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

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 IFYM, Sozopol, 3
Tags: algebra , functional equation
Find all functions \( f:\mathbb{Z} \to \mathbb{Z} \) such that \[ f(x + f(y) - 2y) + f(f(y)) = f(x) \] for all integers \( x \) and \( y \).

KoMaL A Problems 2022/2023, A. 835
Tags: algebra , functional equation , polynomial
Let $f^{(n)}(x)$ denote the $n^{\text{th}}$ iterate of function $f$, i.e $f^{(1)}(x)=f(x)$, $f^{(n+1)}(x)=f(f^{(n)}(x))$. Let $p(n)$ be a given polynomial with integer coefficients, which maps the positive integers into the positive integers. Is it possible that the functional equation $f^{(n)}(n)=p(n)$ has exactly one solution $f$ that maps the positive integers into the positive integers? [i]Submitted by Dávid Matolcsi and Kristóf Szabó, Budapest[/i]

2012 Dutch IMO TST, 5
Tags: functional equation , algebra
Find all functions $f : R \to R$ satisfying $f(x + xy + f(y))=(f(x) + \frac12)(f(y) + \frac12 )$ for all $x, y \in R$.

2010 IMO Shortlist, 5
Tags: function , algebra , functional equation
Denote by $\mathbb{Q}^+$ the set of all positive rational numbers. Determine all functions $f : \mathbb{Q}^+ \mapsto \mathbb{Q}^+$ which satisfy the following equation for all $x, y \in \mathbb{Q}^+:$ \[f\left( f(x)^2y \right) = x^3 f(xy).\] [i]Proposed by Thomas Huber, Switzerland[/i]

2017 Harvard-MIT Mathematics Tournament, 3
Tags: function , algebra , functional equation
Let $f: \mathbb{R}\rightarrow \mathbb{R}$ be a function satisfying $f(x)f(y)=f(x-y)$. Find all possible values of $f(2017)$.

1982 IMO Shortlist, 1
Tags: function , algebra , functional equation
The function $f(n)$ is defined on the positive integers and takes non-negative integer values. $f(2)=0,f(3)>0,f(9999)=3333$ and for all $m,n:$ \[ f(m+n)-f(m)-f(n)=0 \text{ or } 1. \] Determine $f(1982)$.

2013 IMO Shortlist, N6
Tags: function , number theory , functional equation
Determine all functions $f: \mathbb{Q} \rightarrow \mathbb{Z} $ satisfying \[ f \left( \frac{f(x)+a} {b}\right) = f \left( \frac{x+a}{b} \right) \] for all $x \in \mathbb{Q}$, $a \in \mathbb{Z}$, and $b \in \mathbb{Z}_{>0}$. (Here, $\mathbb{Z}_{>0}$ denotes the set of positive integers.)

2010 Albania Team Selection Test, 2
Tags: function , induction , domain , functional equation , algebra
Find all the continuous functions $f : \mathbb{R} \mapsto\mathbb{R}$ such that $\forall x,y \in \mathbb{R}$, $(1+f(x)f(y))f(x+y)=f(x)+f(y)$.

2009 Middle European Mathematical Olympiad, 1
Tags: function , algebra , functional equation , real
Find all functions $ f: \mathbb{R} \to \mathbb{R}$, such that \[ f(xf(y)) \plus{} f(f(x) \plus{} f(y)) \equal{} yf(x) \plus{} f(x \plus{} f(y))\] holds for all $ x$, $ y \in \mathbb{R}$, where $ \mathbb{R}$ denotes the set of real numbers.

2020 GQMO, 5
Tags: functional equation , algebra
Let $\mathbb{Q}$ denote the set of rational numbers. Determine all functions $f:\mathbb{Q}\longrightarrow\mathbb{Q}$ such that, for all $x, y \in \mathbb{Q}$, $$f(x)f(y+1)=f(xf(y))+f(x)$$ [i]Nicolás López Funes and José Luis Narbona Valiente, Spain[/i]

2012 Hanoi Open Mathematics Competitions, 5
Tags: functional equation , algebra
Let $f(x)$ be a function such that $f(x)+2f\left(\frac{x+2010}{x-1}\right)=4020 - x$ for all $x \ne 1$. Then the value of $f(2012)$ is (A) $2010$, (B) $2011$, (C) $2012$, (D) $2014$, (E) None of the above.

2024 Taiwan Mathematics Olympiad, 3
Tags: algebra , functional equation
Find all functions $f$ from real numbers to real numbers such that $$2f((x+y)^2)=f(x+y)+(f(x))^2+(4y-1)f(x)-2y+4y^2$$ holds for all real numbers $x$ and $y$.

Russian TST 2021, P2
Tags: number theory , functional equation , nonnegative integers
Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions: [list] [*] $(i)$ $f(n) \neq 0$ for at least one $n$; [*] $(ii)$ $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$; [*] $(iii)$ there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k<n$. [/list]

1992 Austrian-Polish Competition, 6
Tags: functional , functional equation , algebra , number theory , divisor
A function $f: Z \to Z$ has the following properties: $f (92 + x) = f (92 - x)$ $f (19 \cdot 92 + x) = f (19 \cdot 92 - x)$ ($19 \cdot 92 = 1748$) $f (1992 + x) = f (1992 - x)$ for all integers $x$. Can all positive divisors of $92$ occur as values of f?

2024 ELMO Problems, 3
Tags: functional equation , algebra
For some positive integer $n,$ Elmo writes down the equation \[x_1+x_2+\dots+x_n=x_1+x_2+\dots+x_n.\] Elmo inserts at least one $f$ to the left side of the equation and adds parentheses to create a valid functional equation. For example, if $n=3,$ Elmo could have created the equation \[f(x_1+f(f(x_2)+x_3))=x_1+x_2+x_3.\] Cookie Monster comes up with a function $f: \mathbb{Q}\to\mathbb{Q}$ which is a solution to Elmo's functional equation. (In other words, Elmo's equation is satisfied for all choices of $x_1,\dots,x_n\in\mathbb{Q})$. Is it possible that there is no integer $k$ (possibly depending on $f$) such that $f^k(x)=x$ for all $x$? [i]Srinivas Arun[/i]

2008 IMO Shortlist, 5
Tags: function , number theory , modular arithmetic , divisor , functional equation
For every $ n\in\mathbb{N}$ let $ d(n)$ denote the number of (positive) divisors of $ n$. Find all functions $ f: \mathbb{N}\to\mathbb{N}$ with the following properties: [list][*] $ d\left(f(x)\right) \equal{} x$ for all $ x\in\mathbb{N}$. [*] $ f(xy)$ divides $ (x \minus{} 1)y^{xy \minus{} 1}f(x)$ for all $ x$, $ y\in\mathbb{N}$.[/list] [i]Proposed by Bruno Le Floch, France[/i]

2017 Azerbaijan Team Selection Test, 3
Tags: functional equation , algebra
Find all functions $f : \mathbb R\to\mathbb R $ such that \[f(x+yf(x^2))=f(x)+xf(xy)\] for all real numbers $x$ and $y$.

2022 JHMT HS, 9
Tags: algebra , functional equation , calculus , integration
There is a unique continuous function $f$ over the positive real numbers satisfying $f(4) = 1$ and \[ 9 - (f(x))^4 = \frac{x^2}{(f(x))^2} - 2xf(x) \] for all positive $x$. Compute the value of $\int_{0}^{140} (f(x))^3\,dx$.

2019 Baltic Way, 3
Tags: algebra , functional equation
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f(xf(y)-y^2)=(y+1)f(x-y)$$ holds for all $x,y\in\mathbb{R}$.

2021 Bolivia Ibero TST, 2
Tags: function , number theory , functional equation
Let $f: \mathbb Z^+ \to \mathbb Z$ be a function such that [b]a)[/b] $f(p)=1$ for every prime $p$. [b]b)[/b] $f(xy)=xf(y)+yf(x)$ for every pair of positive integers $x,y$ Find the least number $n \ge 2021$ such that $f(n)=n$

2004 Singapore Team Selection Test, 3
Tags: function , induction , quadratic , functional equation , algebra
Find all functions $ f: \mathbb{R} \to \mathbb{R}$ satisfying \[ f\left(\frac {x \plus{} y}{x \minus{} y}\right) \equal{} \frac {f\left(x\right) \plus{} f\left(y\right)}{f\left(x\right) \minus{} f\left(y\right)} \] for all $ x \neq y$.

2017 Taiwan TST Round 1, 1
Tags: algebra , polynomial , functional equation
Find all polynomials $P$ with real coefficients which satisfy \[P(x)P(x+1)=P(x^2-x+3) \quad \forall x \in \mathbb{R}\]

2019 Taiwan TST Round 1, 1
Tags: function , algebra , functional equation
Find all functions $ f: \mathbb{R} \to \mathbb{R} $ such that $$ f\left(xf\left(y\right)-f\left(x\right)-y\right) = yf\left(x\right)-f\left(y\right)-x $$ holds for all $ x,y \in \mathbb{R} $

2022 Macedonian Team Selection Test, Problem 3
Tags: algebra , function , functional equation
We consider all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that $f(f(n)+n)=n$ and $f(a+b-1) \leq f(a)+f(b)$ for all positive integers $a, b, n$. Prove that there are at most two values for $f(2022)$. $\textit {Proposed by Ilija Jovcheski}$