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: 1513

2024 Vietnam Team Selection Test, 1
Tags: functional equation , polynomial , function , algebra
Let $P(x) \in \mathbb{R}[x]$ be a monic, non-constant polynomial. Determine all continuous functions $f: \mathbb{R} \to \mathbb{R}$ such that $$f(f(P(x))+y+2023f(y))=P(x)+2024f(y),$$ for all reals $x,y$.

2016 Belarus Team Selection Test, 1
Tags: functional equation , algebra
a) Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that\[f(x-f(y))=f(f(x))-f(y)-1\]holds for all $x,y\in\mathbb{Z}$. (It is [url=https://artofproblemsolving.com/community/c6h1268817p6621849]2015 IMO Shortlist A2 [/url]) b) The same question for if \[f(x-f(y))=f(f(x))-f(y)-2\] for all integers $x,y$

2020 GQMO, 4
Tags: algebra , functional equation
For all real numbers $x$, we denote by $\lfloor x \rfloor$ the largest integer that does not exceed $x$. Find all functions $f$ that are defined on the set of all real numbers, take real values, and satisfy the equality \[f(x + y) = (-1)^{\lfloor y \rfloor} f(x) + (-1)^{\lfloor x \rfloor} f(y)\] for all real numbers $x$ and $y$. [i]Navneel Singhal, India[/i]

2024 Kyiv City MO Round 1, Problem 5
Tags: functional equation , number theory
Find all functions $f : \mathbb{N} \to \mathbb{N}$ such that for any positive integers $m, n$ the number $$(f(m))^2+ 2mf(n) + f(n^2)$$ is the square of an integer. [i]Proposed by Fedir Yudin[/i]

2021 Olimphíada, 6
Tags: functional equation , functional equation in n , algebra
Let $\mathbb{Z}_{>0}$ be the set of positive integers. Find all functions $f : \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that, for all $m, n \in \mathbb{Z}_{>0 }$: $$f(mf(n)) + f(n) | mn + f(f(n)).$$

2025 Benelux, 1
Tags: algebra , functional equation
Does there exist a function $f:\mathbb{R}\to \mathbb{R}$ such that $$f(x^2+f(y))=f(x)^2-y$$ for all $x, y\in \mathbb{R}$?

2020 Bulgaria EGMO TST, 2
Tags: function , algebra , functional equation , injectivity , bijection , surjectivity
The function $f:\mathbb{R} \to \mathbb{R}$ is such that $f(f(x+1)) = x^3+1$ for all real numbers $x$. Prove that the equation $f(x) = 0 $ has exactly one real root.

2024 Alborz Mathematical Olympiad, P2
Tags: function , functional equation , algebra
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

2022 Latvia Baltic Way TST, P3
Tags: functional equation
Let $\mathbb R$ be the set of real numbers. Determine all functions $f:\mathbb R\to\mathbb R$ that satisfy the equation\[ f(f(x))+yf(xy+1) = f(x-f(y)) + xf(y)^2. \]for all real numbers $x$ and $y$.

1981 IMO Shortlist, 6
Tags: algebra , polynomial , functional equation
Let $P(z)$ and $Q(z)$ be complex-variable polynomials, with degree not less than $1$. Let \[P_k = \{z \in \mathbb C | P(z) = k \}, Q_k = \{ z \in \mathbb C | Q(z) = k \}.\] Let also $P_0 = Q_0$ and $P_1 = Q_1$. Prove that $P(z) \equiv Q(z).$

2012 Belarus Team Selection Test, 3
Tags: algebra , functional , functional equation
Find all functions $f : Q \to Q$, such that $$f(x + f (y + f(z))) = y + f(x + z)$$ for all $x ,y ,z \in Q$ . (I. Voronovich)

2017 Greece Team Selection Test, 3
Tags: algebra , functional equation
Find all fuctions $f,g:\mathbb{R}\rightarrow \mathbb{R}$ such that: $f(x-3f(y))=xf(y)-yf(x)+g(x) \forall x,y\in\mathbb{R}$ and $g(1)=-8$

2010 Mathcenter Contest, 6
Tags: algebra , functional equation
Find all $a\in\mathbb{N}$ such that exists a bijective function $g :\mathbb{N} \to \mathbb{N}$ and a function $f:\mathbb{N}\to\mathbb{N}$, such that for all $x\in\mathbb{N}$, $$f(f(f(...f(x)))...)=g(x)+a$$ where $f$ appears $2009$ times. [i](tatari/nightmare)[/i]

2008 National Olympiad First Round, 19
Tags: function , quadratic , algebra , functional equation
Let $f:(0,\infty) \rightarrow (0,\infty)$ be a function such that \[ 10\cdot \frac{x+y}{xy}=f(x)\cdot f(y)-f(xy)-90 \] for every $x,y \in (0,\infty)$. What is $f(\frac 1{11})$? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 11 \qquad\textbf{(C)}\ 21 \qquad\textbf{(D)}\ 31 \qquad\textbf{(E)}\ \text{There is more than one solution} $

2020 OMMock - Mexico National Olympiad Mock Exam, 6
Tags: algebra , functional equation
Find all functions $f \colon \mathbb{R} \to \mathbb{R}$ such that \[f(f(x) - y) = f(xy) + f(x)f(-y)\] for any two real numbers $x, y$. [i]Proposed by Pablo Valeriano[/i]

KoMaL A Problems 2024/2025, A. 895
Tags: functional equation , algebra
Let's call a function $f:\mathbb R\to\mathbb R$[i] weakly periodic[/i] if it is continuous and $f(x+1)=f(f(x))+1$ for all $x\in\mathbb R$. a) Does there exist a weakly periodic function such that $f(x)>x$ for all $x\in\mathbb R$? b) Does there exist a weakly periodic function such that $f(x)<x$ for all $x\in\mathbb R$? [i]Proposed by: András Imolay, Budapest[/i]

2006 Thailand Mathematical Olympiad, 6
Tags: algebra , functional equation , functional , trigonometry
A function $f : R \to R$ has $f(1) < 0$, and satisfy the functional equation $$f(\cos (x + y)) = (\cos x)f(\cos y) + 2f(\sin x)f(\sin y)$$ for all reals $x, y$. Compute $f \left(\frac{2006}{2549 }\right)$

III Soros Olympiad 1996 - 97 (Russia), 11.2
Tags: algebra , functional equation
Is there a function $f(x)$ defined and continuous on $R$ such that: a) $f(f(x)) = 1 + 2x$ ? b) $f(f(x)) = 1 - 2x $?

2014 IMAC Arhimede, 1
Tags: functional equation , functional equation in n , algebra
The function $f: N \to N_0$ is such that $f (2) = 0, f (3)> 0, f (6042) = 2014$ and $f (m + n)- f (m) - f (n) \in\{0,1\}$ for all $m,n \in N$. Determine $f (2014)$. $N_0=\{0,1,2,...\}$

2016 Taiwan TST Round 3, 2
Tags: function , functional equation , algebra
Determine all functions $f:\mathbb{R}^+\rightarrow \mathbb{R}^+$ satisfying $f(x+y+f(y))=4030x-f(x)+f(2016y), \forall x,y \in \mathbb{R}^+$.

2010 Belarus Team Selection Test, 2.4
Tags: functional equation , functional equation in q , functional , algebra
Find all functions $f, g : Q \to Q$ satisfying the following equality $f(x + g(y)) = g(x) + 2 y + f(y)$ for all $x, y \in Q$. (I. Voronovich)

2019-IMOC, A1
Tags: algebra , functional equation
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all $x,y\in\mathbb{R}$, $$f(xy+f(x))=f(xf(y))+x$$

2000 Singapore Team Selection Test, 1
Tags: algebra , functional equation , functional
Find all functions $f : R \to R$ such for any $x, y \in R,$ $$(x - y)f(x + y) - (x + y)f(x - y) = 4xy(x^2 - y^2)$$

2024 Kosovo Team Selection Test, P3
Tags: functional equation , algebra
Find all functions $f:\mathbb R\to\mathbb R$ such that $$(x-y)f(x+y) - (x+y)f(x-y) = 2y(f(x)-f(y) - 1)$$for all $x, y\in\mathbb R$.

2012 Cuba MO, 7
Tags: algebra , functional , functional equation
Find all the functions $f : R\to R$ such that $f(x^2 + f(y)) = y - x^2$ for all $x, y$ reals.