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: 23

2014 Contests, 1
Tags: algebra , functional equation in n , functional equation
The function $f: N \to N_0$ is such that $f (2) = 0, f (3)> 0, f (6042) = 2014$ and $f (m + n)- f (m) - f (n) \in\{0,1\}$ for all $m,n \in N$. Determine $f (2014)$. $N_0=\{0,1,2,...\}$

2024 South Africa National Olympiad, 4
Tags: functional equation , functional equation in n , algebra
Find all functions $f$ from integers to integers such that \[ f(m+n) + f(m-n) - 2f(m) = 6mn^2\] for all integers $m$ and $n$.

2011 Baltic Way, 2
Tags: functional equation in n , algebra , functional equation
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.

2018 Azerbaijan BMO TST, 2
Tags: perfect square , functional equation in n , functional equation , number theory
Find all functions $f :Z_{>0} \to Z_{>0}$ such that the number $xf(x) + f ^2(y) + 2xf(y)$ is a perfect square for all positive integers $x,y$.

1996 Nordic, 4
Tags: functional equation in n , algebra
The real-valued function $f$ is defined for positive integers, and the positive integer $a$ satisfies $f(a) = f(1995), f(a+1) = f(1996), f(a+2) = f(1997), f(n + a) = \frac{f(n) - 1}{f(n) + 1}$ for all positive integers $n$. (i) Show that $f(n+ 4a) = f(n)$ for all positive integers $n$. (ii) Determine the smallest possible $a$.

2018 SIMO, Q3
Tags: functional equation in n , functional equation
Suppose $f:\mathbb{N}\rightarrow \mathbb{N}$ is a function such that $$f^n(n) = 2n$$ for all $n\in \mathbb{N}$. Must $f(n) = n+1$ for all $n$?

1998 Czech and Slovak Match, 4
Tags: natural number , functional equation in n , algebra
Find all functions $f : N\rightarrow N - \{1\}$ satisfying $f (n)+ f (n+1)= f (n+2) +f (n+3) -168$ for all $n \in N$ .

2019 Romania Team Selection Test, 3
Tags: functional equation , functional equation in n , algebra
Determine all functions $f$ from the set of non-negative integers to itself such that $f(a + b) = f(a) + f(b) + f(c) + f(d)$, whenever $a, b, c, d$, are non-negative integers satisfying $2ab = c^2 + d^2$.

2008 Dutch IMO TST, 1
Tags: functional equation , functional equation in n , algebra
Find all funtions $f : Z_{>0} \to Z_{>0}$ that satisfy $f(f(f(n))) + f(f(n)) + f(n) = 3n$ for all $n \in Z_{>0}$ .

2021 Olimphíada, 6
Tags: functional equation in n , algebra , functional equation
Let $\mathbb{Z}_{>0}$ be the set of positive integers. Find all functions $f : \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that, for all $m, n \in \mathbb{Z}_{>0 }$: $$f(mf(n)) + f(n) | mn + f(f(n)).$$

2008 Dutch IMO TST, 1
Tags: algebra , functional equation , functional equation in n
Find all funtions $f : Z_{>0} \to Z_{>0}$ that satisfy $f(f(f(n))) + f(f(n)) + f(n) = 3n$ for all $n \in Z_{>0}$ .

2018 SIMO, Q1
Tags: least common multiple , functional equation in n , greatest common divisor , functional equation , number theory
Find all functions $f:\mathbb{N}\setminus\{1\} \rightarrow\mathbb{N}$ such that for all distinct $x,y\in \mathbb{N}$ with $y\ge 2018$, $$\gcd(f(x),y)\cdot \mathrm{lcm}(x,f(y))=f(x)f(y).$$

2017 Balkan MO Shortlist, N2
Tags: functional equation , perfect square , functional equation in n , number theory
Find all functions $f :Z_{>0} \to Z_{>0}$ such that the number $xf(x) + f ^2(y) + 2xf(y)$ is a perfect square for all positive integers $x,y$.

2019 Israel Olympic Revenge, P4
Tags: functional equation in n , functional equation , algebra
Call a function $\mathbb Z_{>0}\rightarrow \mathbb Z_{>0}$ $\emph{M-rugged}$ if it is unbounded and satisfies the following two conditions: $(1)$ If $f(n)|f(m)$ and $f(n)<f(m)$ then $n|m$. $(2)$ $|f(n+1)-f(n)|\leq M$. a. Find all $1-rugged$ functions. b. Determine if the number of $2-rugged$ functions is smaller than $2019$.

2016 Brazil Team Selection Test, 1
Tags: functional equation , functional equation in n , algebra
Determine all functions $f$ from the set of non-negative integers to itself such that $f(a + b) = f(a) + f(b) + f(c) + f(d)$, whenever $a, b, c, d$, are non-negative integers satisfying $2ab = c^2 + d^2$.

1987 Nordic, 3
Tags: algebra , minimum value , increasing function , functional equation in n
Let $f$ be a strictly increasing function defined in the set of natural numbers satisfying the conditions $f(2) = a > 2$ and $f(mn) = f(m)f(n)$ for all natural numbers $m$ and $n$. Determine the smallest possible value of $a$.

2024 South Africa National Olympiad, 6
Tags: number theory , functional equation in n
Let $f:\mathbb{N}\to\mathbb{N}_0$ be a function that satisfies \[ f(mn) = mf(n) + nf(m)\] for all positive integers $m,n$ and $f(2024)=10120$. Prove that there are two integers $m,n$ with $m\ne n$ such that $f(m)=f(n)$.

2008 Indonesia TST, 4
Tags: functional , number theory , functional equation in n , functional equation
Find all pairs of positive integer $\alpha$ and function $f : N \to N_0$ that satisfies (i) $f(mn^2) = f(mn) + \alpha f(n)$ for all positive integers $m, n$. (ii) If $n$ is a positive integer and $p$ is a prime number with $p|n$, then $f(p) \ne 0$ and $f(p)|f(n)$.

2016 Romanian Master of Mathematics Shortlist, A1
Tags: functional equation , algebra , functional equation in n
Determine all functions $f$ from the set of non-negative integers to itself such that $f(a + b) = f(a) + f(b) + f(c) + f(d)$, whenever $a, b, c, d$, are non-negative integers satisfying $2ab = c^2 + d^2$.

2008 Indonesia TST, 4
Tags: functional equation , functional , functional equation in n , number theory
Find all pairs of positive integer $\alpha$ and function $f : N \to N_0$ that satisfies (i) $f(mn^2) = f(mn) + \alpha f(n)$ for all positive integers $m, n$. (ii) If $n$ is a positive integer and $p$ is a prime number with $p|n$, then $f(p) \ne 0$ and $f(p)|f(n)$.

2015 Moldova Team Selection Test, 1
Tags: function , functional equation in n , algebra
Find all functions $f : \mathbb{Z}_{+} \rightarrow \mathbb{Z}_{+}$ that satisfy $f(mf(n)) = n+f(2015m)$ for all $m,n \in \mathbb{Z}_{+}$.

OIFMAT II 2012, 2
Tags: number theory , functional , functional equation in n
Find all functions $ f: N \rightarrow N $ such that: $\bullet$ $ f (m) = 1 \iff m = 1 $; $\bullet$ If $ d = \gcd (m, n) $, then $ f (mn) = \frac {f (m) f (n)} {f (d)} $; and $\bullet$ $ \forall m \in N $, we have $ f ^ {2012} (m) = m $. Clarification: $f^n (a) = f (f^{n-1} (a))$

2014 IMAC Arhimede, 1
Tags: algebra , functional equation , functional equation in n
The function $f: N \to N_0$ is such that $f (2) = 0, f (3)> 0, f (6042) = 2014$ and $f (m + n)- f (m) - f (n) \in\{0,1\}$ for all $m,n \in N$. Determine $f (2014)$. $N_0=\{0,1,2,...\}$