Found problems: 1513
2021 Sharygin Geometry Olympiad, 20
The mapping $f$ assigns a circle to every triangle in the plane so that the following conditions hold. (We consider all nondegenerate triangles and circles of nonzero radius.)
[b](a)[/b] Let $\sigma$ be any similarity in the plane and let $\sigma$ map triangle $\Delta_1$ onto triangle $\Delta_2$. Then $\sigma$ also maps circle $f(\Delta_1)$ onto circle $f(\Delta_2)$.
[b](b)[/b] Let $A,B,C$ and $D$ be any four points in general position. Then circles $f(ABC),f(BCD),f(CDA)$ and $f(DAB)$ have a common point.
Prove that for any triangle $\Delta$, the circle $f(\Delta)$ is the Euler circle of $\Delta$.
2023 China Team Selection Test, P22
Find all functions $f:\mathbb {Z}\to\mathbb Z$, satisfy that for any integer ${a}$, ${b}$, ${c}$,
$$2f(a^2+b^2+c^2)-2f(ab+bc+ca)=f(a-b)^2+f(b-c)^2+f(c-a)^2$$
2023 Israel TST, P3
Find all functions $f:\mathbb{Z}\to \mathbb{Z}_{>0}$ for which
\[f(x+f(y))^2+f(y+f(x))^2=f(f(x)+f(y))^2+1\]
holds for any $x,y\in \mathbb{Z}$.
2019 APMO, 1
Let $\mathbb{Z}^+$ be the set of positive integers. Determine all functions $f : \mathbb{Z}^+\to\mathbb{Z}^+$ such that $a^2+f(a)f(b)$ is divisible by $f(a)+b$ for all positive integers $a,b$.
2016 Postal Coaching, 2
Determine all functions $f : \mathbb R \to \mathbb R$ such that $$f(f(x)- f(y)) = f(f(x)) - 2x^2f(y) + f\left(y^2\right),$$ for all reals $x, y$.
2024 Belarusian National Olympiad, 10.3
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that for every $x,y \in \mathbb{R}$ the following equation holds:$$1+f(xy)=f(x+f(y))+(y-1)f(x-1)$$
[i]M. Zorka[/i]
1985 IMO Longlists, 33
A sequence of polynomials $P_m(x, y, z), m = 0, 1, 2, \cdots$, in $x, y$, and $z$ is defined by $P_0(x, y, z) = 1$ and by
\[P_m(x, y, z) = (x + z)(y + z)P_{m-1}(x, y, z + 1) - z^2P_{m-1}(x, y, z)\]
for $m > 0$. Prove that each $P_m(x, y, z)$ is symmetric, in other words, is unaltered by any permutation of $x, y, z.$
2025 Bulgarian Winter Tournament, 10.4
The function $f: \mathbb{Z}_{>0} \times \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is such that $f(a,b) + f(b,c) = f(ac, b^2) + 1$ for any positive integers $a,b,c$. Assume there exists a positive integer $n$ such that $f(n, m) \leq f(n, m + 1)$ for all positive integers $m$. Determine all possible values of $f(2025, 2025)$.
2008 Indonesia MO, 4
Find all function $ f: \mathbb{N}\rightarrow\mathbb{N}$ satisfy $ f(mn)\plus{}f(m\plus{}n)\equal{}f(m)f(n)\plus{}1$ for all natural number $ n$
2022 Saudi Arabia IMO TST, 3
Find all non-constant functions $f : Q^+ \to Q^+$ satisfying the equation $$f(ab + bc + ca) =f(a)f(b) +f(b)f(c)+f(c)f(a)$$
for all $a, b,c \in Q^+$ .
2022 Middle European Mathematical Olympiad, 7
Determine all functions $f : \mathbb {N} \rightarrow \mathbb {N}$ such that $f$ is increasing (not necessarily strictly) and the numbers $f(n)+n+1$ and $f(f(n))-f(n)$ are both perfect squares for every positive integer $n$.
1967 IMO Longlists, 50
The function $\varphi(x,y,z)$ defined for all triples $(x,y,z)$ of real numbers, is such that there are two functions $f$ and $g$ defined for all pairs of real numbers, such that
\[\varphi(x,y,z) = f(x+y,z) = g(x,y+z)\]
for all real numbers $x,y$ and $z.$ Show that there is a function $h$ of one real variable, such that
\[\varphi(x,y,z) = h(x+y+z)\]
for all real numbers $x,y$ and $z.$
2011 Saudi Arabia IMO TST, 3
Find all functions $f : R \to R$ such that $$2f(x) =f(x+y)+f(x+2y)$$, for all $x \in R$ and for all $y \ge 0$.
2024 Thailand Mathematical Olympiad, 3
Let $c$ be a positive real number. Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ that satisfy $$x^2f(xf(y))f(x)f(y)=c$$ for all positive reals $x$ and $y$.
2001 IMO Shortlist, 1
Let $ T$ denote the set of all ordered triples $ (p,q,r)$ of nonnegative integers. Find all functions $ f: T \rightarrow \mathbb{R}$ satisfying
\[ f(p,q,r) = \begin{cases} 0 & \text{if} \; pqr = 0, \\
1 + \frac{1}{6}(f(p + 1,q - 1,r) + f(p - 1,q + 1,r) & \\
+ f(p - 1,q,r + 1) + f(p + 1,q,r - 1) & \\
+ f(p,q + 1,r - 1) + f(p,q - 1,r + 1)) & \text{otherwise} \end{cases}
\]
for all nonnegative integers $ p$, $ q$, $ r$.
2010 Contests, 3
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(x+y) = \max(f(x),y) + \min(f(y),x)$.
[i]George Xing.[/i]
2005 India National Olympiad, 6
Find all functions $f : \mathbb{R} \longrightarrow \mathbb{R}$ such that \[ f(x^2 + yf(z)) = xf(x) + zf(y) , \] for all $x, y, z \in \mathbb{R}$.
1990 Canada National Olympiad, 5
The function $f : \mathbb N \to \mathbb R$ satisfies $f(1) = 1, f(2) = 2$ and \[f (n+2) = f(n+2 - f(n+1) ) + f(n+1 - f(n) ).\] Show that $0 \leq f(n+1) - f(n) \leq 1$. Find all $n$ for which $f(n) = 1025$.
2010 Thailand Mathematical Olympiad, 6
Let $f : R \to R$ be a function satisfying the functional equation $f(3x + y) + f(3x-y) = f(x + y) + f(x - y) + 16f(x)$ for all reals $x, y$. Show that $f$ is even, that is, $f(-x) = f(x)$ for all reals $x$
2012 India National Olympiad, 6
Let $f : \mathbb{Z} \to \mathbb{Z}$ be a function satisfying $f(0) \ne 0$, $f(1) = 0$ and
$(i) f(xy) + f(x)f(y) = f(x) + f(y)$
$(ii)\left(f(x-y) - f(0)\right ) f(x)f(y) = 0 $
for all $x,y \in \mathbb{Z}$, simultaneously.
$(a)$ Find the set of all possible values of the function $f$.
$(b)$ If $f(10) \ne 0$ and $f(2) = 0$, find the set of all integers $n$ such that $f(n) \ne 0$.
2011 Mathcenter Contest + Longlist, 7 sl9
Find the function $\displaystyle{f : \mathbb{R}-\left\{ 0\,\right\} \rightarrow \mathbb{R} }$ such that $$f(x)+f(1-\frac{1}{x}) = \frac{1}{x},\,\,\, \forall x \in \mathbb{R}- \{ 0, 1\,\}$$
[i](-InnoXenT-)[/i]
2021 Serbia National Math Olympiad, 5
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for every $x,y\in\mathbb{R}$ the following equality holds: $$f(xf(y)+x^2+y)=f(x)f(y)+xf(x)+f(y).$$
2011 Peru IMO TST, 1
Let $\Bbb{Z}^+$ denote the set of positive integers. Find all functions $f:\Bbb{Z}^+\to \Bbb{Z}^+$ that satisfy the following condition: for each positive integer $n,$ there exists a positive integer $k$ such that $$\sum_{i=1}^k f_i(n)=kn,$$ where $f_1(n)=f(n)$ and $f_{i+1}(n)=f(f_i(n)),$ for $i\geq 1. $
2024 Alborz Mathematical Olympiad, P2
Let $\mathbb{Z}$ be the set of integers. Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that for all integers $a$ and $b$, we have: $$f(a^2+ab)+f(b^2+ab)=(a+b)f(a+b).$$
Proposed by Heidar Shushtari
2011 QEDMO 10th, 9
Let $X = Q-\{-1,0,1\}$. We consider the function $f: X\to X$ given by $f (x) = x -\frac{1}{x} .$ Is there an $a \in X$ such that for every natural number n there is a $y \in X$ with $f (f (...( f (y)) ...)) = a$ where $f$ occurs exactly $n$ times on the left side?