This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 98

2024 Switzerland Team Selection Test, 8
Tags: algebra , functional inequality
Let $\mathbb R_{>0}$ be the set of positive real numbers. Determine all functions $f \colon \mathbb R_{>0} \to \mathbb R_{>0}$ such that \[x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)\] for every $x, y \in \mathbb R_{>0}$.

2019 Thailand Mathematical Olympiad, 6
Tags: algebra , functional inequality
Determine all function $f:\mathbb{R}\to\mathbb{R}$ such that $xf(y)+yf(x)\leqslant xy$ for all $x,y\in\mathbb{R}$.

2021 Middle European Mathematical Olympiad, 1
Tags: functional equation , functional inequality , algebra
Determine all functions $f: \mathbb{R} \to \mathbb{R}$ such that the inequality \[ f(x^2)-f(y^2) \le (f(x)+y)(x-f(y)) \] holds for all real numbers $x$ and $y$.

2024 Middle European Mathematical Olympiad, 1
Tags: algebra , monotone , construction , functional inequality , function , sequence
Let $\mathbb{N}_0$ denote the set of non-negative integers. Determine all non-negative integers $k$ for which there exists a function $f: \mathbb{N}_0 \to \mathbb{N}_0$ such that $f(2024) = k$ and $f(f(n)) \leq f(n+1) - f(n)$ for all non-negative integers $n$.

2022 IMO Shortlist, A3
Tags: functional inequality
Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that for each $x \in \mathbb{R}^+$, there is exactly one $y \in \mathbb{R}^+$ satisfying $$xf(y)+yf(x) \leq 2$$

2019 Dutch IMO TST, 3
Tags: functional inequality , find all functions , algebra
Find all functions $f : Z \to Z$ satisfying the following two conditions: (i) for all integers $x$ we have $f(f(x)) = x$, (ii) for all integers $x$ and y such that $x + y$ is odd, we have $f(x) + f(y) \ge x + y$.

2021 Balkan MO Shortlist, A2
Tags: shortlist , algebra , functional inequality
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(x^2 + y) \ge (\frac{1}{x} + 1)f(y)$$ holds for all $x \in \mathbb{R} \setminus \{0\}$ and all $y \in \mathbb{R}$.

2019 Swedish Mathematical Competition, 5
Tags: algebra , inequalities , functional , functional inequality , functional equation
Let $f$ be a function that is defined for all positive integers and whose values are positive integers. For $f$ it also holds that $f (n + 1)> f (n)$ and $f (f (n)) = 3n$, for each positive integer $n$. Calculate $f (2019)$.

2009 Brazil Team Selection Test, 3
Tags: function , algebra , functional inequality
Let $ S\subseteq\mathbb{R}$ be a set of real numbers. We say that a pair $ (f, g)$ of functions from $ S$ into $ S$ is a [i]Spanish Couple[/i] on $ S$, if they satisfy the following conditions: (i) Both functions are strictly increasing, i.e. $ f(x) < f(y)$ and $ g(x) < g(y)$ for all $ x$, $ y\in S$ with $ x < y$; (ii) The inequality $ f\left(g\left(g\left(x\right)\right)\right) < g\left(f\left(x\right)\right)$ holds for all $ x\in S$. Decide whether there exists a Spanish Couple [list][*] on the set $ S \equal{} \mathbb{N}$ of positive integers; [*] on the set $ S \equal{} \{a \minus{} \frac {1}{b}: a, b\in\mathbb{N}\}$[/list] [i]Proposed by Hans Zantema, Netherlands[/i]

2009 Germany Team Selection Test, 2
Tags: function , algebra , functional inequality
Let $ S\subseteq\mathbb{R}$ be a set of real numbers. We say that a pair $ (f, g)$ of functions from $ S$ into $ S$ is a [i]Spanish Couple[/i] on $ S$, if they satisfy the following conditions: (i) Both functions are strictly increasing, i.e. $ f(x) < f(y)$ and $ g(x) < g(y)$ for all $ x$, $ y\in S$ with $ x < y$; (ii) The inequality $ f\left(g\left(g\left(x\right)\right)\right) < g\left(f\left(x\right)\right)$ holds for all $ x\in S$. Decide whether there exists a Spanish Couple [list][*] on the set $ S \equal{} \mathbb{N}$ of positive integers; [*] on the set $ S \equal{} \{a \minus{} \frac {1}{b}: a, b\in\mathbb{N}\}$[/list] [i]Proposed by Hans Zantema, Netherlands[/i]

2024 Indonesia TST, 2
Tags: algebra , functional inequality
Let $\mathbb R_{>0}$ be the set of positive real numbers. Determine all functions $f \colon \mathbb R_{>0} \to \mathbb R_{>0}$ such that \[x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)\] for every $x, y \in \mathbb R_{>0}$.

2009 Peru IMO TST, 6
Tags: function , algebra , functional inequality
Let $ S\subseteq\mathbb{R}$ be a set of real numbers. We say that a pair $ (f, g)$ of functions from $ S$ into $ S$ is a [i]Spanish Couple[/i] on $ S$, if they satisfy the following conditions: (i) Both functions are strictly increasing, i.e. $ f(x) < f(y)$ and $ g(x) < g(y)$ for all $ x$, $ y\in S$ with $ x < y$; (ii) The inequality $ f\left(g\left(g\left(x\right)\right)\right) < g\left(f\left(x\right)\right)$ holds for all $ x\in S$. Decide whether there exists a Spanish Couple [list][*] on the set $ S \equal{} \mathbb{N}$ of positive integers; [*] on the set $ S \equal{} \{a \minus{} \frac {1}{b}: a, b\in\mathbb{N}\}$[/list] [i]Proposed by Hans Zantema, Netherlands[/i]

2017 Saudi Arabia BMO TST, 2
Tags: function , functional inequality , functional , algebra
Let $R^+$ be the set of positive real numbers. Find all function $f : R^+ \to R$ such that, for all positive real number $x$ and $y$, the following conditions are satisfied: i) $2f (x) + 2f (y) \le f (x + y)$ ii) $(x + y)[y f (x) + x f (y)] \ge x y f (x + y)$

2022 Abelkonkurransen Finale, 4a
Tags: functional inequality , algebra , functional equation
Find all functions $f:\mathbb R^+ \to \mathbb R^+$ satisfying \begin{align*} f\left(\frac{1}{x}\right) \geq 1 - \frac{\sqrt{f(x)f\left(\frac{1}{x}\right)}}{x} \geq x^2 f(x), \end{align*} for all positive real numbers $x$.

1994 Austrian-Polish Competition, 1
Tags: functional , functional inequality , functional equation , algebra
A function $f: R \to R$ satisfies the conditions: $f (x + 19) \le f (x) + 19$ and $f (x + 94) \ge f (x) + 94$ for all $x \in R$. Prove that $f (x + 1) = f (x) + 1$ for all $x \in R$.

2025 Iran MO (2nd Round), 5
Tags: inequalities , function , functional inequality
Find all functions $f:\mathbb{R}^+ \to \mathbb{R}$ such that for all $x,y,z>0$ $$ 3(x^3+y^3+z^3)\geq f(x+y+z)\cdot f(xy+yz+xz) \geq (x+y+z)(xy+yz+xz). $$

2009 Ukraine Team Selection Test, 9
Tags: function , algebra , functional inequality
Let $ S\subseteq\mathbb{R}$ be a set of real numbers. We say that a pair $ (f, g)$ of functions from $ S$ into $ S$ is a [i]Spanish Couple[/i] on $ S$, if they satisfy the following conditions: (i) Both functions are strictly increasing, i.e. $ f(x) < f(y)$ and $ g(x) < g(y)$ for all $ x$, $ y\in S$ with $ x < y$; (ii) The inequality $ f\left(g\left(g\left(x\right)\right)\right) < g\left(f\left(x\right)\right)$ holds for all $ x\in S$. Decide whether there exists a Spanish Couple [list][*] on the set $ S \equal{} \mathbb{N}$ of positive integers; [*] on the set $ S \equal{} \{a \minus{} \frac {1}{b}: a, b\in\mathbb{N}\}$[/list] [i]Proposed by Hans Zantema, Netherlands[/i]

2019 Regional Olympiad of Mexico Center Zone, 2
Tags: inequalities , functional inequality , functional , algebra
Find all functions $ f: \mathbb {R} \rightarrow \mathbb {R} $ such that $ f (x + y) \le f (xy) $ for every pair of real $ x $, $ y$.

2021 Peru EGMO TST, 6
Tags: algebra , function , functional inequality , functional equation
Find all functions $f : R \to R$ such that $$f(x + y) \ge xf(x) + yf(y)$$, for all $x, y \in R$ .

2008 IMO Shortlist, 3
Tags: function , algebra , functional inequality
Let $ S\subseteq\mathbb{R}$ be a set of real numbers. We say that a pair $ (f, g)$ of functions from $ S$ into $ S$ is a [i]Spanish Couple[/i] on $ S$, if they satisfy the following conditions: (i) Both functions are strictly increasing, i.e. $ f(x) < f(y)$ and $ g(x) < g(y)$ for all $ x$, $ y\in S$ with $ x < y$; (ii) The inequality $ f\left(g\left(g\left(x\right)\right)\right) < g\left(f\left(x\right)\right)$ holds for all $ x\in S$. Decide whether there exists a Spanish Couple [list][*] on the set $ S \equal{} \mathbb{N}$ of positive integers; [*] on the set $ S \equal{} \{a \minus{} \frac {1}{b}: a, b\in\mathbb{N}\}$[/list] [i]Proposed by Hans Zantema, Netherlands[/i]

1979 Kurschak Competition, 2
Tags: algebra , functional , functional inequality , function
$f$ is a real-valued function defined on the reals such that $f(x) \le x$ and $f(x + y) \le f(x) + f(y)$ for all $x, y$. Prove that $f(x) = x$ for all $x$.

2009 Germany Team Selection Test, 2
Tags: function , algebra , functional inequality
Let $ S\subseteq\mathbb{R}$ be a set of real numbers. We say that a pair $ (f, g)$ of functions from $ S$ into $ S$ is a [i]Spanish Couple[/i] on $ S$, if they satisfy the following conditions: (i) Both functions are strictly increasing, i.e. $ f(x) < f(y)$ and $ g(x) < g(y)$ for all $ x$, $ y\in S$ with $ x < y$; (ii) The inequality $ f\left(g\left(g\left(x\right)\right)\right) < g\left(f\left(x\right)\right)$ holds for all $ x\in S$. Decide whether there exists a Spanish Couple [list][*] on the set $ S \equal{} \mathbb{N}$ of positive integers; [*] on the set $ S \equal{} \{a \minus{} \frac {1}{b}: a, b\in\mathbb{N}\}$[/list] [i]Proposed by Hans Zantema, Netherlands[/i]

1977 IMO Longlists, 2
Tags: algebra , functional inequality , functional equation
Find all functions $f : \mathbb{N}\rightarrow \mathbb{N}$ satisfying following condition: \[f(n+1)>f(f(n)), \quad \forall n \in \mathbb{N}.\]

1994 Czech And Slovak Olympiad IIIA, 1
Tags: functional inequality , function , algebra
Let $f : N \to N$ be a function which satisfies $f(x)+ f(x+2) \le 2 f(x+1)$ for any $x \in N$. Prove that there exists a line in the coordinate plane containing infinitely many points of the form $(n, f(n)), n \in N$.

2023 ISL, A4
Tags: algebra , functional inequality
Let $\mathbb R_{>0}$ be the set of positive real numbers. Determine all functions $f \colon \mathbb R_{>0} \to \mathbb R_{>0}$ such that \[x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)\] for every $x, y \in \mathbb R_{>0}$.