Found problems: 1513
Let $f : \mathbb{Z} \to \mathbb{Z}$ be a function satisfying $f(0) \neq 0 = f(1)$. Assume also that $f$ satisfies equations [b](A)[/b] and [b](B)[/b] below. \begin{eqnarray*}f(xy) = f(x) + f(y) -f(x) f(y)\qquad\mathbf{(A)}\\
f(x-y) f(x) f(y) = f(0) f(x) f(y)\qquad\mathbf{(B)}
\end{eqnarray*} for all integers $x,y$.
[b](i)[/b] Determine explicitly the set $\big\{f(a)~:~a\in\mathbb{Z}\big\}$.
[b](ii)[/b] Assuming that there is a non-zero integer $a$ such that $f(a) \neq 0$, prove that the set $\big\{b~:~f(b) \neq 0\big\}$ is infinite.
Find all functions $f: \mathbb{Q^+} \rightarrow \mathbb{Q}$ satisfying $f(x)+f(y)= \left(f(x+y)+\frac{1}{x+y} \right) (1-xy+f(xy))$ for all $x, y \in \mathbb{Q^+}$.
Find all real functions $f$ defined on $ \mathbb R$, such that \[f(f(x)+y) = f(f(x)-y)+4f(x)y ,\] for all real numbers $x,y$.
Find all functions \(f:\mathbb{R} \rightarrow \mathbb{R}\) such that
\[ f(x^2+y^2f(x)) = xf(y)^2-f(x)^2 \]
for every \(x, y \in \mathbb{R}\)
A function $ f$ defined on the positive integers (and taking positive integers values) is given by:
$ \begin{matrix} f(1) \equal{} 1, f(3) \equal{} 3 \\
f(2 \cdot n) \equal{} f(n) \\
f(4 \cdot n \plus{} 1) \equal{} 2 \cdot f(2 \cdot n \plus{} 1) \minus{} f(n) \\
f(4 \cdot n \plus{} 3) \equal{} 3 \cdot f(2 \cdot n \plus{} 1) \minus{} 2 \cdot f(n), \end{matrix}$
for all positive integers $ n.$ Determine with proof the number of positive integers $ \leq 1988$ for which $ f(n) \equal{} n.$
$f: \mathbb{R} \rightarrow \mathbb{R}$ is a function such that for $\forall x,y\in \mathbb{R}$ the equation
$f(xy+x+y)=f(xy)+f(x)+f(y)$
is true. Prove that $f(x+y)=f(x)+f(y)$ for $\forall$ $x,y\in \mathbb{R}$.
Show that there exists no function $f : Z \to Z$ such that for all $m, n \in Z$
$$f(m + f(n)) = f(m) - n.$$
The function $g : [0, \infty) \to [0, \infty)$ satisfies the functional equation: $g(g(x)) = \frac{3x}{x + 3}$, for all $x \ge 0$.
You are also told that for $2 \le x \le 3$: $g(x) = \frac{x + 1}{2}$.
(a) Find $g(2021)$.
(b) Find $g(1/2021)$.
Find all functions $f$ from the set of reals to itself so that for all reals $x,y,$
$$f(x)f(f(x)+y) = f(x^2) + f(xy).$$
[i]Proposed by Culver Kwan[/i]
Determine all functions $f:\mathbb R\to\mathbb R$ such that for all $x, y \in \mathbb R$
$$f(xf(y) - yf(x)) = f(xy) - xy.$$
Find all functions $f: \mathbb {R}^+ \times \mathbb {R}^+ \to \mathbb {R}^+$ that satisfy the following conditions for all positive real numbers $x,y,z:$
$$f\left ( f(x,y),z \right )=x^2y^2f(x,z)$$
$$f\left ( x,1+f(x,y) \right ) \ge x^2 + xyf(x,x)$$
[i]Proposed by Mojtaba Zare, Ali Daei Nabi[/i]
Let $ \mathbb{R}^\plus{}$ be the set of all non-negative real numbers. Given two positive real numbers $ a$ and $ b,$ suppose that a mapping $ f: \mathbb{R}^\plus{} \mapsto \mathbb{R}^\plus{}$ satisfies the functional equation:
\[ f(f(x)) \plus{} af(x) \equal{} b(a \plus{} b)x.\]
Prove that there exists a unique solution of this equation.
A function $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfies the conditions
\[\begin{cases}f(x+24)\le f(x)+24\\ f(x+77)\ge f(x)+77\end{cases}\quad\text{for all}\ x\in\mathbb{R}\]
Prove that $f(x+1)=f(x)+1$ for all real $x$.
Find all strictly monotonous functions $f : N \to N$ that satisfy $f(f(n)) = 3n$ for all $n \in N$.
Find all functions $f: \mathbb{Z}\rightarrow\mathbb{Z}$ such that for all $x,y \in \mathbb{Z}$:
\[f(x-y+f(y))=f(x)+f(y).\]
Find all continuous functions $f: \mathbb{R}\to \mathbb{R}$ such that
\[ f(x)-f(y)\in \mathbb{Q}\quad \text{ for all }\quad x-y\in\mathbb{Q} \]
Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ which satisfy \[ f(m + f(n)) = f(m) + n \] for all $m,n \in \mathbb{Z}$.
Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(xf(x-y))+yf(x)=x+y+f(x^2),$$ for all real numbers $x$ and $y.$
For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$
[i]Proposed by Warut Suksompong, Thailand[/i]
Find all polynomials $P$ such that $P(x)+3P(x+2)=3P(x+1)+P(x+3)$ for all real numbers $x$.
Find all functions $f : R \to R$ satisfying the equality $f (2^x + 2y) =2^y f ( f (x)) f (y) $for every $x, y \in R$.
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}$.
Determine the least possible value of $f(1998),$ where $f:\Bbb{N}\to \Bbb{N}$ is a function such that for all $m,n\in {\Bbb N}$,
\[f\left( n^{2}f(m)\right) =m\left( f(n)\right) ^{2}. \]
Determine all functions $f\colon\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ such that, for all positive integers $a$ and $b$,
\[
f^{bf(a)}(a+1)=(a+1)f(b).
\]
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]