Found problems: 2349
Find all functions $ f:\mathbb{N} \rightarrow \mathbb{ N }_0 $ satisfy the following conditions:
i) $ f(ab)=f(a)+f(b)-f(\gcd(a,b)), \forall a,b \in \mathbb{N} $
ii) For all primes $ p $ and natural numbers $ a $, $ f(a)\ge f(ap) \Rightarrow f(a)+f(p) \ge f(a)f(p)+1 $
find all functions $f:\mathbb{Z} \to \mathbb{Z} $
such that $f(m+n)+f(m)f(n)=n^2(f(m)+1)+m^2(f(n)+1)+mn(2-mn)$ holds for all $m,n \in \mathbb{Z}$
Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$
Determine all functions $f: \mathbb R \to \mathbb R$ with the property
\[f(f(x)+y)=2x+f(f(y)-x), \quad \forall x,y \in \mathbb R.\]
A real number $a \ne 0$ is given. Determine all functions $f : R \to R$ satisfying $f(x)f(y) + f(x + y) = axy$ for all real numbers $x, y$.
Let $S \subset [0, 1]$ be a set of 5 points with $\{0, 1\} \subset S$. The graph of a real function $f : [0, 1] \to [0, 1]$ is continuous and increasing, and it is linear on every subinterval $I$ in $[0, 1]$ such that the endpoints but no interior points of $I$ are in $S$.
We want to compute, using a computer, the extreme values of $g(x, t) = \frac{f(x+t)-f(x)}{ f(x)-f(x-t)}$ for $x - t, x + t \in [0, 1]$. At how many points $(x, t)$ is it necessary to compute $g(x, t)$ with the computer?
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 $2ab = c^2 + d^2$.
Find all functions $f : R \to R$ that satisfy $f(x + y^2 - f(y)) = f(x)$ for all $x,y \in R$.
(Vo Quoc Ba Can)
Determine all polynomials $p$ with real coefficients for which $p(0)=0$ and
$$f(f(n))+n=4f(n)\qquad\text{for all }n\in\mathbb N,$$where $f(n)=\lfloor p(n)\rfloor$.
Find all functions $f : Z^+ \to Z^+$ such that $f(k + 1) >f(f(k))$ for $k > 1$, where $Z^+$ is the set of positive integers.
Let $f(x)$ be a function such that $f(1) = 1234$, $f(2)=1800$, and $f(x) = f(x-1) + 2f(x-2)-1$ for all integers $x$. Evaluate the number of divisors of
\[\sum_{i=1}^{2022}f(i)\]
[i]2022 CCA Math Bonanza Tiebreaker Round #4[/i]
Let $g:\mathbb{N}\to \mathbb{N}$ be a bijective function and suppose that $f:\mathbb{N}\to \mathbb{N}$ is a function such that:
[list]
[*] For all naturals $x$, $$\underbrace{f(\cdots (f}_{x^{2023}\;f\text{'s}}(x)))=x. $$
[*] For all naturals $x,y$ such that $x|y$, we have $f(x)|g(y)$.
[/list]
Prove that $f(x)=x$.
[i]Proposed by Pulkit Sinha[/i]
Find all functions $f:\mathbb{Q}\rightarrow\mathbb{R} \setminus \{ 0 \}$ such that
\[(f(x))^2f(2y)+(f(y))^2f(2x)=2f(x)f(y)f(x+y)\]
for all $x,y\in\mathbb{Q}$
[color=darkred]Find all functions $f:\mathbb{R}\to\mathbb{R}$ with the following property: for any open bounded interval $I$, the set $f(I)$ is an open interval having the same length with $I$ .[/color]
Find all $ f: Q^ \plus{} \to\ Z$ functions that satisfy $ f \left(\frac {1}{x} \right) \equal{} f(x)$ and $ (x \plus{} 1)f(x \minus{} 1) \equal{} xf(x)$ for all rational numbers that are bigger than 1.
Find all finite and non-empty sets $A$ of functions $f: \mathbb{R} \mapsto \mathbb{R}$ such that for all $f_1, f_2 \in A$, there exists $g \in A$ such that for all $x, y \in \mathbb{R}$
$$f_1 \left(f_2 (y)-x\right)+2x=g(x+y)$$
Let $A=\{1,2,\ldots,2n\}$ and let the function $g:A\to A$ be defined by $g(k)=2n-k+1$. Does there exist a function $f:A\to A$ such that $f(k)\ne g(k)$ and $f(f(f(k)))=g(k)$ for all $k\in A$, if (a) $n=999$; (b) $n=1000$?
It is given function $f : A \rightarrow \mathbb{R}$, $(A\subseteq \mathbb{R})$ such that $$f(x+y)=f(x)\cdot f(y)-f(xy)+1; (\forall x,y \in A)$$ If $f : A \rightarrow \mathbb{R}$, $(\mathbb{N} \subseteq A\subseteq \mathbb{R})$ is solution of given functional equation, prove that: $$f(n)=\begin{cases}
\frac{c^{n+1}-1}{c-1} \text{, } \forall n \in \mathbb{N}, c \neq 1 \\
n+1 \text{, } \forall n \in \mathbb{N}, c = 1
\end{cases}$$
where $c=f(1)-1$
$a)$ Solve given functional equation for $A=\mathbb{N}$
$b)$ With $A=\mathbb{Q}$, find all functions $f$ which are solutions of the given functional equation and also $f(1997) \neq f(1998)$
Find all binary operations $\diamondsuit: \mathbb R_{>0}\times \mathbb R_{>0}\to \mathbb R_{>0}$ (meaning $\diamondsuit$ takes pairs of positive real numbers to positive real numbers) such that for any real numbers $a, b, c > 0$,
[list]
[*] the equation $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$ holds; and
[*] if $a\ge 1$ then $a\,\diamondsuit\, a\ge 1$.
[/list]
[i]Evan Chen[/i]
Prove that there is no polynomial $P(x)$ with real coefficients that satisfies
$$P'(x)P''(x)>P(x)P'''(x)\qquad\text{for all }x\in\mathbb R.$$Is this statement true for all of the thrice differentiable real functions?
Let $n$ be a positive integer, and denote by $\mathbb{Z}_n$ the ring of integers modulo $n$. Suppose that there exists a function $f:\mathbb{Z}_n\to\mathbb{Z}_n$ satisfying the following three properties:
(i) $f(x)\neq x$,
(ii) $f(f(x))=x$,
(iii) $f(f(f(x+1)+1)+1)=x$ for all $x\in\mathbb{Z}_n$.
Prove that $n\equiv 2 \pmod4$.
(Proposed by Ander Lamaison Vidarte, Berlin Mathematical School, Germany)
Find all functions $f\colon\mathbb R\to\mathbb R$ such that for all real numbers $x$ and $y$,
\[f(xf(y))+f(y)=f(x+y)+f(xy).\]
[i]Milan Haiman[/i]
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]
Find all monotonous functions $ f: \mathbb{R} \to \mathbb{R}$ that satisfy the following functional equation:
\[f(f(x)) \equal{} f( \minus{} f(x)) \equal{} f(x)^2.\]
Let $c_1,c_2 \in \mathbb{R^+}$. Find all $f : \mathbb{R^+} \rightarrow \mathbb{R^+}$ such that for all $x,y \in \mathbb{R^+}$ $$f(x+c_1f(y))=f(x)+c_2f(y)$$