Found problems: 141
2008 Miklós Schweitzer, 9
For a given $\alpha >0$ let us consider the regular, non-vanishing $f(z)$ maps on the unit disc $\{ |z|<1 \}$ for which $f(0)=1$ and $\mathrm{Re}\, z\frac{f'(z)}{f(z)}>-\alpha$ ($|z|<1$). Show that the range of
$$g(z)=\frac{1}{(1-z)^{2\alpha}}$$
contains the range of all other such functions. Here we consider that regular branch of $g(z)$ for which $g(0)=1$.
(translated by Miklós Maróti)
2001 Miklós Schweitzer, 8
Let $H$ be a complex Hilbert space. The bounded linear operator $A$ is called [i]positive[/i] if $\langle Ax, x\rangle \geq 0$ for all $x\in H$. Let $\sqrt A$ be the positive square root of $A$, i.e. the uniquely determined positive operator satisfying $(\sqrt{A})^2=A$. On the set of positive operators we introduce the
$$A\circ B=\sqrt A B\sqrt B$$
operation. Prove that for a given pair $A, B$ of positive operators the identity
$$(A\circ B)\circ C=A\circ (B\circ C)$$
holds for all positive operator $C$ if and only if $AB=BA$.
2003 Miklós Schweitzer, 5
Let $d>1$ be integer and $0<r<\frac12$. Show that there exist finitely many (depending only on $d,r$) nonzero vectors in $\mathbb{R}^d$ such that if the distance of a straight line in $\mathbb{R}^d$ from the integer lattice $\mathbb{Z}^d$ is at least $r$, then this line is orthogonal to one of these finitely many vectors.
(translated by L. Erdős)
2014 Miklós Schweitzer, 9
Let $\rho:\mathbb{R}^n\to \mathbb{R}$, $\rho(\mathbf{x})=e^{-||\mathbf{x}||^2}$, and let $K\subset \mathbb{R}^n$ be a convex body, i.e., a compact convex set with nonempty interior. Define the barycenter $\mathbf{s}_K$ of the body $K$ with respect to the weight function $\rho$ by the usual formula
\[\mathbf{s}_K=\frac{\int_K\rho(\mathbf{x})\mathbf{x}d\mathbf{x}}{\int_K\rho(\mathbf{x})d\mathbf{x}}.\]
Prove that the translates of the body $K$ have pairwise distinct barycenters with respect to $\rho$.
2007 Miklós Schweitzer, 4
Let $p$ be a prime number and $a_1, \ldots, a_{p-1}$ be not necessarily distinct nonzero elements of the $p$-element $\mathbb Z_p \pmod{p}$ group. Prove that each element of $\mathbb Z_p$ equals a sum of some of the $a_i$'s (the empty sum is $0$).
(translated by Miklós Maróti)
2011 Miklós Schweitzer, 10
Let $X_0, \xi_{i, j}, \epsilon_k$ (i, j, k ∈ N) be independent, non-negative integer random variables. Suppose that $\xi_{i, j}$ (i, j ∈ N) have the same distribution, $\epsilon_k$ (k ∈ N) also have the same distribution.
$\mathbb{E}(\xi_{1,1})=1$ , $\mathbb{E}(X_0^l)<\infty$ , $\mathbb{E}(\xi_{1,1}^l)<\infty$ , $\mathbb{E}(\epsilon_1^l)<\infty$ for some $l\in\mathbb{N}$
Consider the random variable $X_n := \epsilon_n + \sum_{j=1}^{X_{n-1}} \xi_{n,j}$ (n ∈ N) , where $\sum_{j=1}^0 \xi_{n,j} :=0$
Introduce the sequence $M_n := X_n-X_{n-1}-\mathbb{E}(\epsilon_n)$ (n ∈ N)
Prove that there is a polynomial P of degree $\leq l/2$ such that $\mathbb{E}(M_n^l) = P_l(n)$ (n ∈ N).
2001 Miklós Schweitzer, 6
Let $I\subset \mathbb R$ be a non-empty open interval, $\varepsilon\geq 0$ and $f\colon I\rightarrow\mathbb R$ a function satisfying the
$$f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)+\varepsilon t(1-t)|x-y|$$
inequality for all $x,y\in I$ and $t\in [0,1]$. Prove that there exists a convex $g\colon I\rightarrow\mathbb R$ function, such that the function $l :=f-g$ has the $\varepsilon$-Lipschitz property, that is
$$|l(x)-l(y)|\leq \varepsilon|x-y|\text{ for all }x,y\in I$$
2006 Miklós Schweitzer, 3
G is a complete geometric graph such that for any 4-coloring of its edges, we can find n edges which are pairwise disjoint and have the same color. Prove that the minimum number of vertices of G is 6n-4.
[hide=idea]a graph with 6n-4 vertices has 2n-1 pairwise disjoint edges with 1 of 2 colors. by PP, there exist n pairwise disjoint edges of the same color. [/hide]
2004 Miklós Schweitzer, 7
Suppose that the closed subset $K$ of the sphere
$$S^2=\{ (x,y,z)\in \mathbb{R}^3\colon x^2+y^2+z^2=1 \}$$
is symmetric with respect to the origin and separates any two antipodal points in $S^2 \backslash K$. Prove that for any positive $\varepsilon$ there exists a homogeneous polynomial $P$ of odd degree such that the Hausdorff distance between
$$Z(P)=\{ (x,y,z)\in S^2 \colon P(x,y,z)=0\}$$
and $K$ is less than $\varepsilon$.
2016 Miklós Schweitzer, 5
Does there exist a piecewise linear continuous function $f:\mathbb{R}\to \mathbb{R}$ such that for any two-way infinite sequence $a_n\in[0,1]$, $n\in\mathbb{Z}$, there exists an $x\in\mathbb{R}$ with
\[
\limsup_{K\to \infty} \frac{\#\{k\le K\,:\, k\in\mathbb{N},f^k(x)\in[n,n+1)\}}{K}=a_n
\]
for all $n\in\mathbb{Z}$, where $f^k=f\circ f\circ \dots\circ f$ stands for the $k$-fold iterate of $f$?
2023 Miklós Schweitzer, 2
Let $G_0, G_1,\ldots$ be infinite open subsets of a Hausdorff space. Prove that there exist some infinite pairwise disjoint open sets $V_0,V_1,\ldots$ and some indices $n_0<n_1<\cdots$ such that $V_i\subseteq G_{n_i}$ for every $i\geqslant 0.$
2017 Miklós Schweitzer, 4
Let $K$ be a number field which is neither $\mathbb{Q}$ nor a quadratic imaginary extension of $\mathbb{Q}$. Denote by $\mathcal{L}(K)$ the set of integers $n\ge 3$ for which we can find units $\varepsilon_1,\ldots,\varepsilon_n\in K$ for which
$$\varepsilon_1+\dots+\varepsilon_n=0,$$but $\displaystyle\sum_{i\in I}\varepsilon_i\neq 0$ for any nonempty proper subset $I$ of $\{1,2,\dots,n\}$. Prove that $\mathcal{L}(K)$ is infinite, and that its smallest element can be bounded from above by a function of the degree and discriminant of $K$. Further, show that for infinitely many $K$, $\mathcal{L}(K)$ contains infinitely many even and infinitely many odd elements.
2019 Miklós Schweitzer, 8
Let $f: \mathbb{R} \to \mathbb{R}$ be a measurable function such that $f(x+t) - f(x)$ is locally integrable for every $t$ as a function of $x$. Prove that $f$ is locally integrable.
2012 Miklós Schweitzer, 8
For any function $f: \mathbb{R}^2\to \mathbb{R}$ consider the function $\Phi_f:\mathbb{R}^2\to [-\infty,\infty]$ for which $\Phi_f(x,y)=\limsup_{ z \to y} f(x,z)$ for any $(x,y) \in \mathbb{R}^2$.
[list=a]
[*]Is it true that if $f$ is Lebesgue measurable then $\Phi_f$ is also Lebesgue measurable?[/*]
[*]Is it true that if $f$ is Borel measurable then $\Phi_f$ is also Borel measurable?[/*]
[/list]
2002 Miklós Schweitzer, 1
For an arbitrary ordinal number $\alpha$ let $H(\alpha)$ denote the set of functions $f\colon \alpha \rightarrow \{ -1,0,1\}$ that map all but finitely many elements of $\alpha$ to $0$. Order $H(\alpha)$ according to the last difference, that is, for $f, g\in H(\alpha)$ let $f\prec g$ if $f(\beta) < g(\beta)$ holds for the maximum ordinal number $\beta < \alpha$ with $f(\beta) \neq g(\beta)$. Prove that the ordered set $(H(\alpha), \prec)$ is scattered (i.e. it doesn't contain a subset isomorphic to the set of rational numbers with the usual order), and that any scattered order type can be embedded into some $(H(\alpha), \prec)$.
1985 Miklós Schweitzer, 10
Show that any two intervals $A, B\subseteq \mathbb R$ of positive lengths can be countably disected into each other, that is, they can be written as countable unions $A=A_1\cup A_2\cup\ldots\,$ and $B=B_1\cup B_2\cup\ldots\,$ of pairwise disjoint sets, where $A_i$ and $B_i$ are congruent for every $i\in \mathbb N$ [Gy. Szabo]
2003 Miklós Schweitzer, 6
Show that the recursion $n=x_n(x_{n-1}+x_n+x_{n+1})$, $n=1,2,\ldots$, $x_0=0$ has exaclty one nonnegative solution.
(translated by L. Erdős)
2000 Miklós Schweitzer, 7
Let $H(D)$ denote the space of functions holomorphic on the disc $D=\{ z\colon |z|<1 \}$, endowed with the topology of uniform convergence on each compact subset of $D$. If $f(z)=\sum_{n=0}^{\infty} a_nz^n$, then we shall denote $S_n(f,z)=\sum_{k=0}^n a_kz^k$. A function $f\in H(D)$ is called [i]universal[/i] if, for every continuous function $g\colon\partial D\rightarrow \mathbb{C}$ and for every $\varepsilon >0$, there are partial sums $S_{n(j)}(f,z)$ approximating $g$ uniformly on the arc $\{ e^{it} \colon 0\le t\le 2\pi - \varepsilon\}$. Prove that the set of universal functions contains a dense $G_{\delta}$ subset of $H(D)$.
2008 Miklós Schweitzer, 8
Let $S$ be the Sierpiński triangle. What can we say about the Hausdorff dimension of the elevation sets $f^{-1}(y)$ for typical continuous real functions defined on $S$? (A property is satisfied for typical continuous real functions on $S$ if the set of functions not having this property is of the first Baire category in the metric space of continuous $S\rightarrow\mathbb{R}$ functions with the supremum norm.)
(translated by Miklós Maróti)
2002 Miklós Schweitzer, 3
Put $\mathbb{A}=\{ \mathrm{yes}, \mathrm{no} \}$. A function $f\colon \mathbb{A}^n\rightarrow \mathbb{A}$ is called a [i]decision function[/i] if
(a) the value of the function changes if we change all of its arguments; and
(b) the values does not change if we replace any of the arguments by the function value.
A function $d\colon \mathbb{A}^n \rightarrow \mathbb{A}$ is called a [i]dictatoric function[/i], if there is an index $i$ such that the value of the function equals its $i$th argument.
The [i]democratic function[/i] is the function $m\colon \mathbb{A}^3 \rightarrow \mathbb{A}$ that outputs the majority of its arguments.
Prove that any decision function is a composition of dictatoric and democratic functions.
1995 Miklós Schweitzer, 2
Given $f,g\in L^1[0,1]$ and $\int_0^1 f = \int_0^1 g=1$, prove that there exists an interval I st $\int_I f = \int_I g=\frac12$.
2018 Miklós Schweitzer, 4
Let $P$ be a finite set of points in the plane. Assume that the distance between any two points is an integer. Prove that $P$ can be colored by three colors such that the distance between any two points of the same color is an even number.
2012 Miklós Schweitzer, 3
There is a simple graph which chromatic number is equal to $k$. We painted all of the edges of graph using two colors. Prove that there exist a monochromatic tree with $k$ vertices
1985 Miklós Schweitzer, 8
Let $\frac{2}{\sqrt5+1}\leq p < 1$, and let the real sequence $\{ a_n \}$ have the following property: for every sequence $\{ e_n \}$ of $0$'s and $\pm 1$'s for which $\sum_{n=1}^\infty e_np^n=0$, we also have $\sum_{n=1}^\infty e_na_n=0$. Prove that there is a number $c$ such that $a_n=cp^n$ for all $n$. [Z. Daroczy, I. Katai]
1997 Miklós Schweitzer, 10
Assign independent standard normally distributed random variables to the vertices of an n-dimensional cube. Say one vertex is greater than another if the assigned number is greater. Define a random walk on the vertices according to the following rules:
a) the starting point is chosen from all the vertices with equal probability,
b) during our journey, if we reach a vertex such that there are adjacent vertices which have higher values, we choose the next vertex with equal probability,
c) if there is none, we stop.
Prove that $\forall\varepsilon>0 \,\exists K\, \forall n>1$
$$P(\lambda> K \log n) <\varepsilon$$
where $\lambda$ is the number of steps of the random walk.