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

2007 France Team Selection Test, 2
Tags: function , induction , functional equation , algebra
Find all functions $f: \mathbb{Z}\rightarrow\mathbb{Z}$ such that for all $x,y \in \mathbb{Z}$: \[f(x-y+f(y))=f(x)+f(y).\]

1997 IMO Shortlist, 22
Tags: function , algebra , functional equation , polynomial
Does there exist functions $ f,g: \mathbb{R}\to\mathbb{R}$ such that $ f(g(x)) \equal{} x^2$ and $ g(f(x)) \equal{} x^k$ for all real numbers $ x$ a) if $ k \equal{} 3$? b) if $ k \equal{} 4$?

2023 IFYM, Sozopol, 2
Tags: functional equation , algebra
Does there exist a function $f: \mathbb{Z}_{\geq 0} \to \mathbb{Z}_{\geq 0}$ such that \[ f(ab) = f(a)b + af(b) \] for all $a,b \in \mathbb{Z}_{\geq 0}$ and $f(p) > p^p$ for every prime number $p$? [i] (Here, $\mathbb{Z}_{\geq 0}$ denotes the set of non-negative integers.)[/i]

2018 Iran MO (2nd Round), 4
Tags: algebra , function , functional equation
Find all functions $f:\Bbb {R} \rightarrow \Bbb {R} $ such that: $$f(x+y)f(x^2-xy+y^2)=x^3+y^3$$ for all reals $x, y $.

2018 Costa Rica - Final Round, F2
Tags: functional equation , functional , algebra , sequence , combinatorics
Consider $f (n, m)$ the number of finite sequences of $ 1$'s and $0$'s such that each sequence that starts at $0$, has exactly n $0$'s and $m$ $ 1$'s, and there are not three consecutive $0$'s or three $ 1$'s. Show that if $m, n> 1$, then $$f (n, m) = f (n-1, m-1) + f (n-1, m-2) + f (n-2, m-1) + f (n-2, m-2)$$

2024 Dutch BxMO/EGMO TST, IMO TSTST, 3
Tags: algebra , functional equation
Find all pairs of positive integers $(a, b)$ such that $f(x)=x$ is the only function $f:\mathbb{R}\to \mathbb{R}$ that satisfies $$f^a(x)f^b(y)+f^b(x)f^a(y)=2xy$$ for all $x, y\in \mathbb{R}$.

2015 Turkey Team Selection Test, 7
Tags: functional equation , algebra , function , sophie germain identity
Find all the functions $f:R\to R$ such that \[f(x^2) + 4y^2f(y) = (f(x-y) + y^2)(f(x+y) + f(y))\] for every real $x,y$.

2022 Korea Junior Math Olympiad, 4
Tags: algebra , functional equation , function
Find all function $f:\mathbb{N} \longrightarrow \mathbb{N}$ such that forall positive integers $x$ and $y$, $\frac{f(x+y)-f(x)}{f(y)}$ is again a positive integer not exceeding $2022^{2022}$.

2022 SEEMOUS, 2
Tags: functional equation , continuity , palic , differentiation , integration , analysis
Let $a, b, c \in \mathbb{R}$ be such that $$a + b + c = a^2 + b^2 + c^2 = 1, \hspace{8px} a^3 + b^3 + c^3 \neq 1.$$ We say that a function $f$ is a [i]Palić function[/i] if $f: \mathbb{R} \rightarrow \mathbb{R}$, $f$ is continuous and satisfies $$f(x) + f(y) + f(z) = f(ax + by + cz) + f(bx + cy + az) + f(cx + ay + bz)$$ for all $x, y, z \in \mathbb{R}.$ Prove that any Palić function is infinitely many times differentiable and find all Palić functions.

2024 CAPS Match, 5
Tags: algebra , functional equation , parameterization
Let $\alpha\neq0$ be a real number. Determine all functions $f:\mathbb R\to\mathbb R$ such that \[f\left(x^2+y^2\right)=f(x-y)f(x+y)+\alpha yf(y)\] holds for all $x, y\in\mathbb R.$

2023 District Olympiad, P1
Tags: functional equation , continuity , algebra
Determine all continuous functions $f:\mathbb{R}\to\mathbb{R}$ for which $f(1)=e$ and \[f(x+y)=e^{3xy}\cdot f(x)f(y),\]for all real numbers $x{}$ and $y{}$.

2019 Switzerland - Final Round, 6
Tags: algebra , functional , functional equation
Show that there exists no function $f : Z \to Z$ such that for all $m, n \in Z$ $$f(m + f(n)) = f(m) - n.$$

2019 Swedish Mathematical Competition, 5
Tags: algebra , inequalities , functional , functional inequality , functional equation
Let $f$ be a function that is defined for all positive integers and whose values are positive integers. For $f$ it also holds that $f (n + 1)> f (n)$ and $f (f (n)) = 3n$, for each positive integer $n$. Calculate $f (2019)$.

1963 Putnam, A2
Tags: function , number theory , relatively prime , functional equation
Let $f:\mathbb{N}\rightarrow \mathbb{N}$ be a strictly increasing function such that $f(2)=2$ and $f(mn)=f(m)f(n)$ for every pair of relatively prime positive integers $m$ and $n$. Prove that $f(n)=n$ for every positive integer $n$.

1995 Singapore Team Selection Test, 1
Tags: algebra , functional equation , functional
Let $N =\{1, 2, 3, ...\}$ be the set of all natural numbers and $f : N\to N$ be a function. Suppose $f(1) = 1$, $f(2n) = f(n)$ and $f(2n + 1) = f(2n) + 1$ for all natural numbers $n$. (i) Calculate the maximum value $M$ of $f(n)$ for $n \in N$ with $1 \le n \le 1994$. (ii) Find all $n \in N$, with 1 \le n \le 1994, such that $f(n) = M$.

2003 Switzerland Team Selection Test, 10
Tags: functional equation , functional , algebra
Find all strictly monotonous functions $f : N \to N$ that satisfy $f(f(n)) = 3n$ for all $n \in N$.

2025 Korea Winter Program Practice Test, P1
Tags: algebra , functional equation
Determine all functions $f:\mathbb{R}^{+} \to \mathbb{R}^{+}$ such that for any positive reals $x,y$, $$f(xy+f(xy)) = xf(y) + yf(x)$$

2001 Czech And Slovak Olympiad IIIA, 6
Tags: functional equation , function , algebra
Let be given natural numbers $a_1,a_2,...,a_n$ and a function $f : Z \to R$ such that $f(x) = 1$ for all integers $x < 0$ and $f(x) = 1- f(x-a_1)f(x-a_2)... f(x-a_n)$ for all integers $x \ge 0$. Prove that there exist natural numbers $s$ and $t$ such that for all integers $x > s$ it holds that $f(x+t) = f(x)$.

2012 China Team Selection Test, 3
Tags: function , modular arithmetic , functional equation , number theory , algebra
Given an integer $n\ge 2$, a function $f:\mathbb{Z}\rightarrow \{1,2,\ldots,n\}$ is called [i]good[/i], if for any integer $k,1\le k\le n-1$ there exists an integer $j(k)$ such that for every integer $m$ we have \[f(m+j(k))\equiv f(m+k)-f(m) \pmod{n+1}. \] Find the number of [i]good[/i] functions.

2015 India IMO Training Camp, 2
Tags: algebra , functional equation , function
Find all functions from $\mathbb{N}\cup\{0\}\to\mathbb{N}\cup\{0\}$ such that $f(m^2+mf(n))=mf(m+n)$, for all $m,n\in \mathbb{N}\cup\{0\}$.

2009 Tuymaada Olympiad, 4
Tags: polynomial , floor function , functional equation , algebra
Determine the maximum number $ h$ satisfying the following condition: for every $ a\in [0,h]$ and every polynomial $ P(x)$ of degree 99 such that $ P(0)\equal{}P(1)\equal{}0$, there exist $ x_1,x_2\in [0,1]$ such that $ P(x_1)\equal{}P(x_2)$ and $ x_2\minus{}x_1\equal{}a$. [i]Proposed by F. Petrov, D. Rostovsky, A. Khrabrov[/i]

2015 Postal Coaching, Problem 2
Tags: functional equation , function , algebra
Find all functions $f: \mathbb{Q} \to \mathbb{R}$ such that $f(xy)=f(x)f(y)+f(x+y)-1$ for all rationals $x,y$

2024 China Team Selection Test, 5
Tags: algebra , functional equation
Find all functions $f:\mathbb N_+\to \mathbb N_+,$ such that for all positive integer $a,b,$ $$\sum_{k=0}^{2b}f(a+k)=(2b+1)f(f(a)+b).$$ [i]Created by Liang Xiao, Yunhao Fu[/i]

2024 Vietnam Team Selection Test, 1
Tags: algebra , functional equation , polynomial , function
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 $x,y$.

1988 IMO Shortlist, 26
Tags: functional equation , binary representation , function , algebra
A function $ f$ defined on the positive integers (and taking positive integers values) is given by: $ \begin{matrix} f(1) \equal{} 1, f(3) \equal{} 3 \\ f(2 \cdot n) \equal{} f(n) \\ f(4 \cdot n \plus{} 1) \equal{} 2 \cdot f(2 \cdot n \plus{} 1) \minus{} f(n) \\ f(4 \cdot n \plus{} 3) \equal{} 3 \cdot f(2 \cdot n \plus{} 1) \minus{} 2 \cdot f(n), \end{matrix}$ for all positive integers $ n.$ Determine with proof the number of positive integers $ \leq 1988$ for which $ f(n) \equal{} n.$