Found problems: 141
2012 Miklós Schweitzer, 7
Let $\Gamma$ be a simple curve, lying inside a circle of radius $r$, rectifiable and of length $\ell$. Prove that if $\ell > kr\pi$, then there exists a circle of radius $r$ which intersects $\Gamma$ in at least $k+1$ distinct points.
2004 Miklós Schweitzer, 9
Let $F$ be a smooth (i.e. $C^{\infty}$) closed surface. Call a continuous map $f\colon F\rightarrow \mathbb{R}^2$ an [i]almost-immersion[/i] if there exists a smooth closed embedded curve $\gamma$ (possibly disconnected) in $F$ such that $f$ is smooth and of maximal rank (i.e., rank 2) on $F\backslash \gamma$ and each point $p\in\gamma$ admits local coordinate charts $(x,y)$ and $(u,v)$ about $p$ and $f(p)$, respectively, such taht the coordinates of $p$ and $f(p)$ are zero and the map $f$ is given by $(x,y)\rightarrow (u,v), u=|x|, v=y$.
Determine the genera of those smooth, closed, connected, orientable surfaces $F$ that admit an almost-immersion in the plane with the curve $\gamma$ having a given positive number $n$ of connected components.
2020 Miklós Schweitzer, 9
Let $D\subseteq \mathbb{C}$ be a compact set with at least two elements and consider the space $\Omega=\bigtimes_{i=1}^{\infty} D$ with the product topology. For any sequence $(d_n)_{n=0}^{\infty} \in \Omega$ let $f_{(d_n)}(z)=\sum_{n=0}^{\infty}d_nz^n$, and for each point $\zeta \in \mathbb{C}$ with $|\zeta|=1$ we define $S=S(\zeta,(d_n))$ to be the set of complex numbers $w$ for which there exists a sequence $(z_k)$ such that $|z_k|<1$, $z_k \to \zeta$, and $f_{d_n}(z_k) \to w$. Prove that on a residual set of $\Omega$, the set $S$ does not depend on the choice of $\zeta$.
2004 Miklós Schweitzer, 2
Write $t(G)$ for the number of complete quadrilaterals in the graph $G$ and $e_G(S)$ for the number of edges spanned by a subset $S$ of vertices of $G$. Let $G_1, G_2$ be two (simple) graphs on a common underlying set $V$ of vertices, $|V|-n$, and assume that $|e_{G_1}(S)-e_{G_2}(S)|<\frac{n^2}{1000}$ holds for any subset $S\subseteq V$. Prove that $|t(G_1)-t(G_2)|\le \frac{n^4}{1000}$.
2008 Miklós Schweitzer, 5
Let $A$ be an infinite subset of the set of natural numbers, and denote by $\tau_A(n)$ the number of divisors of $n$ in $A$. Construct a set $A$ for which
$$\sum_{n\le x}\tau_A(n)=x+O(\log\log x)$$
and show that there is no set for which the error term is $o(\log\log x)$ in the above formula.
(translated by Miklós Maróti)
1986 Miklós Schweitzer, 7
Prove that the series $\sum_p c_p f(px)$, where the summation is over all primes, unconditionally converges in $L^2[0,1]$ for every $1$-periodic function $f$ whose restriction to $[0,1]$ is in $L^2[0,1]$ if and only if $\sum_p |c_p|<\infty$. ([i]Unconditional convergence[/i] means convergence for all rearrangements.) [G. Halasz]
2019 Miklós Schweitzer, 1
Prove that if every subspace of a Hausdorff space $X$ is $\sigma$-compact, then $X$ is countable.
2004 Miklós Schweitzer, 3
Prove that there is a constant $c>0$ such that for any $n>3$ there exists a planar graph $G$ with $n$ vertices such that every straight-edged plane embedding of $G$ has a pair of edges with ratio of lengths at least $cn$.
1986 Miklós Schweitzer, 9
Consider a latticelike packing of translates of a convex region $K$. Let $t$ be the area of the fundamental parallelogram of the lattice defining the packing, and let $t_{\min} (K)$ denote the minimal value of $t$ taken for all latticelike packings. Is there a natural number $N$ such that for any $n>N$ and for any $K$ different from a parallelogram, $nt_{\min} (K)$ is smaller that the area of any convex domain in which $n$ translates to $K$ can be placed without overlapping? (By a [i]latticelike packing[/i] of $K$ we mean a set of nonoverlapping translates of $K$ obtained from $K$ by translations with all vectors of a lattice.) [G. and L. Fejes-Toth]
2013 Miklós Schweitzer, 12
There are ${n}$ tokens in a pack. Some of them (at least one, but not all) are white and the rest are black. All tokens are extracted randomly from the pack, one by one, without putting them back. Let ${X_i}$ be the ratio of white tokens in the pack before the ${i^{\text{th}}}$ extraction and let
\[ \displaystyle T =\max \{ |X_i-X_j| : 1 \leq i \leq j \leq n\}.\]
Prove that ${\Bbb{E}(T) \leq H(\Bbb{E}(X_1))},$ where ${H(x)=-x\ln x -(1-x)\ln(1-x)}.$
[i]Proposed by Tamás Móri[/i]
2001 Miklós Schweitzer, 1
Let $f\colon 2^S\rightarrow \mathbb R$ be a function defined on the subsets of a finite set $S$. Prove that if $f(A)=F(S\backslash A)$ and $\max \{ f(A), f(B)\}\geq f(A\cup B)$ for all subsets $A, B$ of $S$, then $f$ assumes at most $|S|$ distinct values.
2004 Miklós Schweitzer, 10
Let $\mathcal{N}_p$ stand for a $p$ dimensional random variable of standard normal distribution. For $a\in\mathbb{R}^p$, let $H_p(a)$ stand for the expectation $E|\mathcal{N}_p+a|$. For $p>1$, prove that
$$H_p(a)=(p-1)\int_0^{\infty} H_1\left( \frac{|a|}{\sqrt{r^2+1}}\right) \frac{r^{p-2}}{\sqrt{(r^2+1)^p}} \mathrm{d}r$$
1986 Miklós Schweitzer, 1
If $(A, <)$ is a partially ordered set, its dimension, $\dim (A, <)$, is the least cardinal $\kappa$ such that there exist $\kappa$ total orderings $\{ <_{\alpha} \colon \alpha < \kappa \}$ on $A$ with $<=\cap_{\alpha < \kappa} <_\alpha$. Show that if $\dim (A, <)>\aleph_0$, then there exist disjoint $A_0, A_1\subseteq A$ with $\dim (A_0, <)$, $\dim (A_1, <)>\aleph_0$. [D. Kelly, A. Hajnal, B. Weiss]
1996 Miklós Schweitzer, 6
Let $\{a_n\}$ be a bounded real sequence.
(a) Prove that if X is a positive-measure subset of $\mathbb R$, then for almost all $x\in X$, there exist a subsequence $\{y_n\}$ of X such that $$\sum_{n=1}^\infty (n(y_n-x)-a_n)=1$$
(b) construct an unbounded sequence $\{a_n\}$ for which the above equation is also true.
2007 Miklós Schweitzer, 5
Let $D=\{ (x,y) \mid x>0, y\neq 0\}$ and let $u\in C^1(\overline {D})$ be a bounded function that is harmonic on $D$ and for which $u=0$ on the $y$-axis. Prove that $u$ is identically zero.
(translated by Miklós Maróti)
1986 Miklós Schweitzer, 3
(a) Prove that for every natural number $k$, there are positive integers $a_1<a_2<\ldots <a_k$ such that $a_i-a_j$ divides $a_i$ for all $1\leq i, j\leq k, i\neq j$.
(b) Show that there is an absolute constant $C>0$ such that $a_1>k^{Ck}$ for every sequence $a_1,\ldots, a_k$ of numbers that satisfy the above divisibility condition.
[A. Balogh, I. Z. Ruzsa]
2008 Miklós Schweitzer, 11
Let $\zeta_1, \ldots, \zeta_n$ be (not necessarily independent) random variables with normal distribution for which $E\zeta_j=0$ and $E\zeta_j^2\le 1$ for all $1\le j\le n$. Prove that
$$E\left( \max_{1\le j\le n} \zeta_j \right)\le\sqrt{2\log n}$$
(translated by Miklós Maróti)
2004 Miklós Schweitzer, 4
Determine all totally multiplicative and non-negative functions $f\colon\mathbb{Z}\rightarrow \mathbb{Z}$ with the property that if $a, b\in \mathbb{Z}$ and $b\neq 0$, then there exist integers $q$ and $r$ such that $a-qb+r$ and $f(r)<f(b)$.
2023 Miklós Schweitzer, 9
Let $C[-1,1]$ be the space of continuous real functions on the interval $[-1,1]$ with the usual supremum norm, and let $V{}$ be a closed, finite-codimensional subspace of $C[-1,1].$ Prove that there exists a polynomial $p\in V$ with norm at most one, which satisfies $p'(0)>2023.$
2016 Miklós Schweitzer, 2
Let $K=(V,E)$ be a finite, simple, complete graph. Let $d$ be a positive integer. Let $\phi:E\to \mathbb{R}^d$ be a map from the edge set to Euclidean space, such that the preimage of any point in the range defines a connected graph on the entire vertex set $V$, and the points assigned to the edges of any triangle in $K$ are collinear. Show that the range of $\phi$ is contained in a line.
2001 Miklós Schweitzer, 11
Let $\xi_{(k_1, k_2)}, k_1, k_2 \in\mathbb N$ be random variables uniformly bounded. Let $c_l, l\in\mathbb N$ be a positive real strictly increasing infinite sequence such that $c_{l+1}/ c_l$ is bounded. Let $d_l=\log \left(c_{l+1}/c_l\right), l\in\mathbb N$ and suppose that $D_n=\sum_{l=1}^n d_l\uparrow \infty$ when $n\to\infty$
Suppose there exist $C>0$ and $\varepsilon>0$ such that
$$\left| \mathbb E \left\{ \xi_{(k_1,k_2)}\xi_{(l_1,l_2)}\right\}\right| \leq C\prod_{i=1}^2 \left\{ \log_+\log_+\left( \frac{c_{\max\{ k_i, l_i\}}}{c_{\min\{ k_i, l_i\}}}\right)\right\}^{-(1+\varepsilon)}$$
for each $(k_1, k_2), (l_1,l_2)\in\mathbb N^2$ ($\log_+$ is the positive part of the natural logarithm). Show that
$$\lim_{\substack{n_1\to\infty \\ n_2\to\infty}} \frac{1}{D_{n_1}D_{n_2}}\sum_{k_1=1}^{n_1} \sum_{k_2=1}^{n_2} d_{k_1}d_{k_2}\xi_{(k_1,k_2)}=0$$
almost surely.
(translated by j___d)
2019 Miklós Schweitzer, 7
Given a polynomial $P$, assume that $L = \{z \in \mathbb{C}: |P(z)| = 1\}$ is a Jordan curve. Show that the zeros of $P'$ are in the interior of $L$.
2008 Miklós Schweitzer, 10
Let $V$ be the set of non-collinear pairs of vectors in $\mathbb{R}^3$, and $H$ be the set of lines passing through the origin. Is is true that for every continuous map $f\colon V\rightarrow H$ there exists a continuous map $g\colon V\rightarrow \mathbb{R}^3\,\backslash\,\{ 0\}$ such that $g(v)\in f(v)$ for all $v\in V$?
(translated by Miklós Maróti)
2001 Miklós Schweitzer, 9
Let $H$ be the hyperbolic plane, $I(H)$ be the isometry group of $H$, and $O\in H$ be a fixed starting point. Determine those continuous $\sigma\colon H\rightarrow I(H)$ mappings that satisfty the following three conditions:
(a) $\sigma(O)=\mathrm{id}$, and $\sigma (X)O=X$ for all $X\in H$;
(b) for every $X\in H\backslash \{ O\}$ point, the $\sigma(X)$ isometry is a paracyclic shift, i.e. every member of a system of paracycles through a common infinitely far point is left invariant;
(c) for any pair $P,Q\in H$ of points there exists a point $X\in H$ such that $\sigma(X)P=Q$.
Prove that the $\sigma\colon H\rightarrow I(H)$ mappings satisfying the above conditions are differentiable with the exception of a point.
2001 Miklós Schweitzer, 10
Show that if a connected, nowhere zero sectional curvature of Riemannian manifold, where symmetric (1,1)-tensor of the Levi-Civita connection covariant derivative vanishes, then the tensor is constant times the unit tensor.
(translated by j___d)