Found problems: 98
2024 Azerbaijan IMO TST, 3
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 $x, y \in \mathbb R_{>0}$.
2021 Balkan MO Shortlist, A2
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 Philippine TST, 3
Determine all ordered triples $(a, b, c)$ of real numbers such that whenever a function $f : \mathbb{R} \to \mathbb{R}$ satisfies $$|f(x) - f(y)| \le a(x - y)^2 + b(x - y) + c$$ for all real numbers $x$ and $y$, then $f$ must be a constant function.
2021 Polish MO Finals, 2
Let $n$ be an integer. For pair of integers $0 \leq i,$ $j\leq n$ there exist real number $f(i,j)$ such that:
1) $ f(i,i)=0$ for all integers $0\leq i \leq n$
2) $0\leq f(i,l) \leq 2\max \{ f(i,j), f(j,k), f(k,l) \}$ for all integers $i$, $j$, $k$, $l$ satisfying $0\leq i\leq j\leq k\leq l\leq n$.
Prove that $$f(0,n) \leq 2\sum_{k=1}^{n}f(k-1,k)$$
2008 Ukraine Team Selection Test, 8
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]
2017 Miklós Schweitzer, 6
Let $I$ and $J$ be intervals. Let $\varphi,\psi:I\to\mathbb{R}$ be strictly increasing continuous functions and let $\Phi,\Psi:J\to\mathbb{R}$ be continuous functions. Suppose that $\varphi(x)+\psi(x)=x$ and $\Phi(u)+\Psi(u)=u$ holds for all $x\in I$ and $u\in J$. Show that if $f:I\to J$ is a continuous solution of the functional inequality
$$f\big(\varphi(x)+\psi(y)\big)\le \Phi\big(f(x)\big)+\Psi\big(f(y)\big)\qquad (x,y\in I),$$then $\Phi\circ f\circ \varphi^{-1}$ and $\Psi\circ f\circ \psi^{-1}$ are convex functions.
2009 Switzerland - Final Round, 9
Find all injective functions $f : N\to N$ such that holds for all natural numbers $n$:
$$f(f(n)) \le \frac{f(n) + n}{2}$$
2009 IMO Shortlist, 5
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]
2018 Dutch IMO TST, 2
Find all functions $f : R \to R$ such that $f(x^2)-f(y^2) \le (f(x)+y) (x-f(y))$ for all $x, y \in R$.
2023 ISL, A4
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 $x, y \in \mathbb R_{>0}$.
2018 Dutch IMO TST, 2
Find all functions $f : R \to R$ such that $f(x^2)-f(y^2) \le (f(x)+y) (x-f(y))$ for all $x, y \in R$.
2022 IMO, 2
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 $$xf(y)+yf(x) \leq 2$$
2001 Switzerland Team Selection Test, 6
A function $f : [0,1] \to R$ has the following properties:
(a) $f(x) \ge 0$ for $0 < x < 1$,
(b) $f(1) = 1$,
(c) $f(x+y) \ge f(x)+ f(y) $ whenever $x,y,x+y \in [0,1]$.
Prove that $f(x) \le 2x$ for all $x \in [0,1]$.
2021 Peru EGMO TST, 6
Find all functions $f : R \to R$ such that $$f(x + y) \ge xf(x) + yf(y)$$, for all $x, y \in R$ .
2024 Romania Team Selection Tests, P4
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 $x, y \in \mathbb R_{>0}$.
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]
2011 IMO, 3
Let $f : \mathbb R \to \mathbb R$ be a real-valued function defined on the set of real numbers that satisfies
\[f(x + y) \leq yf(x) + f(f(x))\]
for all real numbers $x$ and $y$. Prove that $f(x) = 0$ for all $x \leq 0$.
[i]Proposed by Igor Voronovich, Belarus[/i]
2022 Abelkonkurransen Finale, 4a
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 $x$.
2019 Dutch IMO TST, 3
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 $f(x) + f(y) \ge x + y$.
2009 Brazil Team Selection Test, 3
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]
2008 Brazil Team Selection Test, 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]
2009 Ukraine Team Selection Test, 9
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
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 $x, y \in \mathbb R_{>0}$.
2021-IMOC qualification, A3
Find all injective function $f: N \to N$ satisfying that for all positive integers $m,n$, we have: $f(n(f(m)) \le nm$
1997 Slovenia Team Selection Test, 2
Find all polynomials $p$ with real coefficients such that for all real $x$ , $xp(x)p(1-x)+x^3 +100 \ge 0$.