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

1979 AMC 12/AHSME, 26
Tags: function , algebra , functional equation
The function $f$ satisfies the functional equation \[f(x) +f(y) = f(x + y ) - xy - 1\] for every pair $x,~ y$ of real numbers. If $f( 1) = 1$, then the number of integers $n \neq 1$ for which $f ( n ) = n$ is $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }\text{infinite}$

2024 Belarus Team Selection Test, 3.2
Tags: functional equation , algebra
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that for any reals $x \neq y$ the following equality is true: $$f(x+y)^2=f(x+y)+f(x)+f(y)$$ [i]D. Zmiaikou[/i]

2023 Balkan MO Shortlist, A3
Tags: algebra , functional equation
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$, such that $$f(xy+f(x^2))=xf(x+y)$$ for all reals $x, y$.

2017 Thailand TSTST, 4
Tags: function , functional equation , algebra
Find all function $f:\mathbb{N}^*\rightarrow \mathbb{N}^*$ that satisfy: $(f(1))^3+(f(2))^3+...+(f(n))^3=(f(1)+f(2)+...+f(n))^2$

1998 Belarus Team Selection Test, 3
Tags: continuous , functional equation , functional , algebra
Find all continuous functions $f: R \to R$ such that $g(g(x)) = g(x)+2x$ for all real $x$.

2009 Thailand Mathematical Olympiad, 2
Tags: injective , function , functional , functional equation , algebra
Is there an injective function $f : Z^+ \to Q$ satisfying the equation $f(xy) = f(x) + f(y)$ for all positive integers $x$ and $y$?

2022 Switzerland Team Selection Test, 12
Tags: functional equation , algebra
Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ such that \[x+f(yf(x)+1)=xf(x+y)+yf(yf(x))\] for all $x,y>0.$

2025 Turkey Team Selection Test, 3
Tags: functional equation , algebra
Find all $f: \mathbb{R} \rightarrow \mathbb{R}$ such that, for all $x,y \in \mathbb{R}-\{0\}$, $$ f(x) \neq 0 \text{ and } \frac{f(x)}{f(y)} + \frac{f(y)}{f(x)} - f \left( \frac{x}{y}-\frac{y}{x} \right) =2 $$

1999 Ukraine Team Selection Test, 9
Tags: function , functional equation , algebra
Find all functions $u : R \to R$ for which there is a strictly increasing function $f : R \to R$ such that $f(x+y) = f(x)u(y)+ f(y)$ for all $x,y \in R$.

2012 Switzerland - Final Round, 2
Tags: algebra , functional , functional equation
Determine all functions $f : R \to R$ such that for all $x, y\in R$ holds $$f (f(x) + 2f(y)) = f(2x) + 8y + 6.$$

2020 Dutch IMO TST, 3
Tags: functional equation , functional equation in z , functional , algebra
Find all functions $f: Z \to Z$ that satisfy $$f(-f (x) - f (y))= 1 -x - y$$ for all $x, y \in Z$

2018 China Team Selection Test, 4
Tags: function , algebra , functional equation , fe
Functions $f,g:\mathbb{Z}\to\mathbb{Z}$ satisfy $$f(g(x)+y)=g(f(y)+x)$$ for any integers $x,y$. If $f$ is bounded, prove that $g$ is periodic.

1999 Nordic, 1
Tags: functional equation , algebra
The function $f$ is defined for non-negative integers and satisfies the condition $f(n) = f(f(n + 11))$, if $n \le 1999$ and $f(n) = n - 5$, if $n > 1999$. Find all solutions of the equation $f(n) = 1999$.

1963 Putnam, B3
Tags: function , algebra , functional equation
Find every twice-differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfies the functional equation $$ f(x)^2 -f(y)^2 =f(x+y)f(x-y)$$ for all $x,y \in \mathbb{R}. $

2019 BAMO, 4
Tags: function , functional equation , algebra
Let $S$ be a finite set of nonzero real numbers, and let $f : S\to S$ be a function with the following property: for each $x \in S$, either $f ( f (x)) = x+ f (x)$ or $f ( f (x)) = \frac{x+ f (x)}{2}$. Prove that $f (x) = x$ for all $x \in S$.

2025 Korea - Final Round, P2
Tags: algebra , function , functional equation
Let $\mathbb{R}$ be the set of real numbers. Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ that satisfy the following condition. Here, $f^{100}(x)$ is the function obtained by composing $f(x)$ $100$ times, that is, $(\underbrace{f \circ f \circ \cdots \circ f}_{100 \ \text{times}})(x).$ [b](Condition)[/b] For all $x, y \in \mathbb{R}$, $$f(x + f^{100}(y)) = x + y \ \ \ \text{or} \ \ \ f(f^{100}(x) + y) = x + y$$

1968 German National Olympiad, 3
Tags: algebra , functional , functional equation
Specify all functions $y = f(x)$, each in the largest possible domain (within the range of real numbers) of the equation $$a \cdot f(x^n) + f(-x^n) = bx$$ suffice, where $b$ is any real number, $n$ is any odd natural number and $a$ is a real number with $|a| \ne 1$.

2018-IMOC, A1
Tags: fe , functional equation , algebra
Find all functions $f:\mathbb Q\to\mathbb Q$ such that for all $x,y,z,w\in\mathbb Q$, $$f(f(xyzw)+x+y)+f(z)+f(w)=f(f(xyzw)+z+w)+f(x)+f(y).$$

2018 Korea Winter Program Practice Test, 1
Tags: functional equation , function , floor function , algebra
Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ satisfying the following conditions : 1) $f(x+y)-f(x)-f(y) \in \{0,1\} $ for all $x,y \in \mathbb{R}$ 2) $\lfloor f(x) \rfloor = \lfloor x \rfloor $ for all real $x$.

2017 Ukraine Team Selection Test, 7
Tags: number theory , polynomial , functional equation , sum of digits
For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$ [i]Proposed by Warut Suksompong, Thailand[/i]

2019 Romanian Master of Mathematics, 5
Tags: fe , algebra , functional equation
Determine all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying \[f(x + yf(x)) + f(xy) = f(x) + f(2019y),\] for all real numbers $x$ and $y$.

1979 Austrian-Polish Competition, 4
Tags: functional equation , function , algebra
Determine all functions $f : N_0 \to R$ satisfying $f (x+y)+ f (x-y)= f (3x)$ for all $x,y$.

2004 IMO, 2
Tags: algebra , polynomial , functional equation
Find all polynomials $f$ with real coefficients such that for all reals $a,b,c$ such that $ab+bc+ca = 0$ we have the following relations \[ f(a-b) + f(b-c) + f(c-a) = 2f(a+b+c). \]

2009 Turkey Team Selection Test, 1
Tags: function , algorithm , number theory , euclidean algorithm , continued fraction , algebra , functional equation
Find all $ f: Q^ \plus{} \to\ Z$ functions that satisfy $ f \left(\frac {1}{x} \right) \equal{} f(x)$ and $ (x \plus{} 1)f(x \minus{} 1) \equal{} xf(x)$ for all rational numbers that are bigger than 1.

2020 Kürschák Competition, P2
Tags: functional equation , algebra
Find all functions $f\colon \mathbb{Q}\to \mathbb{R}_{\geq 0}$ such that for any two rational numbers $x$ and $y$ the following conditions hold [list] [*] $f(x+y)\leq f(x)+f(y)$, [*]$f(xy)=f(x)f(y)$, [*]$f(2)=1/2$. [/list]