Olympiad problems

VMEO III 2006 Shortlist, A1

Find all functions $f:R \to R$ such that $$f(x^2+f(y)-y) =(f(x))^2-f(y)$$ for all $x,y \in R$

1988 China Team Selection Test, 2

Tags: function , functional equation , algebra

Find all functions $f: \mathbb{Q} \mapsto \mathbb{C}$ satisfying (i) For any $x_1, x_2, \ldots, x_{1988} \in \mathbb{Q}$, $f(x_{1} + x_{2} + \ldots + x_{1988}) = f(x_1)f(x_2) \ldots f(x_{1988})$. (ii) $\overline{f(1988)}f(x) = f(1988)\overline{f(x)}$ for all $x \in \mathbb{Q}$.

2020 Hong Kong TST, 1

Tags: number theory , functional equation , function

Find all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that for every positive integer $n$ the following is valid: If $d_1,d_2,\ldots,d_s$ are all the positive divisors of $n$, then $$f(d_1)f(d_2)\ldots f(d_s)=n.$$

2004 Switzerland - Final Round, 4

Tags: algebra , functional , functional equation

Determine all functions $f : R \to R$ such that for all $x, y \in R$ holds $$f(xf(x) + f(y)) = y + f(x)^2$$

2023 USAMO, 2

Tags: functional equation

Let $\mathbb{R}^+$ be the set of positive real numbers. Find all functions $f \colon \mathbb{R}^+ \to \mathbb{R}^+$ such that, for all $x,y \in \mathbb{R}^+$, $$f(xy+f(x))=xf(y)+2.$$

2024 Bulgaria National Olympiad, 3

Tags: algebra , functional equation , bijectivity , reciprocal

Find all functions $f:\mathbb {R}^{+} \rightarrow \mathbb{R}^{+}$, such that $$f(af(b)+a)(f(bf(a))+a)=1$$ for any positive reals $a, b$.

2013 USA TSTST, 6

Tags: function , number theory , algebra , functional equation

Let $\mathbb N$ be the set of positive integers. Find all functions $f: \mathbb N \to \mathbb N$ that satisfy the equation \[ f^{abc-a}(abc) + f^{abc-b}(abc) + f^{abc-c}(abc) = a + b + c \] for all $a,b,c \ge 2$. (Here $f^1(n) = f(n)$ and $f^k(n) = f(f^{k-1}(n))$ for every integer $k$ greater than $1$.)

2012 Grand Duchy of Lithuania, 1

Tags: algebra , functional equation , functional

Find all functions $g : R \to R$, for which there exists a strictly increasing function $f : R \to R$ such that $f(x + y) = f(x)g(y) + f(y)$.

2015 Dutch BxMO/EGMO TST, 5

Tags: algebra , functional equation

Find all functions $f : R \to R$ satisfying $(x^2 + y^2)f(xy) = f(x)f(y)f(x^2 + y^2)$ for all real numbers $x$ and $y$.

2016 Israel National Olympiad, 7

Tags: algebra , functional equation , function

Find all functions $f:\mathbb{Z}\rightarrow\mathbb{C}$ such that $f(x(2y+1))=f(x(y+1))+f(x)f(y)$ holds for any two integers $x,y$.

2009 Balkan MO Shortlist, A6

Tags: algebra , functional equation

We denote the set of nonzero integers and the set of non-negative integers with $\mathbb Z^*$ and $\mathbb N_0$, respectively. Find all functions $f:\mathbb Z^* \to \mathbb N_0$ such that: $a)$ $f(a+b)\geq min(f(a), f(b))$ for all $a,b$ in $\mathbb Z^*$ for which $a+b$ is in $\mathbb Z^*$. $b)$ $f(ab)=f(a)+f(b)$ for all $a,b$ in $\mathbb Z^*$.

2003 Cuba MO, 8

Tags: algebra , complex number , functional equation

Find all the functions $f : C \to R^+$ such that they fulfill simultaneously the following conditions: $$(i) \ \ f(uv) = f(u)f(v) \ \ \forall u, v \in C$$ $$(ii) \ \ f(au) = |a | f(u) \ \ \forall a \in R, u \in C$$ $$(iii) \ \ f(u) + f(v) \le |u| + |v| \ \ \forall u, v \in C$$

1998 Belarus Team Selection Test, 1

Tags: functional , functional equation , algebra , periodic

Do there exist functions $f : R \to R$ and $g : R \to R$, $g$ being periodic, such that $$x^3= f(\lfloor x \rfloor ) + g(x)$$ for all real $x$ ?

2019 USAJMO, 2

Tags: function , algebra , functional equation

Let $\mathbb{Z}$ be the set of all integers. Find all pairs of integers $(a,b)$ for which there exist functions $f \colon \mathbb{Z}\rightarrow \mathbb{Z}$ and $g \colon \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying \[ f(g(x))=x+a \quad\text{and}\quad g(f(x))=x+b \] for all integers $x$. [i]Proposed by Ankan Bhattacharya[/i]

2021-IMOC qualification, A2

Tags: algebra , polynomial , functional equation , functional

Find all functions $f:R \to R$, such that $f(x)+f(y)=f(x+y)$, and there exists non-constant polynomials $P(x)$, $Q(x)$ such that $P(x)f(Q(x))=f(P(x)Q(x))$

1986 Polish MO Finals, 4

Tags: algebra , polynomial , derivative , functional equation

Find all $n$ such that there is a real polynomial $f(x)$ of degree $n$ such that $f(x) \ge f'(x)$ for all real $x$.

2002 Singapore Team Selection Test, 3

Tags: algebra , functional , functional equation

Find all functions $f : [0,\infty) \to [0,\infty)$ such that $f(f(x)) +f(x) = 12x$, for all $x \ge 0$.

2021 Peru EGMO TST, 6

Tags: algebra , function , functional inequality , functional equation

Find all functions $f : R \to R$ such that $$f(x + y) \ge xf(x) + yf(y)$$, for all $x, y \in R$ .

2010 Indonesia TST, 2

Tags: function , induction , algebra , functional equation

Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying \[ f(x^3\plus{}y^3)\equal{}xf(x^2)\plus{}yf(y^2)\] for all real numbers $ x$ and $ y$. [i]Hery Susanto, Malang[/i]

2017 International Zhautykov Olympiad, 2

Tags: functional equation , function , algebra

Find all functions $f:R \rightarrow R$ such that $$(x+y^2)f(yf(x))=xyf(y^2+f(x))$$, where $x,y \in \mathbb{R}$

2015 IFYM, Sozopol, 8

Tags: algebra , functional equation , function

Let $\mathbb{N} = \{1, 2, 3, \ldots\}$ be the set of positive integers. Find all functions $f$, defined on $\mathbb{N}$ and taking values in $\mathbb{N}$, such that $(n-1)^2< f(n)f(f(n)) < n^2+n$ for every positive integer $n$.

2019 Iran MO (3rd Round), 2

Tags: functional equation

Find all function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for any three real number $a,b,c$ , if $ a + f(b) + f(f(c)) = 0$ : $$ f(a)^3 + bf(b)^2 + c^2f(c) = 3abc $$. [i]Proposed by Amirhossein Zolfaghari [/i]

2018 China Team Selection Test, 6

Tags: function , divisibility , iteration , algebra , functional equation

Let $M,a,b,r$ be non-negative integers with $a,r\ge 2$, and suppose there exists a function $f:\mathbb{Z}\rightarrow\mathbb{Z}$ satisfying the following conditions: (1) For all $n\in \mathbb{Z}$, $f^{(r)}(n)=an+b$ where $f^{(r)}$ denotes the composition of $r$ copies of $f$ (2) For all $n\ge M$, $f(n)\ge 0$ (3) For all $n>m>M$, $n-m|f(n)-f(m)$ Show that $a$ is a perfect $r$-th power.

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 $c_0, c_1, \cdots , c_d$

2012 Balkan MO, 4

Tags: function , inequalities , functional equation

Let $\mathbb{Z}^+$ be the set of positive integers. Find all functions $f:\mathbb{Z}^+ \rightarrow\mathbb{Z}^+$ such that the following conditions both hold: (i) $f(n!)=f(n)!$ for every positive integer $n$, (ii) $m-n$ divides $f(m)-f(n)$ whenever $m$ and $n$ are different positive integers.