Found problems: 73
1963 Miklós Schweitzer, 1
Show that the perimeter of an arbitrary planar section of a tetrahedron is less than the perimeter of one of the faces of the tetrahedron. [Gy. Hajos]
1975 Miklós Schweitzer, 12
Assume that a face of a convex polyhedron $ P$ has a common edge with every other face. Show that there exists a simple closed polygon that consists of edges of $ P$ and passes through all vertices.
[i]L .Lovasz[/i]
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]
1983 Miklós Schweitzer, 11
Let $ M^n \subset \mathbb{R}^{n\plus{}1}$ be a complete, connected hypersurface embedded into the Euclidean space. Show that $ M^n$ as a Riemannian manifold decomposes to a nontrivial global metric direct product if and only if it is a real cylinder, that is, $ M^n$ can be decomposed to a direct product of the form $ M^n\equal{}M^k \times \mathbb{R}^{n\minus{}k} \;(k<n)$ as well, where $ M^k$ is a hypersurface in some $ (k\plus{}1)$-dimensional subspace $ E^{k\plus{}1} \subset \mathbb{R}^{n\plus{}1} , \mathbb{R}^{n\minus{}k}$ is the orthogonal complement of $ E^{k\plus{}1}$.
[i]Z. Szabo[/i]
1966 Miklós Schweitzer, 2
Characterize those configurations of $ n$ coplanar straight lines for which the sum of angles between all pairs of lines is maximum.
[i]L.Fejes-Toth, A. Heppes[/i]
1972 Miklós Schweitzer, 3
Let $ \lambda_i \;(i=1,2,...)$ be a sequence of distinct positive numbers tending to infinity. Consider the set of all numbers representable in the form \[ \mu= \sum_{i=1}^{\infty}n_i\lambda_i ,\] where $ n_i \geq 0$ are integers and all but finitely many $ n_i$ are $ 0$. Let \[ L(x)= \sum _{\lambda_i \leq x} 1 \;\textrm{and}\ \;M(x)= \sum _{\mu \leq x} 1 \ .\] (In the latter sum, each $ \mu$ occurs as many times as its number of representations in the above form.) Prove that if \[ \lim_{x\rightarrow \infty} \frac{L(x+1)}{L(x)}=1,\] then \[ \lim_{x\rightarrow \infty} \frac{M(x+1)}{M(x)}=1.\]
[i]G. Halasz[/i]
1965 Miklós Schweitzer, 1
Let $ p$ be a prime, $ n$ a natural number, and $ S$ a set of cardinality $ p^n$ . Let $ \textbf{P}$ be a family of partitions of $ S$ into nonempty parts of sizes divisible by $ p$ such that the intersection of any two parts that occur in any of the partitions has at most one element. How large can $ |\textbf{P}|$ be?
1979 Miklós Schweitzer, 5
Give an example of ten different noncoplanar points $ P_1,\ldots ,P_5,Q_1,\ldots ,Q_5$ in $ 3$-space such that connecting each $ P_i$ to each $ Q_j$ by a rigid rod results in a rigid system.
[i]L. Lovasz[/i]
2006 IMS, 4
Assume that $X$ is a seperable metric space. Prove that if $f: X\longrightarrow\mathbb R$ is a function that $\lim_{x\rightarrow a}f(x)$ exists for each $a\in\mathbb R$. Prove that set of points in which $f$ is not continuous is countable.
1974 Miklós Schweitzer, 8
Prove that there exists a topological space $ T$ containing the real line as a subset, such that the Lebesgue-measurable functions, and only those, extend continuously over $ T$. Show that the real line cannot be an everywhere-dense subset of such a space $ T$.
[i]A. Csaszar[/i]
1977 Miklós Schweitzer, 1
Consider the intersection of an ellipsoid with a plane $ \sigma$ passing through its center $ O$. On the line through the point $ O$ perpendicular to $ \sigma$, mark the two points at a distance from $ O$ equal to the area of the intersection. Determine the loci of the marked points as $ \sigma$ runs through all such planes.
[i]L. Tamassy[/i]
1981 Miklós Schweitzer, 5
Let $ K$ be a convex cone in the $ n$-dimensional real vector space $ \mathbb{R}^n$, and consider the sets $ A\equal{}K \cup (\minus{}K)$ and $ B\equal{}(\mathbb{R}^n \setminus A) \cup \{ 0 \}$ ($ 0$ is the origin). Show that one can find two subspaces in $ \mathbb{R}^n$ such that together they span $ \mathbb{R}^n$, and one of them lies in $ A$ and the other lies in $ B$.
[i]J. Szucs[/i]
2009 Miklós Schweitzer, 7
Let $ H$ be an arbitrary subgroup of the diffeomorphism group $ \mathsf{Diff}^\infty(M)$ of a differentiable manifold $ M$. We say that an $ \mathcal C^\infty$-vector field $ X$ is [i]weakly tangent[/i] to the group $ H$, if there exists a positive integer $ k$ and a $ \mathcal C^\infty$-differentiable map $ \varphi \mathrel{: } \mathord{]} \minus{} \varepsilon,\varepsilon\mathord{[}^k\times M\to M$ such that
(i) for fixed $ t_1,\dots,t_k$ the map
\[ \varphi_{t_1,\dots,t_k} : x\in M\mapsto \varphi(t_1,\dots,t_k,x)\]
is a diffeomorphism of $ M$, and $ \varphi_{t_1,\dots,t_k}\in H$;
(ii) $ \varphi_{t_1,\dots,t_k}\in H \equal{} \mathsf{Id}$ whenever $ t_j \equal{} 0$ for some $ 1\leq j\leq k$;
(iii) for any $ \mathcal C^\infty$-function $ f: M\to \mathbb R$
\[ X f \equal{} \left.\frac {\partial^k(f\circ\varphi_{t_1,\dots,t_k})}{\partial t_1\dots\partial t_k}\right|_{(t_1,\dots,t_k) \equal{} (0,\dots,0)}.\]
Prove, that the commutators of $ \mathcal C^\infty$-vector fields that are weakly tangent to $ H\subset \textsf{Diff}^\infty(M)$ are also weakly tangent to $ H$.
1968 Miklós Schweitzer, 10
Let $ h$ be a triangle of perimeter $ 1$, and let $ H$ be a triangle of perimeter $ \lambda$ homothetic to $ h$. Let $ h_1,h_2,...$ be translates of $ h$ such that , for all $ i$, $ h_i$ is different from $ h_{i\plus{}2}$ and touches $ H$ and $ h_{i\plus{}1}$ (that is, intersects without overlapping). For which values of $ \lambda$ can these triangles be chosen so that the sequence $ h_1,h_2,...$ is periodic? If $ \lambda \geq 1$ is such a value, then determine the number of different triangles in a periodic
chain $ h_1,h_2,...$ and also the number of times such a chain goes around the triangle $ H$.
[i]L. Fejes-Toth[/i]
1966 Miklós Schweitzer, 1
Show that a segment of length $ h$ can go through or be tangent to at most $ 2\lfloor h/\sqrt{2}\rfloor\plus{}2$ nonoverlapping unit
spheres.
[i]L.Fejes-Toth, A. Heppes[/i]
2011 Pre-Preparation Course Examination, 3
prove that $1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+...=\frac{\pi}{4}$
1970 Miklós Schweitzer, 3
The traffic rules in a regular triangle allow one to move only along segments parallel to one of the altitudes of the triangle. We define the distance between two points of the triangle to be the length of the shortest such path between them. Put $ \binom{n\plus{}1}{2}$ points into the triangle in such a way that the minimum distance between pairs of points is maximal.
[i]L. Fejes-Toth[/i]
2011 Pre-Preparation Course Examination, 4
represent a way to calculate $\sum_{k=0}^{\infty}\frac{(-1)^k}{(2k+1)^3}=1-\frac{1}{3^3}+\frac{1}{5^3}-\frac{1}{7^3}+...$.
1951 Miklós Schweitzer, 15
Let the line
$ z\equal{}x, \, y\equal{}0$
rotate at a constant speed about the $ z$-axis; let at the same time the point of intersection of this line with the $ z$-axis be displaced along the $ z$-axis at constant speed.
(a) Determine that surface of rotation upon which the resulting helical surface can be developed (i.e. isometrically mapped).
(b) Find those lines of the surface of rotation into which the axis and the generators of the helical surface will be mapped by this development.
1972 Miklós Schweitzer, 6
Let $ P(z)$ be a polynomial of degree $ n$ with complex coefficients, \[ P(0)\equal{}1, \;\textrm{and}\ \;|P(z)|\leq M\ \;\textrm{for}\ \;|z| \leq 1\ .\] Prove that every root of $ P(z)$ in the closed unit disc has multiplicity at most $ c\sqrt{n}$, where $ c\equal{}c(M) >0$ is a constant depending only on $ M$.
[i]G. Halasz[/i]
1964 Miklós Schweitzer, 5
Is it true that on any surface homeomorphic to an open disc there exist two congruent curves homeomorphic to a circle?
1980 Miklós Schweitzer, 9
Let us divide by straight lines a quadrangle of unit area into $ n$ subpolygons and draw a circle into each subpolygon. Show that the sum of the perimeters of the circles is at most $ \pi \sqrt{n}$ (the lines are not allowed to cut the interior of a subpolygon).
[i]G. and L. Fejes-Toth[/i]
1962 Miklós Schweitzer, 4
Show that \[ \prod_{1\leq x < y \leq \frac{p\minus{}1}{2}} (x^2\plus{}y^2) \equiv (\minus{}1)^{\lfloor\frac{p\plus{}1}{8}\rfloor} \;(\textbf{mod}\;p\ ) \] for every prime $ p\equiv 3 \;(\textbf{mod}\;4\ )$. [J. Suranyi]