Found problems: 98
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 $x, y \in \mathbb R_{>0}$.
2015 Taiwan TST Round 3, 2
Consider all polynomials $P(x)$ with real coefficients that have the following property: for any two real numbers $x$ and $y$ one has \[|y^2-P(x)|\le 2|x|\quad\text{if and only if}\quad |x^2-P(y)|\le 2|y|.\] Determine all possible values of $P(0)$.
[i]Proposed by Belgium[/i]
2008 IMO Shortlist, 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]
1977 IMO, 3
Let $\mathbb{N}$ be the set of positive integers. Let $f$ be a function defined on $\mathbb{N}$, which satisfies the inequality $f(n + 1) > f(f(n))$ for all $n \in \mathbb{N}$. Prove that for any $n$ we have $f(n) = n.$
2010 Saudi Arabia BMO TST, 3
Let $a > 0$ be a real number and let $f : R \to R$ be a function satisfying $$f(x_1) + f(x_2) \ge a f(x_1 + x_2), \forall x_1 ,x_2 \in R.$$ Prove that $$f(x_1) + f(x_2) +(x_3) \ge \frac{3a^2}{a+2} f(x_1+ x_2 + x_3), \forall x_1 ,x_2,x_3 \in R$$.
2017 IFYM, Sozopol, 6
Find all functions $f: \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$, for which
$f(k+1)>f(f(k)) \quad \forall k \geq 1$.
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 $$f(0,n) \leq 2\sum_{k=1}^{n}f(k-1,k)$$
1972 Dutch Mathematical Olympiad, 2
Prove that there exists exactly one function $ƒ$ which is defined for all $x \in R$, and for which holds:
$\bullet$ $x \le y \Rightarrow f(x) \le f(y)$, for all $x, y \in R$, and
$\bullet$ $f(f(x)) = x$, for all $x \in R$.
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 $f(0)=0$.
2018 Costa Rica - Final Round, 4
Determine if there exists a function f: $N^*\to N^*$ that satisfies that for all $n \in N^*$, $$10^{f (n)} <10n + 1 <10^{f (n) +1}.$$ Justify your answer.
Note: $N^*$ denotes the set of positive integers.
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 $y \in \mathbb{R}$.
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]
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 $$xf(y)+yf(x) \leq 2$$
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]
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 $x, y \in \mathbb{R}$.
2010 Brazil Team Selection Test, 4
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]
1972 IMO, 2
$f$ and $g$ are real-valued functions defined on the real line. For all $x$ and $y, f(x+y)+f(x-y)=2f(x)g(y)$. $f$ is not identically zero and $|f(x)|\le1$ for all $x$. Prove that $|g(x)|\le1$ for all $x$.
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 $x$.
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 $x,y \in \mathbb{R}.$
2021-IMOC qualification, A3
Find all injective function $f: N \to N$ satisfying that for all positive integers $m,n$, we have: $f(n(f(m)) \le nm$
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]
1977 IMO Shortlist, 1
Find all functions $f : \mathbb{N}\rightarrow \mathbb{N}$ satisfying following condition:
\[f(n+1)>f(f(n)), \quad \forall n \in \mathbb{N}.\]
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]
2016 HMIC, 3
Denote by $\mathbb{N}$ the positive integers. Let $f:\mathbb{N} \rightarrow \mathbb{N}$ be a function such that, for any $w,x,y,z \in \mathbb{N}$, \[ f(f(f(z)))f(wxf(yf(z)))=z^{2}f(xf(y))f(w). \] Show that $f(n!) \ge n!$ for every positive integer $n$.
[i]Pakawut Jiradilok[/i]
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.