Olympiad problems

2019 Kosovo Team Selection Test, 2

Tags: functional equation , algebra , function , bounded , Reals , TST , Kosovo

Determine all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that for every $x,y \in \mathbb{R}$ $$f(x^{4}-y^{4})+4f(xy)^{2}=f(x^{4}+y^{4})$$

2020 Hong Kong TST, 4

Tags: algebra , TST , 2020 , hong Kong TST , fuctional equation , fe , Reals

Find all real-valued functions $f$ defined on the set of real numbers such that $$f(f(x)+y)+f(x+f(y))=2f(xf(y))$$ for any real numbers $x$ and $y$.

2022 Middle European Mathematical Olympiad, 1

Tags: functional equation , 2022 , P1 , Reals , Surjective , memo , MEMO 2022

Find all functions $f: \mathbb R \to \mathbb R$ such that $$f(x+f(x+y))=x+f(f(x)+y)$$ holds for all real numbers $x$ and $y$.

2025 JBMO TST - Turkey, 8

Tags: combinatorics , Reals , arcs , Junior

Pairwise distinct points $P_1,\dots,P_{1024}$, which lie on a circle, are marked by distinct reals $a_1,\dots,a_{1024}$. Let $P_i$ be $Q-$good for a $Q$ on the circle different than $P_1,\dots,P_{1024}$, if and only if $a_i$ is the greatest number on at least one of the two arcs $P_iQ$. Let the score of $Q$ be the number of $Q-$good points on the circle. Determine the greatest $k$ such that regardless of the values of $a_1,\dots,a_{1024}$, there exists a point $Q$ with score at least $k$.

2024 Pan-American Girls’ Mathematical Olympiad, 5

Tags: PAGMO , algebra , functional equation , Reals , constant , PAGMO 2024 , 2024

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(f(x+y) - f(x)) + f(x)f(y) = f(x^2) - f(x+y),$ for all real numbers $x, y$.

2009 Middle European Mathematical Olympiad, 1

Tags: function , algebra proposed , algebra , functional equation , Reals , 2009 , memo

Find all functions $ f: \mathbb{R} \to \mathbb{R}$, such that \[ f(xf(y)) \plus{} f(f(x) \plus{} f(y)) \equal{} yf(x) \plus{} f(x \plus{} f(y))\] holds for all $ x$, $ y \in \mathbb{R}$, where $ \mathbb{R}$ denotes the set of real numbers.

2025 Kosovo National Mathematical Olympiad`, P2

Tags: functional equation , Reals , Kosovo , algebra

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ with the property that for every real numbers $x$ and $y$ it holds that $$f(x+yf(x+y))=f(x)+f(xy)+y^2.$$

2024 Mexico National Olympiad, 5

Tags: 2024 , Mexico , algebra , Reals

Let $A$ and $B$ infinite sets of positive real numbers such that: 1. For any pair of elements $u \ge v$ in $A$, it follows that $u+v$ is an element of $B$. 2. For any pair of elements $s>t$ in $B$, it follows that $s-t$ is an element of $A$. Prove that $A=B$ or there exists a real number $r$ such that $B=\{2r, 3r, 4r, 5r, \dots\}$.

2014 Contests, 1

Tags: function , algebra , functional equation , fe , memo , memo 2014 , Reals

Determine all functions $f:\mathbb{R} \to \mathbb{R}$ such that \[ xf(y) + f(xf(y)) - xf(f(y)) - f(xy) = 2x + f(y) - f(x+y)\] holds for all $x,y \in \mathbb{R}$.

2014 Middle European Mathematical Olympiad, 1

Tags: function , algebra , functional equation , fe , memo , memo 2014 , Reals

Determine all functions $f:\mathbb{R} \to \mathbb{R}$ such that \[ xf(y) + f(xf(y)) - xf(f(y)) - f(xy) = 2x + f(y) - f(x+y)\] holds for all $x,y \in \mathbb{R}$.