 \).