Found problems: 4776
Find the number of ordered triples $(x,y,z)$ of non-negative integers satisfying
(i) $x \leq y \leq z$
(ii) $x + y + z \leq 100.$
Let $f(x)$ be a quadratic function with integer coefficients. If we know that $f(0)$, $f(3)$ and $f(4)$ are all different and elements of the set $\{2, 20, 202, 2022\}$, determine all possible values of $f(1)$.
Denote by $ S$ the set of all positive integers. Find all functions $ f: S \rightarrow S$ such that
\[ f (f^2(m) \plus{} 2f^2(n)) \equal{} m^2 \plus{} 2 n^2\]
for all $ m,n \in S$.
[i]Bulgaria[/i]
$f(n+f(n))=f(n)$ for every $n \in \mathbb{N}$.
a)Prove, that if $f(n)$ is finite, then $f$ is periodic.
b) Give example nonperiodic function.
PS. $0 \not \in \mathbb{N}$
Find all functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that verify the equality
$$ \int_a^b f(x)dx=f(b)-f(a), $$
for any real numbers $ a,b. $
[i]Cosmin Nitu[/i]
Let $f:[0,\infty)\rightarrow [0,\infty)$ be a continuous and bijective function, such that $$\lim_{x\rightarrow\infty}\frac{f^{-1}(f(x)/x)}{x}=1.$$
[list=a]
[*] Show that $\lim_{x\rightarrow\infty}\frac{f(x)}{x}=\infty$ and $\lim_{x\rightarrow\infty}\frac{f^{-1}(ax)}{f^{-1}(x)}=1$ for any $a>0$.
[*] Give an example of function which satisfies the hypothesis.
[b]a)[/b] Prove that there exist two functions $ f,g:\mathbb{R}\longrightarrow\mathbb{R} $ having the properties:
$ \text{(i)}\quad f\circ g=g\circ f $
$\text{(ii)}\quad f\circ f=g\circ g $
$ \text{(iii)}\quad f(x)\neq g(x), \quad \forall x\in\mathbb{R} $
[b]b)[/b] Show that if there are two functions $ f_1,g_1:\mathbb{R}\longrightarrow\mathbb{R} $ with the properties $ \text{(i)} $ and $ \text{(iii)} $ from above, then $ \left( f_1\circ f_1\right)(x) \neq \left( g_1\circ g_1 \right)(x) , $ for all real numbers $ x. $
The function $f(x)$ is defined for all $x\ge 0$ and is always nonnegative. It has the additional property that if any line is drawn from the origin with any positive slope $m$, it intersects the graph $y=f(x)$ at precisely one point, which is $\frac{1}{\sqrt{m}}$ units from the origin. Let $a$ be the unique real number for which $f$ takes on its maximum value at $x=a$ (you may assume that such an $a$ exists). Find $\int_{0}^{a}f(x) \, dx$.
A function $f_n(x)\ (n=1,\ 2,\ \cdots)$ is defined by $f_1(x)=x$ and
\[f_n(x)=x+\frac{e}{2}\int_0^1 f_{n-1}(t)e^{x-t}dt\ (n=2,\ 3,\ \cdots)\].
Find $f_n(x)$.
Find all functions $f : \mathbb{N} \to \mathbb{N}$ with
$$f(x) + yf(f(x)) < x(1 + f(y)) + 2021$$
holds for all positive integers $x,y.$
$\exists ? f: R\to R$ continuos function that:
$\forall x_0\in R \lim\limits_{x \to x_0} \frac{|f(x)-f(x_0)|}{|x-x_0|}=+\infty$
Let $S = \{1, 2, \dots, 9\}.$ Compute the number of functions $f : S \rightarrow S$ such that, for all $s \in S, f(f(f(s))) =s$ and $f(s) - s$ is not divisible by $3$.
Denote the set of positive rational numbers by $\mathbb{Q}_{+}$. Find all functions $f: \mathbb{Q}_{+}\rightarrow \mathbb{Q}_{+}$ that satisfy
$$\underbrace{f(f(f(\dots f(f}_{n}(q))\dots )))=f(nq)$$
for all integers $n\ge 1$ and rational numbers $q>0$.
Determine all strictly increasing functions $f: \mathbb{N}\to\mathbb{N}$ satisfying $nf(f(n))=f(n)^2$ for all positive integers $n$.
[i]Carl Lian and Brian Hamrick.[/i]
a) Prove that there are no one-to-one (injective) functions $f: \mathbb{N} \to \mathbb{N}\cup \{0\}$ such that
\[ f(mn) = f(m)+f(n) , \ \forall \ m,n \in \mathbb{N}. \]
b) Prove that for all positive integers $k$ there exist one-to-one functions $f: \{1,2,\ldots,k\}\to\mathbb{N}\cup \{0\}$ such that $f(mn) = f(m)+f(n)$ for all $m,n\in \{1,2,\ldots,k\}$ with $mn\leq k$.
[i]Mihai Baluna[/i]
Let $ABC$ be an isosceles triangle with right angle at point $A$. Find the minimum of the function $F$ given by
\[F(M) = BM +CM-\sqrt{3}AM\]
Functions $f$ and $g$ are defined so that $f(1) = 4$, $g(1) = 9$, and for each integer $n \ge 1$, $f(n+1) = 2f(n) + 3g(n) + 2n $ and $g(n+1) = 2g(n) + 3 f(n) + 5$. Find $f(2005) - g(2005)$.
Let $f$ be a real-valued differentiable function on the real line $\mathbb{R}$ such that
$\lim_{x\to 0} \frac{f(x)}{x^2}$ exists, and is finite . Prove that $f'(0)=0$.
Let $b = \tfrac 12 (-1 + 3\sqrt{5})$. Determine the number of rational numbers which can be written in the form \[ a_{2014}b^{2014} + a_{2013}b^{2013} + \dots + a_1b + a_0 \] where $a_0, a_1, \dots, a_{2014}$ are nonnegative integers less than $b$.
[i]Proposed by Michael Kural and Evan Chen[/i]
The number $0$ or $1$ is to be assigned to each of the $n$ vertices of a regular polygon. In how many different ways can this be done (if we consider two assignments that can be obtained one from the other through rotation in the plane of the polygon to be identical)?
Let $ f: (1,\infty)\to\mathbb{R}$ be a differentiable function such that
\[ f'(x)\equal{}\frac{x^2\minus{}\left(f(x)\right)^2}{x^2\left(\left(f(x)\right)^2\plus{}1\right)}\quad\text{for all }x>1.\]
Prove that $ \displaystyle\lim_{x\to\infty}f(x)\equal{}\infty.$
Let $A$ be the set of all polynomials $f(x)$ of order $3$ with integer coefficients and cubic coefficient $1$, so that for every $f(x)$ there exists a prime number $p$ which does not divide $2004$ and a number $q$ which is coprime to $p$ and $2004$, so that $f(p)=2004$ and $f(q)=0$.
Prove that there exists a infinite subset $B\subset A$, so that the function graphs of the members of $B$ are identical except of translations
Differentiable functions $f_1,\ldots,f_n:\mathbb R\to\mathbb R$ are linearly independent. Prove that there exist at least $n-1$ linearly independent functions among $f_1',\ldots,f_n'$.
Let $f:[0, \infty) \to \mathbb{R}$ a continuous function, constant on $\mathbb{Z}_{\geq 0}.$ For any $0 \leq a < b < c < d$ which satisfy $f(a)=f(c)$ and $f(b)=f(d)$ we also have $f \left( \frac{a+b}{2} \right) = f \left( \frac{c+d}{2} \right).$
Prove that $f$ is constant.
Let $a,b,c,x,y,z$ be six positive real numbers satisfying $x+y+z=a+b+c$ and $xyz=abc.$ Further, suppose that $a\leq x<y<z\leq c$ and $a<b<c.$ Prove that $a=x,b=y$ and $c=z.$