Find all functions $f: \mathbb{Q^+} \rightarrow \mathbb{Q}$ satisfying $f(x)+f(y)= \left(f(x+y)+\frac{1}{x+y} \right) (1-xy+f(xy))$ for all $x, y \in \mathbb{Q^+}$.
Problem
Source: Turkey TST 2022 P2 Day 1
Tags: functional equation, Functional equation in R, algebra, algebra proposed
13.03.2022 14:20
07.04.2022 19:37
This problem is not completely bashy actually and Q+ can be changed to R+. Here is another solution for $f(1)=0$ (sorry for not writing completely, I do not remember the equations and I am lazy to rediscover them): 1_$P(x,1)$ gives the relation between $f(x)$ and $f(x+1)$. 2_ $P(x,1-x)$ gives the relation between $x,1-x$ and $x(1-x)$. 3_ $P(x,1/x)$ gives the realtion between $x$ and $1/x$. Now $P(1/x,1/(1-x))$ gives (by using 2-3) *$f(x) \in \{0,x-1/x\}$ for all $x \ge 4$ (because we found that $f(1/(x(1-x))=1/(x(1-x)) -x(1-x)$). If $f(a)=0$ for $a \neq 1$ then $P(a,a)$ contradicts with *. Thus $f(x)=x-1/x$ for all $x \ge 4$. Also by 1, $f(x)=x-1/x$ for any $x$. REMARK: $1/x+1/(1-x)=1/x \cdot 1/(1-x)$ is the key thing in this solution.