Math Gold Medalist

Lor

2023 USAMO 

Problem 2

Let $\mathbb{R}^{+}$ be the set of positive real numbers. Find all functions $f:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ such that, for all $x, y \in \mathbb{R}^{+}$,\[f(xy + f(x)) = xf(y) + 2\]


Important Ideas in Functional Equations

 

x=y=1

x=1

y=1

f(1)=c

f(x+c)=f(x)+2

f(x+f(x))=xc+2

If x+c=xy then y=1+c/x

f(1+c/x)=c+2/x

c=2

2019 IMO Problem 1

   Solution