Found problems: 141
2001 Miklós Schweitzer, 5
Prove that if the function $f$ is defined on the set of positive real numbers, its values are real, and $f$ satisfies the equation
$$f\left( \frac{x+y}{2}\right) + f\left(\frac{2xy}{x+y} \right) =f(x)+f(y)$$
for all positive $x,y$, then
$$2f(\sqrt{xy})=f(x)+f(y)$$
for every pair $x,y$ of positive numbers.
1985 Miklós Schweitzer, 11
Let $\xi (E, \pi, B)\, (\pi\colon E\rightarrow B)$ be a real vector bundle of finite rank, and let
$$\tau_E=V\xi \oplus H\xi\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (*)$$
be the tangent bundle of $E$, where $V\xi=\mathrm{Ker}\, d\pi$ is the vertical subbundle of $\tau_E$. Let us denote the projection operators corresponding to the splitting $(*)$ by $v$ and $h$. Construct a linear connection $\nabla$ on $V\xi$ such that
$$\nabla_X\lor Y - \nabla_Y \lor X=v[X,Y] - v[hX,hY]$$
($X$ and $Y$ are vector fields on $E$, $[.,\, .]$ is the Lie bracket, and all data are of class $\mathcal C^\infty$. [J. Szilasi]
2007 Miklós Schweitzer, 9
Let $A$ and $B$ be two triangles on the plane such that the interior of both contains the origin and for each circle $C_r$ centered at the origin $|C_r \cap A|=|C_r\cap B|$ (where $|\cdot |$ is the arc-length measure). Prove that $A$ and $B$ are congruent. Does this statement remain true if the origin is on the border of $A$ or $B$?
(translated by Miklós Maróti)
2007 Miklós Schweitzer, 7
Prove that there exist natural numbers $n_k, m_k, k=0,1,2,\ldots$, such that the numbers $n_k+m_k, k=1,2,\ldots$ are pairwise distinct primes and the set of linear combination of the polynomials $x^{n_k}y^{m_k}$ is dense in $C([0,1] \times [0,1])$ under the supremum norm.
(translated by Miklós Maróti)
2002 Miklós Schweitzer, 7
Let the complex function $F(z)$ be regular on the punctuated disk $\{ 0<|z| < R\}$. By a [i]level curve[/i] we mean a component of the level set of $\mathrm{Re}F(z)$, that is, a maximal connected set on which $\mathrm{Re}F(z)$ is constant. Denote by $A(r)$ the union of those level curves that are entirely contained in the punctuated disk $\{ 0<|z|<r\}$. Prove that if the number of components of $A(r)$ has an upper bound independent of $r$ then $F(z)$ can only have a pole type singularity at $0$.
2004 Miklós Schweitzer, 6
Is is true that if the perfect set $F\subseteq [0,1]$ is of zero Lebesgue measure then those functions in $C^1[0,1]$ which are one-to-one on $F$ form a dense subset of $C^1[0,1]$?
(We use the metric
$$d(f,g)=\sup_{x\in[0,1]} |f(x)-g(x)| + \sup_{x\in[0,1]} |f'(x)-g'(x)|$$
to define the topology in the space $C^1[0,1]$ of continuously differentiable real functions on $[0,1]$.)
2010 Miklós Schweitzer, 11
For problem 11 , i couldn’t find the correct translation , so i just posted the hungarian version . If anyone could translate it ,i would be very thankful .
[tip=see hungarian]Az $X$ ́es$ Y$ valo ́s ́ert ́eku ̋ v ́eletlen v ́altoz ́ok maxim ́alkorrel ́acio ́ja az $f(X)$ ́es $g(Y )$ v ́altoz ́ok korrela ́cio ́j ́anak szupr ́emuma az olyan $f$ ́es $g$ Borel m ́erheto ̋, $\mathbb{R} \to \mathbb{R}$ fu ̈ggv ́enyeken, amelyekre $f(X)$ ́es $g(Y)$ v ́eges sz ́ora ́su ́. Legyen U a $[0,2\pi]$ interval- lumon egyenletes eloszl ́asu ́ val ́osz ́ınu ̋s ́egi v ́altozo ́, valamint n ́es m pozit ́ıv eg ́eszek. Sz ́am ́ıtsuk ki $\sin(nU)$ ́es $\sin(mU)$ maxim ́alkorrela ́ci ́oja ́t. [/tip]
Edit:
[hide=Translation thanks to @tintarn] The maximal correlation of two random variables $X$ and $Y$ is defined to be the supremum of the correlations of $f(X)$ and $g(Y)$ where $f,g:\mathbb{R} \to \mathbb{R}$ are measurable functions such that $f(X)$ and $g(Y)$ is (almost surely?) finite.
Let $U$ be the uniformly distributed random variable on $[0,2\pi]$ and let $m,n$ be positive integers. Compute the maximal correlation of $\sin(nU)$ and $\sin(mU)$.
(Remark: It seems that to make sense we should require that $E[f(X)]$ and $E[g(Y)]$ as well as $E[f(X)^2]$ and $E[g(Y)^2]$ are finite.
In fact, we may then w.l.o.g. assume that $E[f(X)]=E[g(Y)]=0$ and $E[f(Y)^2]=E[g(Y)^2]=1$.)[/hide]
1960 Miklós Schweitzer, 4
[b]4.[/b] Let $\left (H_{\alpha} \right ) $ be a system of sets of integers having the property that for any $\alpha _1 \neq \alpha _2 , H_{\alpha _1}\cap H_{\alpha _2}$ is a finite set and $H_{{\alpha} _1} \neq H_{{\alpha} _2}$. Prove that there exists a system $\left (H_{\alpha} \right )$ of this kind whose cardinality is that of the continuum. Prove further that if none of the intersections of two sets $H_\alpha$ contains more than $K$ elements, then the system $\left (H_{\alpha} \right ) $ is countable ($K$ is an arbitrary fixed integer). [b](St. 4)[/b]
2017 Miklós Schweitzer, 7
Characterize all increasing sequences $(s_n)$ of positive reals for which there exists a set $A\subset \mathbb{R}$ with positive measure such that $\lambda(A\cap I)<\frac{s_n}{n}$ holds for every interval $I$ with length $1/n$, where $\lambda$ denotes the Lebesgue measure.
2003 Miklós Schweitzer, 7
Let $r$ be a nonnegative continuous function on the real line. Show that there exists a function $f\in C^1(\mathbb{R})$, not identically zero, such that $f'(x)=f(x-r(f(x)))$, $x\in\mathbb{R}$.
(translated by L. Erdős)
2008 Miklós Schweitzer, 7
Let $f\colon \mathbb{R}^1\rightarrow \mathbb{R}^2$ be a continuous function such that $f(x)=f(x+1)$ for all $x$, and let $t\in [0,\frac14]$. Prove that there exists $x\in\mathbb{R}$ such that the vector from $f(x-t)$ to $f(x+t)$ is perpendicular to the vector from $f(x)$ to $f(x+\frac12)$.
(translated by Miklós Maróti)
2000 Miklós Schweitzer, 4
Let $a_1<a_2<a_3$ be positive integers. Prove that there are integers $x_1,x_2,x_3$ such that $\sum_{i=1}^3 |x_i | >0$, $\sum_{i=1}^3 a_ix_i= 0$ and
$$\max_{1\le i\le 3} | x_i|<\frac{2}{\sqrt{3}}\sqrt{a_3}+1$$.
2009 Miklós Schweitzer, 8
Let $ \{A_n\}_{n \in \mathbb{N}}$ be a sequence of measurable subsets of the real line which covers almost every point infinitely often. Prove, that there exists a set $ B \subset \mathbb{N}$ of zero density, such that $ \{A_n\}_{n \in B}$ also covers almost every point infinitely often. (The set $ B \subset \mathbb{N}$ is of zero density if $ \lim_{n \to \infty} \frac {\#\{B \cap \{0, \dots, n \minus{} 1\}\}}{n} \equal{} 0$.)
2016 Miklós Schweitzer, 9
For $p_0,\dots,p_d\in\mathbb{R}^d$, let
\[
S(p_0,\dots,p_d)=\left\{ \alpha_0p_0+\dots+\alpha_dp_d : \alpha_i\le 1, \sum_{i=0}^d \alpha_i =1 \right\}.
\]
Let $\pi$ be an arbitrary probability distribution on $\mathbb{R}^d$, and choose $p_0,\dots,p_d$ independently with distribution $\pi$. Prove that the expectation of $\pi(S(p_0,\dots,p_d))$ is at least $1/(d+2)$.
2007 Miklós Schweitzer, 1
Prove that there exist subfields of $\mathbb R$ that are
a) non-measurable and
b) of measure zero and continuum cardinality.
(translated by Miklós Maróti)
1961 Miklós Schweitzer, 4
[b]4.[/b] Let $f(x)$ be a real- or complex-value integrable function on $(0,1)$ with $\mid f(x) \mid \leq 1 $. Set
$ c_k = \int_0^1 f(x) e^{-2 \pi i k x} dx $
and construct the following matrices of order $n$:
$ T= (t_{pq})_{p,q=0}^{n-1}, T^{*}= (t_{pq}^{*})_{p,q =0}^{n-1} $
where $t_{pq}= c_{q-p}, t^{*}= \overline {c_{p-q}}$ . Further, consider the following hyper-matrix of order $m$:
$
S= \begin{bmatrix}
E & T & T^2 & \dots & T^{m-2} & T^{m-1} \\
T^{*} & E & T & \dots & T^{m-3} & T^{m-2} \\
T^{*2} & T^{*} & E & \dots & T^{m-3} & T^{m-2} \\
\dots & \dots & \dots & \dots & \dots & \dots \\
T^{*m-1} & T^{*m-2} & T^{*m-3} & \dots & T^{*} & E
\end{bmatrix} $
($S$ is a matrix of order $mn$ in the ordinary sense; E denotes the unit matrix of order $n$).
Show that for any pair $(m , n) $ of positive integers, $S$ has only non-negative real eigenvalues. [b](R. 19)[/b]
1985 Miklós Schweitzer, 6
Determine all finite groups $G$ that have an automorphism $f$ such that $H\not\subseteq f(H)$ for all proper subgroups $H$ of $G$. [B. Kovacs]
2003 Miklós Schweitzer, 9
Given fi nitely many open half planes on the Euclidean plane. The boundary lines of these half planes divide the plane into convex domains. Find a polynomial $C(q)$ of degree two so that the following holds: for any $q\ge 1$ integer, if the half planes cover each point of the plane at least $q$ times, then the set of points covered exactly $q$ times is the union of at most $C(q)$ domains.
(translated by L. Erdős)
1995 Miklós Schweitzer, 5
Let A be a subset of the set $\{1,2, ...,n\}$ with at least $100\sqrt n$ elements. Prove that there is a four-element arithmetic sequence in which each element is the sum of two different elements of the set A.
2018 Miklós Schweitzer, 3
We call an $n\times n$ matrix [i]well groomed[/i] if it only contains elements $0$ and $1$, and it does not contain the submatrix $\begin{pmatrix}
1& 0\\
0 & 1
\end{pmatrix}.$ Show that there exists a constant $c>0$ such that every well groomed, $n\times n$ matrix contains a submatrix of size at least $cn\times cn$ such that all of the elements of the submatrix are equal. (A well groomed matrix may contain the submatrix $\begin{pmatrix}
0& 1\\
1 & 0
\end{pmatrix}.$ )
2003 Miklós Schweitzer, 8
Let $f_1, f_2, \ldots$ be continuous real functions on the real line. Is it true that if the series $\sum_{n=1}^{\infty} f_n(x)$ is divergent for every $x$, then this holds also true for any typical choice of the signs in the sum (i.e. the set of those $\{ \epsilon _n\}_{n=1}^{\infty} \in \{ +1, -1\}^{\mathbb{N}}$ sequences, for which there series $\sum_{n=1}^{\infty} \epsilon_nf_n(x)$ is convergent at least at one point $x$, forms a subset of first category within the set $\{+1,-1\}^{\mathbb{N}} $)?
(translated by L. Erdős)
1976 Miklós Schweitzer, 1
Assume that $ R$, a recursive, binary relation on $ \mathbb{N}$ (the set of natural numbers), orders $ \mathbb{N}$ into type $ \omega$. Show that if $ f(n)$ is the $ n$th element of this order, then $ f$ is not necessarily recursive.
[i]L. Posa[/i]
2000 Miklós Schweitzer, 2
Let $n$ red and $n$ blue subarcs of a circle be given such that each red subarc intersects each blue subarc. Prove that there is a point which is covered by at least $n$ of the given (red or blue) subarcs.
1985 Miklós Schweitzer, 5
Let $F(x,y)$ and $G(x,y)$ be relatively prime homogeneous polynomials of degree at least one having integer coefficients. Prove that there exists a number $c$ depending only on the degrees and the maximum of the absolute values of the coefficients of $F$ and $G$ such that $F(x,y)\neq G(x,y)$ for any integers $x$ and $y$ that are relatively prime and satisfy $\max \{ |x|,|y|\} > c$. [K. Gyory]
2002 Miklós Schweitzer, 9
Let $M$ be a connected, compact $C^{\infty}$-differentiable manifold, and denote the vector space of smooth real functions on $M$ by $C^{\infty}(M)$. Let the subspace $V\le C^{\infty}(M)$ be invariant under $C^{\infty}$-diffeomorphisms of $M$, that is, let $f\circ h\in V$ for every $f\in V$ and for every $C^{\infty}$-diffeomorphism $h\colon M\rightarrow M$. Prove that if $V$ is different from the subspaces $\{ 0\}$ and $C^{\infty}(M)$ then $V$ only contains the constant functions.