Found problems: 2349
Show that there is no continuous function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for every real number $x$
\[f(x-f(x)) = \dfrac x2.\]
Let $f$ be a real-valued function defined for all real numbers, such that for some $a>0$ we have \[ f(x+a)={1\over2}+\sqrt{f(x)-f(x)^2} \] for all $x$.
Prove that $f$ is periodic, and give an example of such a non-constant $f$ for $a=1$.
Find all functions $f$ from reals to reals which satisfy $f (f(x) + f(y)) = f(x^2) + 2x^2 f(y) + (f(y))^2$ for all real numbers $x$ and $y$.
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(x+y) = \max(f(x),y) + \min(f(y),x)$.
[i]George Xing.[/i]
Let $\mathbb{R}$ denote the set of the reals. Find all $f : \mathbb{R} \to \mathbb{R}$ such that
$$ f(x)f(y) = xf(f(y-x)) + xf(2x) + f(x^2) $$
for all real $x, y$.
Let $f : N \to N$ be a function such that $f(f(1995)) = 95, f(xy) = f(x)f(y)$ and $f(x) \le x$ for all $x,y$.
Find all possible values of $f(1995)$.
Find all functions $f:\mathbb Q\to\mathbb Q$ such that for all $x,y,z,w\in\mathbb Q$,
$$f(f(xyzw)+x+y)+f(z)+f(w)=f(f(xyzw)+z+w)+f(x)+f(y).$$
Find all functions f : R-->R such that
f (f (x) + y^2) = x −1 + (y + 1)f (y)
holds for all real numbers x, y
Let $N_0=\{0, 1, 2 \cdots \}$. Find all functions: $N_0 \to N_0$ such that:
(1) $f(n) < f(n+1)$, all $n \in N_0$;
(2) $f(2)=2$;
(3) $f(mn)=f(m)f(n)$, all $m, n \in N_0$.
Let $\mathbb{R}$ be the set of real numbers. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that \[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\] for every $x,y\in\mathbb{R}$. Assume that the inequality is strict for some $x_0,y_0\in\mathbb{R}$.
Prove that either $f(x)\geqslant 0$ for every $x\in\mathbb{R}$ or $f(x)\leqslant 0$ for every $x\in\mathbb{R}$.
Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(n+1) > f(f(n)).\]
Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ which satisfy the following conditions:
$(\text{i})$ $f(x+f(y))=f(x)f(y)$ for all $x,y>0;$
$(\text{ii})$ there are at most finitely many $x$ with $f(x)=1$.
Let $ {\mathbb Q}^ \plus{}$ be the set of positive rational numbers. Construct a function $ f : {\mathbb Q}^ \plus{} \rightarrow {\mathbb Q}^ \plus{}$ such that
\[ f(xf(y)) \equal{} \frac {f(x)}{y}
\]
for all $ x$, $ y$ in $ {\mathbb Q}^ \plus{}$.
Determine the number of functions $f : Z^+ \to Z^+$ so that for all positive integers $x$ we have $f(f(x)) = f(x + 1)$, and $\max (f(2), . . . , f(14)) \le f(1) - 2 = 12$.
Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that
\[ m^2 + f(n) \mid mf(m) +n \]
for all positive integers $m$ and $n$.
Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that for all $x,y,z\in\mathbb{R}^+$
$$f(x+f(y)+f(f(z)))=z+f(y+f(x))$$
Find all functions $ f: \mathbb{Z} \to \mathbb{Z} $ such that $$ f\left(x+f\left(y\right)\right)f\left(y+f\left(x\right)\right)=\left(2x+f\left(y-x\right)\right)\left(2y+f\left(x-y\right)\right) $$ holds for all integers $ x,y $
Define the function $f: \mathbb{N} \rightarrow \mathbb{N}$ which satisfies, for any $s, n \in \mathbb{N}$, the following conditions:
$f(1) = f(2^s)$ and if $n < 2^s$, then $f(2^s + n) = f(n) + 1.$
Calculate the maximum value of $f(n)$ when $n \leq 2001$ and find the smallest natural number $n$ such that $f(n) = 2001.$
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 $f(x)\equiv x$?
Find all functions $f: \mathbb R^+ \rightarrow \mathbb R^+$ satisfying the following condition: for any three distinct real numbers $a,b,c$, a triangle can be formed with side lengths $a,b,c$, if and only if a triangle can be formed with side lengths $f(a),f(b),f(c)$.
Determine all functions $f: \mathbb R \to \mathbb R$, such that $$ f (f (x + y) - f (x - y)) = xy$$ for all real $x$ and $y$.
Let be two nonzero real numbers $ a,b, $ and a function $ f:\mathbb{R}\longrightarrow [0,\infty ) $ satisfying the functional equation
$$ f(x+a+b)+f(x)=f(x+a)+f(x+b) . $$
[b]1)[/b] Prove that $ f $ is periodic if $ a/b $ is rational.
[b]2)[/b] If $ a/b $ is not rational, could $ f $ be nonperiodic?
Find all functions $f:\mathbb{N} \to \mathbb{N}$ such that for each natural integer $n>1$ and for all $x,y \in \mathbb{N}$ the following holds:
$$f(x+y) = f(x) + f(y) + \sum_{k=1}^{n-1} \binom{n}{k}x^{n-k}y^k$$
In the following, let $N_0$ denotes the set of non-negative integers.
Find all polynomials $P(x)$ that fulfill the following two properties:
(1) All coefficients of $P(x)$ are from $N_0$.
(2) Exists a function $f : N_0 \to N_0$ such as $f (f (f (n))) = P (n)$ for all $n \in N_0$.
A function $f$ is defined on the positive integers by \[\left\{\begin{array}{rcl}f(1) &=& 1, \\ f(3) &=& 3, \\ f(2n) &=& f(n), \\ f(4n+1) &=& 2f(2n+1)-f(n), \\ f(4n+3) &=& 3f(2n+1)-2f(n), \end{array}\right.\] for all positive integers $n$. Determine the number of positive integers $n$, less than or equal to 1988, for which $f(n) = n$.