Found problems: 2349
Prove that there exists a function $f : \mathbb{N} \mapsto \mathbb{N}$ that satisfies the following:
[color=#FFFFFF]___[/color]1. For all positive integers $m, n$ we have \[\gcd(|f(m)-f(n)|, f(mn)) = f(\gcd(m, n))\]
[color=#FFFFFF]___[/color]2. For all positive integers $m$, we have $f(f(m)) = f(m)$.
[color=#FFFFFF]___[/color]3. For all positive integers $k$, there exists a positive integer $n$ with $2024^{k} \mid f(n)$.
[i]Proposed by MV Adhitya, Archit Manas[/i]
Let $a, b, c \in \mathbb{R}$ be such that
$$a + b + c = a^2 + b^2 + c^2 = 1, \hspace{8px} a^3 + b^3 + c^3 \neq 1.$$
We say that a function $f$ is a [i]Palić function[/i] if $f: \mathbb{R} \rightarrow \mathbb{R}$, $f$ is continuous and satisfies
$$f(x) + f(y) + f(z) = f(ax + by + cz) + f(bx + cy + az) + f(cx + ay + bz)$$
for all $x, y, z \in \mathbb{R}.$
Prove that any Palić function is infinitely many times differentiable and find all Palić functions.
Let $a$ be rational and $b,c,d$ are real numbers, and let $f: \mathbb{R}\to [-1.1]$ be a function satisfying $f(x+a+b)-f(x+b)=c[x+2a+[x]-2[x+a]-[b]]+d$ for all $x$. Show that $f$ is periodic.
Find all functions $f: Z \to Z$ that satisfy $$f(-f (x) - f (y))= 1 -x - y$$ for all $x, y \in Z$
Determine all functions $f: \mathbb{Q} \to \mathbb{Q}$ such that
$$f(2xy + \frac{1}{2}) + f(x-y) = 4f(x)f(y) + \frac{1}{2}$$
for all $x,y \in \mathbb{Q}$.
Find all functions \( f : \mathbb{N} \to \mathbb{N} \) that satisfy the following condition: for any positive integers \( m \) and \( n \) such that \( m > n \) and \( m \) is not divisible by \( n \), if we denote by \( r \) the remainder of the division of \( m \) by \( n \), then the remainder of the division of \( f(m) \) by \( n \) is \( f(r) \).
[i]Proposed by Mykyta Kharin[/i]
Determine all real-valued functions $f$ on the set of real numbers satisfying
\[2f(x)=f(x+y)+f(x+2y)\]
for all real numbers $x$ and all non-negative real numbers $y$.
Find all positive integers $ k$ for which the following statement is true: If $ F(x)$ is a polynomial with integer coefficients satisfying the condition $ 0 \leq F(c) \leq k$ for each $ c\in \{0,1,\ldots,k \plus{} 1\}$, then $ F(0) \equal{} F(1) \equal{} \ldots \equal{} F(k \plus{} 1)$.
We call a function $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ [i]good[/i] if for all $x,y \in \mathbb{Q}^+$ we have: $$f(x)+f(y)\geq 4f(x+y).$$
a) Prove that for all good functions $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ and $x,y,z \in \mathbb{Q}^+$ $$f(x)+f(y)+f(z) \geq 8f(x+y+z)$$
b) Does there exists a good functions $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ and $x,y,z \in \mathbb{Q}^+$ such that $$f(x)+f(y)+f(z) < 9f(x+y+z) ?$$
Determine all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying $f(x+y)\le f(x)+f(y)\le x+y$ for all $x,y\in\mathbb{R}$.
Let $\mathbb R$ be the set of real numbers. We denote by $\mathcal F$ the set of all functions $f\colon\mathbb R\to\mathbb R$ such that
$$f(x + f(y)) = f(x) + f(y)$$
for every $x,y\in\mathbb R$ Find all rational numbers $q$ such that for every function $f\in\mathcal F$, there exists some $z\in\mathbb R$ satisfying $f(z)=qz$.
$\textbf{N5.}$ Find all $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for all $a,b,c \in \mathbb{N}$
$f(a)+f(b)+f(c)-ab-bc-ca \mid af(a)+bf(b)+cf(c)-3abc$
Find all injective functions $f\colon \mathbb{R}^* \to \mathbb{R}^* $ from the non-zero reals to the non-zero reals, such that \[f(x+y) \left(f(x) + f(y)\right) = f(xy)\] for all non-zero reals $x, y$ such that $x+y \neq 0$.
Find all differentiable functions $f: (0,\infty)\mapsto (0,\infty)$ for which there is a positive real number $a$ such that
\[ f'\left(\frac ax\right)=\frac x{f(x)} \]
for all $x>0.$
Let $f:\mathbb{R}\to\mathbb{R}^+$ be a continuous and periodic function. Prove that for all $\alpha\in\mathbb{R}$ the following inequality holds:
$\int_0^T\frac{f(x)}{f(x+\alpha)}dx\ge T$,
where $T$ is the period of $f(x)$.
Find all polynomials $P(x)$ with real coefficieints such that $P\left(x^2\right)=P(x)P(x-1)$ for all $x\in\mathbb R$.
Find all functions $~$ $f: \mathbb{R} \to \mathbb{R}$ $~$ which satisfy the equation
\[ f(x+yf(x))\, =\, f(f(x)) + xf(y)\, . \]
Find all functions $f: Z \to Z$ that satisfy $$f(-f (x) - f (y))= 1 -x - y$$ for all $x, y \in Z$
Find all functions $f \colon \mathbb{R} \to \mathbb{R}$ that satisfy the inequality
\[ f(y) - \left(\frac{z-y}{z-x} f(x) + \frac{y-x}{z-x}f(z)\right) \leq f\left(\frac{x+z}{2}\right) - \frac{f(x)+f(z)}{2} \]
for all real numbers $x < y < z$.
[i]Proposed by Gabriel Carroll[/i]
For a given natural number $k > 1$, find all functions $f:\mathbb{R} \to \mathbb{R}$ such that for all $x, y \in \mathbb{R}$, $f[x^k + f(y)] = y +[f(x)]^k$.
Find all functions $f$ defined on the integers greater than or equal to $1$ that satisfy:
(a) for all $n,f(n)$ is a positive integer.
(b) $f(n + m) =f(n)f(m)$ for all $m$ and $n$.
(c) There exists $n_0$ such that $f(f(n_0)) = [f(n_0)]^2$ .
Find all functions $f:\mathbb{N}^* \rightarrow \mathbb{N}$ such that
[list]
[*] $f(x \cdot y) = f(x) + f(y)$
[*] $f(30) = 0$
[*] $f(x)=0$ always when the units digit of $x$ is $7$
[/list]
Prove that there is a function $f$ from the set of all natural numbers into itself such that $f(f(n))=n^2$ for all $n \in \mathbb{N}$.
Find all functions $f: Q \to R$ satisfying $f (x + y) = f (x)f (y) - f(xy) + 1$ for all $x,y \in Q$
If $f$ is a function from the set of positive integers to itself such that $f(x) \leq x^2$ for all natural $x$, and $f\left( f(f(x)) f(f(y))\right) = xy$ for all naturals $x$ and $y$. Find the number of possible values of $f(30)$.
[i]Author: Alex Zhu[/i]