Found problems: 98
2019 Saudi Arabia IMO TST, 1
Find all functions $f : Z^+ \to Z^+$ such that $n^3 - n^2 \le f(n) \cdot (f(f(n)))^2 \le n^3 + n^2$ for every $n$ in positive integers
2010 Germany Team Selection Test, 3
Let $f$ be any function that maps the set of real numbers into the set of real numbers. Prove that there exist real numbers $x$ and $y$ such that \[f\left(x-f(y)\right)>yf(x)+x\]
[i]Proposed by Igor Voronovich, Belarus[/i]
2013 Stars Of Mathematics, 1
Let $\mathcal{F}$ be the family of bijective increasing functions $f\colon [0,1] \to [0,1]$, and let $a \in (0,1)$. Determine the best constants $m_a$ and $M_a$, such that for all $f \in \mathcal{F}$ we have
\[m_a \leq f(a) + f^{-1}(a) \leq M_a.\]
[i](Dan Schwarz)[/i]
2008 Thailand Mathematical Olympiad, 6
Let $f : R \to R$ be a function satisfying the inequality $|f(x + y) -f(x) - f(y)| < 1$ for all reals $x, y$.
Show that $\left| f\left( \frac{x}{2008 }\right) - \frac{f(x)}{2008} \right| < 1$ for all real numbers $x$.
2022 IFYM, Sozopol, 6
For the function $f : Z^2_{\ge0} \to Z_{\ge 0}$ it is known that
$$f(0, j) = f(i, 0) = 1, \,\,\,\,\, \forall i, j \in N_0$$
$$f(i, j) = if (i, j - 1) + jf(i - 1, j),\,\,\,\,\, \forall i, j \in N$$
Prove that for every natural number $n$ the following inequality holds:
$$\sum_{0\le i+j\le n+1} f(i, j) \le 2 \left(\sum^n_{k=0}\frac{1}{k!}\right)\left(\sum^n_{p=1}p!\right)+ 3$$
2000 Mongolian Mathematical Olympiad, Problem 4
Suppose that a function $f:\mathbb R\to\mathbb R$ satisfies the following conditions:
(i) $\left|f(a)-f(b)\right|\le|a-b|$ for all $a,b\in\mathbb R$;
(ii) $f(f(f(0)))=0$.
Prove that $f(0)=0$.
2022 Romania National Olympiad, P3
Determine all functions $f:\mathbb{R}\to\mathbb{R}$ which are differentiable in $0$ and satisfy the following inequality for all real numbers $x,y$ \[f(x+y)+f(xy)\geq f(x)+f(y).\][i]Dan Ștefan Marinescu and Mihai Piticari[/i]
2025 Iran MO (2nd Round), 5
Find all functions $f:\mathbb{R}^+ \to \mathbb{R}$ such that for all $x,y,z>0$
$$
3(x^3+y^3+z^3)\geq f(x+y+z)\cdot f(xy+yz+xz) \geq (x+y+z)(xy+yz+xz).
$$
2019 Regional Olympiad of Mexico Center Zone, 2
Find all functions $ f: \mathbb {R} \rightarrow \mathbb {R} $ such that $ f (x + y) \le f (xy) $ for every pair of real $ x $, $ y$.
2019 District Olympiad, 1
Find the functions $f: \mathbb{R} \to (0, \infty)$ which satisfy $$2^{-x-y} \le \frac{f(x)f(y)}{(x^2+1)(y^2+1)} \le \frac{f(x+y)}{(x+y)^2+1},$$ for all $x,y \in \mathbb{R}.$
2013 Benelux, 2
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that
\[f(x + y) + y \le f(f(f(x)))\]
holds for all $x, y \in \mathbb{R}$.
2008 Germany Team Selection Test, 1
Consider those functions $ f: \mathbb{N} \mapsto \mathbb{N}$ which satisfy the condition
\[ f(m \plus{} n) \geq f(m) \plus{} f(f(n)) \minus{} 1
\]
for all $ m,n \in \mathbb{N}.$ Find all possible values of $ f(2007).$
[i]Author: Nikolai Nikolov, Bulgaria[/i]
2007 IMO Shortlist, 2
Consider those functions $ f: \mathbb{N} \mapsto \mathbb{N}$ which satisfy the condition
\[ f(m \plus{} n) \geq f(m) \plus{} f(f(n)) \minus{} 1
\]
for all $ m,n \in \mathbb{N}.$ Find all possible values of $ f(2007).$
[i]Author: Nikolai Nikolov, Bulgaria[/i]
2023 Dutch BxMO TST, 2
Find all functions $f : \mathbb R \to \mathbb R$ for which
\[f(a - b) f(c - d) + f(a - d) f(b - c) \leq (a - c) f(b - d),\]
for all real numbers $a, b, c$ and $d$. Note that there is only one occurrence of $f$ on the right hand side!
2021 Indonesia TST, A
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that
\[f(x + y) + y \le f(f(f(x)))\]
holds for all $x, y \in \mathbb{R}$.
2015 Costa Rica - Final Round, 5
Let $f: N^+ \to N^+$ be a function that satisfies that
$$kf(n) \le f (kn) \le kf(n)+ k- 1, \,\, \forall k,n \in N^+$$
Prove that
$$f(a) + f(b) \le f (a + b) \le f(a) + f(b) + 1, \,\, \forall a, b \in N^+$$
2021 Indonesia TST, A
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that
\[f(x + y) + y \le f(f(f(x)))\]
holds for all $x, y \in \mathbb{R}$.
2009 Germany Team Selection Test, 2
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]
2022 German National Olympiad, 6
Consider functions $f$ satisfying the following four conditions:
(1) $f$ is real-valued and defined for all real numbers.
(2) For any two real numbers $x$ and $y$ we have $f(xy)=f(x)f(y)$.
(3) For any two real numbers $x$ and $y$ we have $f(x+y) \le 2(f(x)+f(y))$.
(4) We have $f(2)=4$.
Prove that:
a) There is a function $f$ with $f(3)=9$ satisfying the four conditions.
b) For any function $f$ satisfying the four conditions, we have $f(3) \le 9$.
2025 Ukraine National Mathematical Olympiad, 11.3
Find all functions \(f: \mathbb{R} \rightarrow \mathbb{R}\) such that for any real numbers \(x\) and \(y\), the following inequality holds:
\[
f\left(x^2+2y f(x)\right) + (f(y))^2 \leq f\left((x+y)^2\right)
\]
[i]Proposed by Anton Trygub[/i]
2017-IMOC, A4
Show that for all non-constant functions $f:\mathbb R\to\mathbb R$, there are two real numbers $x,y$ such that
$$f(x+f(y))>xf(y)+x.$$
2017 Saudi Arabia BMO TST, 2
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) $(x + y)[y f (x) + x f (y)] \ge x y f (x + y)$
2024 Dutch IMO TST, 2
Find all functions $f:\mathbb{R}_{\ge 0} \to \mathbb{R}$ with
\[2x^3zf(z)+yf(y) \ge 3yz^2f(x)\]
for all $x,y,z \in \mathbb{R}_{\ge 0}$.
1999 Korea - Final Round, 2
Suppose $f(x)$ is a function satisfying $\left | f(m+n)-f(m) \right | \leq \frac{n}{m}$ for all positive integers $m$,$n$. Show that for all positive integers $k$:
\[\sum_{i=1}^{k}\left |f(2^k)-f(2^i) \right |\leq \frac{k(k-1)}{2}\].
2014 All-Russian Olympiad, 2
Given a function $f\colon \mathbb{R}\rightarrow \mathbb{R} $ with $f(x)^2\le f(y)$ for all $x,y\in\mathbb{R} $, $x>y$, prove that $f(x)\in [0,1] $ for all $x\in \mathbb{R}$.