Found problems: 1513
2015 Mexico National Olympiad, 3
Let $\mathbb{N} =\{1, 2, 3, ...\}$ be the set of positive integers. Let $f : \mathbb{N} \rightarrow \mathbb{N}$ be a function that gives a positive integer value, to every positive integer. Suppose that $f$ satisfies the following conditions:
$f(1)=1$
$f(a+b+ab)=a+b+f(ab)$
Find the value of $f(2015)$
Proposed by Jose Antonio Gomez Ortega
2019 Azerbaijan IMO TST, 1
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that
\[ f(xy) = yf(x) + x + f(f(y) - f(x)) \]
for all $x,y \in \mathbb{R}$.
2023 European Mathematical Cup, 4
Let $f\colon\mathbb{N}\rightarrow\mathbb{N}$ be a function such that for all positive integers $x$ and $y$, the number $f(x)+y$ is a perfect square if and only if $x+f(y)$ is a perfect square. Prove that $f$ is injective.
[i]Remark.[/i] A function $f\colon\mathbb{N}\rightarrow\mathbb{N}$ is injective if for all pairs $(x,y)$ of distinct positive integers, $f(x)\neq f(y)$ holds.
[i]Ivan Novak[/i]
1989 Chile National Olympiad, 6
The function $f$, with domain on the set of non-negative integers, is defined by the following :
$\bullet$ $f (0) = 2$
$\bullet$ $(f (n + 1) -1)^2 + (f (n)-1) ^2 = 2f (n) f (n + 1) + 4$, taking $f (n)$ the largest possible value.
Determine $f (n)$.
2017 OMMock - Mexico National Olympiad Mock Exam, 5
Let $k$ be a positive real number. Determine all functions $f:[-k, k]\rightarrow[0, k]$ satisfying the equation
$$f(x)^2+f(y)^2-2xy=k^2+f(x+y)^2$$
for any $x, y\in[-k, k]$ such that $x+y\in[-k, k]$.
[i]Proposed by Maximiliano Sánchez[/i]
2015 IMO Shortlist, A5
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 $x, y \in \mathbb{Z}$.
2017 Benelux, 1
Find all functions $f : \Bbb{Q}_{>0}\to \Bbb{Z}_{>0}$ such that $$f(xy)\cdot \gcd\left( f(x)f(y), f(\frac{1}{x})f(\frac{1}{y})\right)
= xyf(\frac{1}{x})f(\frac{1}{y}),$$ for all $x, y \in \Bbb{Q}_{>0,}$ where $\gcd(a, b)$ denotes the greatest common divisor of $a$ and $b.$
2004 Austrian-Polish Competition, 7
Determine all functions $f:\mathbb{Z}^+\to \mathbb{Z}$ which satisfy the following condition for all pairs $(x,y)$ of [i]relatively prime[/i] positive integers:
\[f(x+y) = f(x+1) + f(y+1).\]
2018 Abels Math Contest (Norwegian MO) Final, 3b
Find all real functions $f$ defined on the real numbers except zero, satisfying
$f(2019) = 1$ and $f(x)f(y)+ f\left(\frac{2019}{x}\right) f\left(\frac{2019}{y}\right) =2f(xy)$ for all $x,y \ne 0$
2020 Peru Iberoamerican Team Selection Test, P2
Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ that satisfy the conditions:
$i) f(f(x)) = xf(x) - x^2 + 2,\forall x\in\mathbb{Z}$
$ii) f$ takes all integer values
1972 Vietnam National Olympiad, 1
Let $\alpha$ be an arbitrary angle and let $x = cos\alpha, y = cosn\alpha$ ($n \in Z$).
i) Prove that to each value $x \in [-1, 1]$ corresponds one and only one value of $y$.
Thus we can write $y$ as a function of $x, y = T_n(x)$.
Compute $T_1(x), T_2(x)$ and prove that $T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)$.
From this it follows that $T_n(x)$ is a polynomial of degree $n$.
ii) Prove that the polynomial $T_n(x$) has $n$ distinct roots in $[-1, 1]$.
2015 Federal Competition For Advanced Students, P2, 1
Let $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}$ be a function with the following properties:
(i) $f(1) = 0$
(ii) $f(p) = 1$ for all prime numbers $p$
(iii) $f(xy) = y \cdot f(x) + x \cdot f(y)$ for all $x,y$ in $\mathbb{Z}_{>0}$
Determine the smallest integer $n \ge 2015$ that satisfies $f(n) = n$.
(Gerhard J. Woeginger)
2021 Thailand TSTST, 2
Let $f:\mathbb{R}^+\to\mathbb{R}^+$ be such that $$f(x+f(y))^2\geq f(x)\left(f(x+f(y))+f(y)\right)$$ for all $x,y\in\mathbb{R}^+$. Show that $f$ is [i]unbounded[/i], i.e. for each $M\in\mathbb{R}^+$, there exists $x\in\mathbb{R}^+$ such that $f(x)>M$.
Revenge EL(S)MO 2024, 4
Determine all triples of positive integers $(A,B,C)$ for which some function $f \colon \mathbb Z_{\geq 0} \to \mathbb Z_{\geq 0}$ satisfies
\[ f^{f(y)} (y + f(2x)) + f^{f(y)} (2y) = (Ax+By)^{C} \]
for all nonnegative integers $x$ and $y$, where $f^k$ as usual denotes $f$ composed $k$ times.
Proposed by [i]Benny Wang[/i]
2001 IMO Shortlist, 1
Let $ T$ denote the set of all ordered triples $ (p,q,r)$ of nonnegative integers. Find all functions $ f: T \rightarrow \mathbb{R}$ satisfying
\[ f(p,q,r) = \begin{cases} 0 & \text{if} \; pqr = 0, \\
1 + \frac{1}{6}(f(p + 1,q - 1,r) + f(p - 1,q + 1,r) & \\
+ f(p - 1,q,r + 1) + f(p + 1,q,r - 1) & \\
+ f(p,q + 1,r - 1) + f(p,q - 1,r + 1)) & \text{otherwise} \end{cases}
\]
for all nonnegative integers $ p$, $ q$, $ r$.
2018 Saudi Arabia BMO TST, 2
Find all functions $f : R \to R$ such that $f( 2x^3 + f (y)) = y + 2x^2 f (x)$ for all real numbers $x, y$.
2007 Cuba MO, 4
Find all functions $f : R_+ \to R_+$ such that $$x^2(f(x)+f(y)) = (x+y)f(f(x)y)$$ for all positive real $x, y$.
PEN K Problems, 18
Find all functions $f: \mathbb{Q}\to \mathbb{R}$ such that for all $x,y\in \mathbb{Q}$: \[f(xy)=f(x)f(y)-f(x+y)+1.\]
2019 IFYM, Sozopol, 2
Does there exist a strictly increasing function $f:\mathbb{N}\rightarrow \mathbb{N}$, such that for $\forall$ $n\in \mathbb{N}$:
$f(f(f(n)))=n+2f(n)$?
PEN K Problems, 13
Find all functions $f: \mathbb{Z}\to \mathbb{Z}$ such that for all $m\in \mathbb{Z}$: \[f(f(m))=m+1.\]
2022-IMOC, A3
Find all functions $f:\mathbb R\to \mathbb R$ such that $$xy(f(x+y)-f(x)-f(y))=2f(xy)$$ for all $x,y\in \mathbb R.$
[i]Proposed by USJL[/i]
2009 Ukraine National Mathematical Olympiad, 4
Find all functions $f : \mathbb R \to \mathbb R$ such that
\[f\left(x+xy+f(y)\right)= \left( f(x)+\frac 12 \right) \left( f(y)+\frac 12 \right) \qquad \forall x,y \in \mathbb R.\]
2020 IMEO, Problem 3
Find all functions $f:\mathbb{R^+} \to \mathbb{R^+}$ such that for all positive real $x, y$ holds
$$xf(x)+yf(y)=(x+y)f\left(\frac{x^2+y^2}{x+y}\right)$$.
[i]Fedir Yudin[/i]
VMEO III 2006 Shortlist, A6
The symbol $N_m$ denotes the set of all integers not less than the given integer $m$. Find all functions $f: N_m \to N_m$ such that $f(x^2+f(y))=y^2+f(x)$ for all $x,y \in N_m$.
2011 Peru IMO TST, 1
Let $\Bbb{Z}^+$ denote the set of positive integers. Find all functions $f:\Bbb{Z}^+\to \Bbb{Z}^+$ that satisfy the following condition: for each positive integer $n,$ there exists a positive integer $k$ such that $$\sum_{i=1}^k f_i(n)=kn,$$ where $f_1(n)=f(n)$ and $f_{i+1}(n)=f(f_i(n)),$ for $i\geq 1. $