Found problems: 126
2022 Miklós Schweitzer, 5
Is it possible to select a non-degenerate segment from each line of the plane such that any two selected segments are disjoint?
1975 Putnam, B4
Does a circle have a subset which is topologically closed and which contains exactly one point of each pair of diametrically opposite points?
2012 Pre-Preparation Course Examination, 4
Suppose that $X$ and $Y$ are metric spaces and $f:X \longrightarrow Y$ is a continious function. Also $f_1: X\times \mathbb R \longrightarrow Y\times \mathbb R$ with equation $f_1(x,t)=(f(x),t)$ for all $x\in X$ and $t\in \mathbb R$ is a closed function. Prove that for every compact set $K\subseteq Y$, its pre-image $f^{pre}(K)$ is a compact set in $X$.
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 $\zeta$.
2007 Pre-Preparation Course Examination, 1
a) There is an infinite sequence of $0,1$, like $\dots,a_{-1},a_{0},a_{1},\dots$ (i.e. an element of $\{0,1\}^{\mathbb Z}$). At each step we make a new sequence. There is a function $f$ such that for each $i$, $\mbox{new }a_{i}=f(a_{i-100},a_{i-99},\dots,a_{i+100})$. This operation is mapping $F: \{0,1\}^{\mathbb Z}\longrightarrow\{0,1\}^{\mathbb Z}$. Prove that if $F$ is 1-1, then it is surjective.
b) Is the statement correct if we have an $f_{i}$ for each $i$?
2010 Paenza, 4
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function with the following property: for all $\alpha \in \mathbb{R}_{>0}$, the sequence $(a_n)_{n \in \mathbb{N}}$ defined as $a_n = f(n\alpha)$ satisfies $\lim_{n \to \infty} a_n = 0$. Is it necessarily true that $\lim_{x \to +\infty} f(x) = 0$?
2000 Miklós Schweitzer, 9
Let $M$ be a closed, orientable $3$-dimensional differentiable manifold, and let $G$ be a finite group of orientation preserving diffeomorphisms of $M$. Let $P$ and $Q$ denote the set of those points of $M$ whose stabilizer is nontrivial (that is, contains a nonidentity element of $G$) and noncyclic, respectively. Let $\chi (P)$ denote the Euler characteristic of $P$. Prove that the order of $G$ divides $\chi (P)$, and $Q$ is the union of $-2\frac{\chi(P)}{|G|}$ orbits of $G$.
1970 Miklós Schweitzer, 5
Prove that two points in a compact metric space can be joined with a rectifiable arc if and only if there exists a positive number $ K$ such that, for any $ \varepsilon>0$, these points can be connected with an $ \varepsilon$-chain not longer that $ K$.
[i]M. Bognar[/i]
1978 Miklós Schweitzer, 9
Suppose that all subspaces of cardinality at most $ \aleph_1$ of a topological space are second-countable. Prove that the whole space is second-countable.
[i]A. Hajnal, I. Juhasz[/i]
2011 Iran MO (3rd Round), 2
Prove that these three statements are equivalent:
(a) For every continuous function $f:S^n \to \mathbb R^n$, there exists an $x\in S^n$ such that $f(x)=f(-x)$.
(b) There is no antipodal mapping $f:S^n \to S^{n-1}$.
(c) For every covering of $S^n$ with closed sets $A_0,\dots,A_n$, there exists an index $i$ such that $A_i\cap -A_i\neq \emptyset$.
2005 Iran MO (3rd Round), 6
Suppose $A\subseteq \mathbb R^m$ is closed and non-empty. Let $f:A\to A$ is a lipchitz function with constant less than 1. (ie there exist $c<1$ that $|f(x)-f(y)|<c|x-y|,\ \forall x,y \in A)$. Prove that there exists a unique point $x\in A$ such that $f(x)=x$.
2009 IMS, 3
Let $ A\subset \mathbb C$ be a closed and countable set. Prove that if the analytic function $ f: \mathbb C\backslash A\longrightarrow \mathbb C$ is bounded, then $ f$ is constant.
2010 All-Russian Olympiad, 1
Let $a \neq b a,b \in \mathbb{R}$ such that $(x^2+20ax+10b)(x^2+20bx+10a)=0$ has no roots for $x$. Prove that $20(b-a)$ is not an integer.
2011 Pre-Preparation Course Examination, 2
prove that $\pi_1 (X,x_0)$ is not abelian. $X$ is like an eight $(8)$ figure.
[b]comments:[/b] eight figure is the union of two circles that have one point $x_0$ in common.
we call a group $G$ abelian if: $\forall a,b \in G:ab=ba$.
2009 All-Russian Olympiad, 6
Given a finite tree $ T$ and isomorphism $ f: T\rightarrow T$. Prove that either there exist a vertex $ a$ such that $ f(a)\equal{}a$ or there exist two neighbor vertices $ a$, $ b$ such that $ f(a)\equal{}b$, $ f(b)\equal{}a$.
1964 Miklós Schweitzer, 8
Let $ F$ be a closed set in the $ n$-dimensional Euclidean space. Construct a function that is $ 0$ on $ F$, positive outside $ F$ , and whose partial derivatives all exist.
1959 Miklós Schweitzer, 2
[b]2.[/b] Omit the vertices of a closed rectangle; the configuration obtained in such a way will be called a reduced rectangle. Prove tha the set-union of any system of reduced rectangles with parallel sides is equal to the union of countably many elements of the system. [b](St. 3)[/b]
1976 Miklós Schweitzer, 10
Suppose that $ \tau$ is a metrizable topology on a set $ X$ of cardinality less than or equal to continuum. Prove that there exists a separable and metrizable topology on $ X$ that is coarser that $ \tau$.
[i]L. Juhasz[/i]
2001 SNSB Admission, 5
Find the fundamental group of the topology of $ \text{SL}_2\left(\mathbb{R}\right) $ on $ \mathbb{R}^4. $
2007 IMS, 8
Let \[T=\{(tq,1-t) \in\mathbb R^{2}| t \in [0,1],q\in\mathbb Q\}\]Prove that each continuous function $f: T\longrightarrow T$ has a fixed point.
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 $[0,1]$.)
2005 Alexandru Myller, 3
Let $f:[0,\infty)\to\mathbb R$ be a continuous function s.t. $\lim_{x\to\infty}\frac {f(x)}x=0$. Let $(x_n)_n$ be a sequence of positive real numbers s.t. $\left(\frac{x_n}n\right)_n$ is bounded. Prove that $\lim_{n\to\infty}\frac{f(x_n)}n=0$.
[i]Dorin Andrica, Eugen Paltanea[/i]
2012 Pre - Vietnam Mathematical Olympiad, 2
Compute $\mathop {\lim }\limits_{n \to \infty } \left\{ {{{\left( {2 + \sqrt 3 } \right)}^n}} \right\}$
2000 Romania Team Selection Test, 3
Let $S$ be the set of interior points of a sphere and $C$ be the set of interior points of a circle. Find, with proof, whether there exists a function $f:S\rightarrow C$ such that $d(A,B)\le d(f(A),f(B))$ for any two points $A,B\in S$ where $d(X,Y)$ denotes the distance between the points $X$ and $Y$.
[i]Marius Cavachi[/i]
1981 Miklós Schweitzer, 9
Let $ n \geq 2$ be an integer, and let $ X$ be a connected Hausdorff space such that every point of $ X$ has a neighborhood homeomorphic to the Euclidean space $ \mathbb{R}^n$. Suppose that any discrete (not necessarily closed ) subspace $ D$ of $ X$ can be covered by a family of pairwise disjoint, open sets of $ X$ so that each of these open sets contains precisely one element of $ D$. Prove that $ X$ is a union of at most $ \aleph_1$ compact subspaces.
[i]Z. Balogh[/i]