Found problems: 4776
Fix an integer $ b \geq 2$. Let $ f(1) \equal{} 1$, $ f(2) \equal{} 2$, and for each $ n \geq 3$, define $ f(n) \equal{} n f(d)$, where $ d$ is the number of base-$ b$ digits of $ n$. For which values of $ b$ does
\[ \sum_{n\equal{}1}^\infty \frac{1}{f(n)}
\]
converge?
Let $f$ be a function with the following properties:
1) $f(n)$ is defined for every positive integer $n$;
2) $f(n)$ is an integer;
3) $f(2)=2$;
4) $f(mn)=f(m)f(n)$ for all $m$ and $n$;
5) $f(m)>f(n)$ whenever $m>n$.
Prove that $f(n)=n$.
Let $ f:\mathbb{R}\longrightarrow\mathbb{R} ,f(x)=\left\{\begin{matrix} \sin x , & x\not\in\mathbb{Q} \\ 0, & x\in\mathbb{Q}\end{matrix}\right. . $
[b]a)[/b] Determine the maximum length of an interval $ I\subset\mathbb{R} $ such that $ f|_I $ is discontinuous everywhere, yet has the intermediate value property.
[b]b)[/b] Study the convergence of the sequence $ \left( x_n\right)_{n\in\mathbb{N}\cup\{ 0\}} $ defined by $ x_0\in (0,\pi /2),x_{n+1}=f\left( x_n\right),\forall n\ge 0. $
Let $ f:[0,1]\longrightarrow\mathbb{R} $ a continuous function. Prove that
$$ \int_0^1 f^2\left( x^2 \right) dx\ge \frac{3}{4}\left( \int_0^1 f(x)dx \right)^2 , $$
and find the circumstances under which equality happens.
A $(2n + 1) \times (2n + 1)$ grid, with $n> 0$, is colored in such a way that each of the cell is white or black. A cell is called [i]special[/i] if there are at least $n$ other cells of the same color in its row, and at least another $n$ cells of the same color in its column.
(a) Prove that there are at least $2n + 1$ special boxes.
(b) Provide an example where there are at most $4n$ special cells.
(c) Determine, as a function of $n$, the minimum possible number of special cells.
([b]5[/b]) Find $ p$ so that $ \lim_{x\rightarrow\infty}x^p\left(\sqrt[3]{x\plus{}1}\plus{}\sqrt[3]{x\minus{}1}\minus{}2\sqrt[3]{x}\right)$ is some non-zero real number.
Set $\mathbb Q_1=\{x\in\mathbb Q\mid x\ge1\}$. Suppose that a function $f:\mathbb Q_1\to\mathbb R$ satisfies the inequality $\left|f(x+y)-f(x)-f(y)\right|<\epsilon$ for all $x,y\in\mathbb Q_1$, where $\epsilon>0$ is given. Prove that there exists a real number $q$ such that
$$\left|\frac{f(x)}x-q\right|<2\epsilon\qquad\text{for all }x\in\mathbb Q_1.$$
Let $ ABC$ be an acute-angled triangle. Let $ L$ be any line in the plane of the triangle $ ABC$. Denote by $ u$, $ v$, $ w$ the lengths of the perpendiculars to $ L$ from $ A$, $ B$, $ C$ respectively. Prove the inequality $ u^2\cdot\tan A \plus{} v^2\cdot\tan B \plus{} w^2\cdot\tan C\geq 2\cdot S$, where $ S$ is the area of the triangle $ ABC$. Determine the lines $ L$ for which equality holds.
Find all surjective functions $f: \mathbb{N}\rightarrow \mathbb{N}$ such that for all $m,n\in \mathbb{N}$ \[f(m)\mid f(n) \mbox{ if and only if }m\mid n.\]
Find all functions $f:\mathbb{Z}\rightarrow\mathbb{C}$ such that $f(x(2y+1))=f(x(y+1))+f(x)f(y)$ holds for any two integers $x,y$.
Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
$$x^2f(x)+yf(y^2)=f(x+y)f(x^2-xy+y^2)$$
for all $x,y\in\mathbb{R}$.
Determine all functions $f:\mathbb{R}\to\mathbb{R}$, where $\mathbb{R}$ is the set of all real numbers, satisfying the following two conditions:
1) There exists a real number $M$ such that for every real number $x,f(x)<M$ is satisfied.
2) For every pair of real numbers $x$ and $y$,
\[ f(xf(y))+yf(x)=xf(y)+f(xy)\]
is satisfied.
Let $F$ be the set of all polynomials $\Gamma$ such that all the coefficients of $\Gamma (x)$ are integers and $\Gamma (x) = 1$ has integer roots. Given a positive intger $k$, find the smallest integer $m(k) > 1$ such that there exist $\Gamma \in F$ for which $\Gamma (x) = m(k)$ has exactly $k$ distinct integer roots.
Find all functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that verify, for any nonzero real number $ x $ the relation
$$ xf(x/a)-f(a/x)=b, $$
where $ a\neq 0,b $ are two real numbers.
[i]Dan Popescu[/i]
Find all functions $ f : Z\rightarrow Z$ for which we have $ f (0) \equal{} 1$ and $ f ( f (n)) \equal{} f ( f (n\plus{}2)\plus{}2) \equal{} n$, for every natural number $ n$.
We say that a set $S$ of positive integers is special when there exists a function $f : \mathbb{N}\to \mathbb{N}$ satisfying that:
$\bullet$ $f(k)\in S, \forall k\in\mathbb{N}$
$\bullet$ No integer $k$ with $2\le k \le 2021$ can be written as $\frac{af(a)}{bf(b)}$ with $a,b\in \mathbb{N}$
Find the smallest positive integer $n$ such that the set $S = \{ 1, 2021, 2021^2, \ldots , 2021^n \}$ is special or prove that no such integer exists.
Note: $\mathbb{N}$ represents the set $\{ 1, 2, 3, \ldots \}$
Let $r_1,r_2,\dots ,r_n$ be real numbers. Given $n$ reals $a_1,a_2,\dots ,a_n$ that are not all equal to $0$, suppose that inequality
\[r_1(x_1-a_1)+ r_2(x_2-a_2)+\dots + r_n(x_n-a_n)\leq\sqrt{x_1^2+ x_2^2+\dots + x_n^2}-\sqrt{a_1^2+a_2^2+\dots +a_n^2}\]
holds for arbitrary reals $x_1,x_2,\dots ,x_n$. Find the values of $r_1,r_2,\dots ,r_n$.
Let $f_0:[0,1]\to\mathbb R$ be a continuous function. Define the sequence of functions $f_n:[0,1]\to\mathbb R$ by
$$f_n(x)=\int^x_0f_{n-1}(t)dt$$
for all integers $n\ge1$.
a) Prove that the series $\sum_{n=1}^\infty f_n(x)$ is convergent for every $x\in[0,1]$.
b) Find an explicit formula for the sum of the series $\sum_{n=1}^\infty f_n(x),x\in[0,1]$.
Suppose that $x$ is measured in radians. Find the maximum value of \[\frac{\sin2x+\sin4x+\sin6x}{\cos2x+\cos4x+\cos6x}\] for $0\le x\le \frac{\pi}{16}$
Find all functions $f: \mathbb{N}\to\mathbb{N}$ which satisfy the inequality:
\[f(3x+2y)=f(x)f(y)\]
for all non-negative integers $x,y$.
Let $\mathbb N$ denote the set of positive integers. Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that:
(i) The greatest common divisor of the sequence $f(1), f(2), \dots$ is $1$.
(ii) For all sufficiently large integers $n$, we have $f(n) \neq 1$ and \[ f(a)^n \mid f(a+b)^{a^{n-1}} - f(b)^{a^{n-1}} \] for all positive integers $a$ and $b$.
[i]Proposed by Yang Liu[/i]
Find the derivatives of the lengths of the semiaxes of the ellipsoid $x^2 + y^2 + z^2 + xy + yz + zx = 1 + \epsilon xy$ with respect to $\epsilon$ at $\epsilon = 0$.
Prove that there exists a positive real number $C$ with the following property: for any integer $n\ge 2$ and any subset $X$ of the set $\{1,2,\ldots,n\}$ such that $|X|\ge 2$, there exist $x,y,z,w \in X$(not necessarily distinct) such that
\[0<|xy-zw|<C\alpha ^{-4}\]
where $\alpha =\frac{|X|}{n}$.
Prove that infinitely many positive integers cannot be written in the form \[{x_{1}}^{3}+{x_{2}}^{5}+{x_{3}}^{7}+{x_{4}}^{9}+{x_{5}}^{11},\] where $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\in \mathbb{N}$.
Let be a continuous function $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R} $ that has a root, and for which the line $ y=0 $ in the Cartesian plane is an horizontal asymptote. Show that $ f $ is bounded and touches its boundaries.
[i]Mihai Piticari[/i] and [i]Vladimir Cerbu[/i]