Mathematics Olympiad - Functional Equations

Problem 1

Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that for all \( x, y \in \mathbb{R} \), \[ f(x+y) = f(x) + f(y) + xy. \]

Problem 2

Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that for all \( x, y \in \mathbb{R} \), \[ f(f(x) + y) = f(f(x) - y) + 4f(x)y. \]

Problem 3

Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that for all \( x, y \in \mathbb{R} \), \[ f(x^2 + f(y)) = y + f(x)^2. \]

Problem 4

Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that for all \( x, y \in \mathbb{R} \), \[ f(x+y) = f(x)f(y). \]

Problem 5

Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that for all \( x, y \in \mathbb{R} \), \[ f(f(x) + y) = f(x + f(y)). \]

Problem 6

Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that for all \( x, y \in \mathbb{R} \), \[ f(x^2 + y) = f(x)^2 + f(y). \]

Problem 7

Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that for all \( x, y \in \mathbb{R} \), \[ f(x + f(y)) = y + f(x). \]