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 Switzerland - Final Round, 5
Tags: function , algebra , functional equation
Does there exist any function $f: \mathbb{R}^+ \to \mathbb{R}$ such that for every positive real number $x,y$ the following is true : $$f(xf(x)+yf(y)) = xy$$

2021 Dutch IMO TST, 3
Tags: functional , functional equation , algebra
Find all functions $f : R \to R$ with $f (x + yf(x + y))= y^2 + f(x)f(y)$ for all $x, y \in R$.

2011 Turkey Team Selection Test, 1
Tags: function , algorithm , induction , number theory , euclidean algorithm , algebra , functional equation
Let $\mathbb{Q^+}$ denote the set of positive rational numbers. Determine all functions $f: \mathbb{Q^+} \to \mathbb{Q^+}$ that satisfy the conditions \[ f \left( \frac{x}{x+1}\right) = \frac{f(x)}{x+1} \qquad \text{and} \qquad f \left(\frac{1}{x}\right)=\frac{f(x)}{x^3}\] for all $x \in \mathbb{Q^+}.$

1987 Austrian-Polish Competition, 3
Tags: sequence , functional , functional equation , limit
A function $f: R \to R$ satisfies $f (x + 1) = f (x) + 1$ for all $x$. Given $a \in R$, define the sequence $(x_n)$ recursively by $x_0 = a$ and $x_{n+1} = f (x_n)$ for $n \ge 0$. Suppose that, for some positive integer m, the difference $x_m - x_0 = k$ is an integer. Prove that the limit $\lim_{n\to \infty}\frac{x_n}{n}$ exists and determine its value.

2019 USA TSTST, 7
Tags: number theory , greatest common divisor , algebra , functional equation
Let $f: \mathbb Z\to \{1, 2, \dots, 10^{100}\}$ be a function satisfying $$\gcd(f(x), f(y)) = \gcd(f(x), x-y)$$ for all integers $x$ and $y$. Show that there exist positive integers $m$ and $n$ such that $f(x) = \gcd(m+x, n)$ for all integers $x$. [i]Ankan Bhattacharya[/i]

2005 Thailand Mathematical Olympiad, 3
Tags: functional , functional equation , algebra
Does there exist a function $f : Z^+ \to Z^+$ such that $f(f(n)) = 2n$ for all positive integers $n$? Justify your answer, and if the answer is yes, give an explicit construction.

2015 IMO Shortlist, A5
Tags: function , algebra , functional equation
Let $2\mathbb{Z} + 1$ denote the set of odd integers. Find all functions $f:\mathbb{Z} \mapsto 2\mathbb{Z} + 1$ satisfying \[ f(x + f(x) + y) + f(x - f(x) - y) = f(x+y) + f(x-y) \] for every $x, y \in \mathbb{Z}$.

1986 IMO, 2
Tags: function , algebra , functional equation
Find all functions $f$ defined on the non-negative reals and taking non-negative real values such that: $f(2)=0,f(x)\ne0$ for $0\le x<2$, and $f(xf(y))f(y)=f(x+y)$ for all $x,y$.

1988 IMO Shortlist, 19
Tags: function , algebra , functional equation
Let $ f(n)$ be a function defined on the set of all positive integers and having its values in the same set. Suppose that $ f(f(n) \plus{} f(m)) \equal{} m \plus{} n$ for all positive integers $ n,m.$ Find the possible value for $ f(1988).$

2018 China Team Selection Test, 4
Tags: function , functional equation , fe , algebra
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.

1994 Abels Math Contest (Norwegian MO), 3b
Tags: functional equation , functional , algebra
Prove that there is no function $f : Z \to Z$ such that $f(f(x)) = x+1$ for all $x$.

2013 Romanian Masters In Mathematics, 2
Tags: function , floor function , induction , functional equation , algebra
Does there exist a pair $(g,h)$ of functions $g,h:\mathbb{R}\rightarrow\mathbb{R}$ such that the only function $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying $f(g(x))=g(f(x))$ and $f(h(x))=h(f(x))$ for all $x\in\mathbb{R}$ is identity function $f(x)\equiv x$?

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 $$f (f(x) + 2f(y)) = f(2x) + 8y + 6.$$

2023 CMI B.Sc. Entrance Exam, 2
Tags: functional equation
Solve for $g : \mathbb{Z}^+ \to \mathbb{Z}^+$ such that $$g(m + n) = g(m) + mn(m + n) + g(n)$$ Show that $g(n)$ is of the form $\sum_{i=0}^{d} {c_i n^i}$ \\ and find necessary and sufficient conditions on $d$ and $c_0, c_1, \cdots , c_d$

2014 Brazil Team Selection Test, 4
Tags: function , algebra , functional equation
Let $\mathbb{Z}_{\ge 0}$ be the set of all nonnegative integers. Find all the functions $f: \mathbb{Z}_{\ge 0} \rightarrow \mathbb{Z}_{\ge 0} $ satisfying the relation \[ f(f(f(n))) = f(n+1 ) +1 \] for all $ n\in \mathbb{Z}_{\ge 0}$.

2016 Postal Coaching, 2
Tags: function , functional equation , algebra
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 Argentina Iberoamerican TST, 5
Tags: algebra , functional equation
Let \( \mathbb R \) be the set of real numbers. Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that, for all real numbers \( x \) and \( y \), the following equation holds:$$\big (x^2-y^2\big )f\big (xy\big )=xf\big (x^2y\big )-yf\big (xy^2\big ).$$

2015 Belarus Team Selection Test, 1
Tags: functional equation , algebra , functional
Do there exist numbers $a,b \in R$ and surjective function $f: R \to R$ such that $f(f(x)) = bx f(x) +a$ for all real $x$? I.Voronovich

2008 VJIMC, Problem 1
Tags: fe , functional equation , algebra
Find all functions $f:\mathbb Z\to\mathbb Z$ such that $$19f(x)-17f(f(x))=2x$$for all $x\in\mathbb Z$.

2020 DMO Stage 1, 3.
Tags: functional equation , algebra
[b]Q.[/b] Determine all the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(x) \geqslant x+1, \forall\ x \in \mathbb{R}\quad \text{and}\quad f(x+y) \geqslant f(x) f(y), \forall\ x, y \in \mathbb{R}$$ [i]Proposed by TuZo[/i]

2011 Baltic Way, 2
Tags: algebra , functional equation , functional equation in n
Let $f:\mathbb{Z}\to\mathbb{Z}$ be a function such that for all integers $x$ and $y$, the following holds: \[f(f(x)-y)=f(y)-f(f(x)).\] Show that $f$ is bounded.

1993 Poland - Second Round, 6
Tags: function , continuous function , algebra , functional equation
A continuous function $f : R \to R$ satisfies the conditions $f(1000) = 999$ and $f(x)f(f(x)) = 1$ for all real $x$. Determine $f(500)$.

2015 IMO, 5
Tags: functional equation , algebra , dorlir ahmeti , imo proposal
Let $\mathbb R$ be the set of real numbers. Determine all functions $f:\mathbb R\to\mathbb R$ that satisfy the equation\[f(x+f(x+y))+f(xy)=x+f(x+y)+yf(x)\]for all real numbers $x$ and $y$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

2003 Singapore Team Selection Test, 3
Tags: algebra , functional equation , functional
Determine all functions $f : Z\to Z$, where $Z$ is the set of integers, such that $$f(m + f(f(n))) = -f(f(m + 1)) - n$$ for all integers $m$ and $n$.

1996 IMO Shortlist, 5
Tags: function , number theory , functional equation
Show that there exists a bijective function $ f: \mathbb{N}_{0}\to \mathbb{N}_{0}$ such that for all $ m,n\in \mathbb{N}_{0}$: \[ f(3mn \plus{} m \plus{} n) \equal{} 4f(m)f(n) \plus{} f(m) \plus{} f(n). \]