Found problems: 79
Let $ f(x,y,z)$ be a nonnegative harmonic function in the unit ball of $ \mathbb{R}^3$ for which the inequality $ f(x_0,0,0) \leq \varepsilon^2$ holds for some $ 0\leq x_0 \leq 1$ and $ 0<\varepsilon<(1\minus{}x_0)^2$. Prove that $ f(x,y,z) \leq \varepsilon$ in the ball with center at the origin an radius $ (1\minus{}3\varepsilon^{1/4}).$
[i]P. Turan[/i]
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.
Suppose that $X$ is a compact metric space and $T: X\rightarrow X$ is a continous function. Prove that $T$ has a returning point. It means there is a strictly increasing sequence $n_i$ such that $\lim_{k\rightarrow \infty} T^{n_k}(x_0)=x_0$ for some $x_0$.
Let us defined a pseudo-Riemannian metric on the set of points of the Euclidean space $ \mathbb{E}^3$ not lying on the $ z$-axis by the metric tensor \[ \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & \minus{}\sqrt{x^2\plus{}y^2} \\ \end{array} \right),\] where $ (x,y,z)$ is a Cartesian coordinate system $ \mathbb{E}^3$. Show that the orthogonal projections of the geodesic curves of this Riemannian space onto the $ (x,y)$-plane are straight lines or conic sections with focus at the origin
[i]P. Nagy[/i]
Consider an arc of a planar curve such that the total curvature of the arc is less than $ \pi$. Suppose, further, that the curvature and its derivative with respect to the arc length exist at every point of the arc and the latter nowhere equals zero. Let the osculating circles belonging to the endpoints of the arc and one of these points be given. Determine the possible positions of the other endpoint.
Let $ f$ be a complex-valued, completely multiplicative,arithmetical function. Assume that there exists an infinite increasing sequence $ N_k$ of natural numbers such that \[ f(n)\equal{}A_k \not\equal{} 0 \;\textrm{provided}\ \; N_k \leq n \leq N_k\plus{}4 \sqrt{N_k}\
.\] Prove that $ f$ is identically $ 1$.
[i]I. Katai[/i]
Let $ K$ be a compact convex body in the $ n$-dimensional Euclidean space. Let $ P_1,P_2,...,P_{n\plus{}1}$ be the vertices of a simplex having maximal volume among all simplices inscribed in $ K$. Define the points $ P_{n\plus{}2},P_{n\plus{}3},...$ successively so that $ P_k \;(k>n\plus{}1)$ is a point of $ K$ for which the volume of the convex hull of $ P_1,...,P_k$ is maximal. Denote this volume by $ V_k$. Decide, for different values of $ n$, about the truth of the statement "the sequence $ V_{n\plus{}1},V_{n\plus{}2},...$ is concave."
[i]L. Fejes- Toth, E. Makai[/i]
Let a neighborhood basis of a point $ x$ of the real line consist of all Lebesgue-measurable sets containing $ x$ whose density at $ x$ equals $ 1$. Show that this requirement defines a topology that is regular but not normal.
[i]A. Csaszar[/i]
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.
Let $ T$ be a surjective mapping of the hyperbolic plane onto itself which maps collinear points into collinear points. Prove that $ T$ must be an isometry.
[i]M. Bognar[/i]
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]
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]
Given four points $ A_1,A_2,A_3,A_4$ in the plane in such a way that $ A_4$ is the centroid of the $ \bigtriangleup A_1A_2A_3$,
find a point $ A_5$ in the plane that maximizes the ratio \[ \frac{\min_{1 \leq i < j < k \leq 5}T(A_iA_jA_k)}{\max_{1 \leq i < j < k \leq 5}T(A_iA_jA_k)}.\] ($ T(ABC)$ denotes the area of the triangle $ \bigtriangleup ABC.$ )
[i]J. Suranyi[/i]
Is it true that on any surface homeomorphic to an open disc there exist two congruent curves homeomorphic to a circle?
Let $ \mathcal{T}_1$ and $ \mathcal{T}_2$ be second-countable topologies on the set $ E$. We would like to find a real function $ \sigma$ defined on $ E \times E$ such that \[ 0 \leq \sigma(x,y) <\plus{}\infty, \;\sigma(x,x)\equal{}0 \ ,\] \[ \sigma(x,z) \leq
\sigma(x,y)\plus{}\sigma(y,z) \;(x,y,z \in E) \ ,\] and, for any $ p \in E$, the sets \[ V_1(p,\varepsilon)\equal{}\{ x : \;\sigma(x,p)< \varepsilon \ \} \;(\varepsilon >0) \] form a neighborhood base of $ p$ with respect to $ \mathcal{T}_1$, and the sets \[ V_2(p,\varepsilon)\equal{}\{ x : \;\sigma(p,x)< \varepsilon \ \} \;(\varepsilon >0) \] form a neighborhood base of $ p$ with respect to $ \mathcal{T}_2$. Prove that such a function $ \sigma$ exists if and only if, for any $ p \in E$ and $ \mathcal{T}_i$-open set $ G \ni p \;(i\equal{}1,2) $, there exist a $ \mathcal{T}_i$-open set $ G'$ and a $ \mathcal{T}_{3\minus{}i}$-closed set $ F$ with $ p \in G' \subset F \subset G.$
[i]A. Csaszar[/i]
Suppose that a bounded subset $ S$ of the plane is a union of congruent, homothetic, closed triangles. Show that the boundary of $ S$ can be covered by a finite number of rectifiable arcs.
[i]L. Geher[/i]
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]
Let $ F$ be a surface of nonzero curvature that can be represented around one of its points $ P$ by a power series and is symmetric around the normal planes parallel to the principal directions at $ P$. Show that the derivative with respect to the arc length of the curvature of an arbitrary normal section at $ P$ vanishes at $ P$. Is it possible to replace the above symmetry condition by a weaker one?
[i]A. Moor[/i]
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$.
represent a way to calculate $\sum_{k=0}^{\infty}\frac{(-1)^k}{(2k+1)^3}=1-\frac{1}{3^3}+\frac{1}{5^3}-\frac{1}{7^3}+...$.
Prove that if a sequence of Mikusinski operators of the form $ \mu e^{\minus{}\lambda s}$ ( $ \lambda$ and $ \mu$ nonnegative real
numbers, $ s$ the differentiation operator) is convergent in the sense of Mikusinski, then its limit is also of this form.
[i]E. Geaztelyi[/i]
Suppose that $ V$ is a locally compact topological space that admits no countable covering with compact sets. Let $ \textbf{C}$
denote the set of all compact subsets of the space $ V$ and $ \textbf{U}$ the set of open subsets that are not contained in any compact set. Let $ f$ be a function from $ \textbf{U}$ to $ \textbf{C}$ such that $ f(U)\subseteq U$ for all $ U \in \textbf{U}$. Prove that either
(i) there exists a nonempty compact set $ C$ such that $ f(U)$ is not a proper subset of $ C$ whenever $ C \subseteq U \in \textbf{U}$,
(ii) or for some compact set $ C$, the set \[ f^{-1}(C)= \bigcup \{U \in \textbf{U}\;: \ \;f(U)\subseteq C\ \}\] is an element of $ \textbf{U}$, that is, $ f^{-1}(C)$ is not contained in any compact set.
[i]A. Mate[/i]
Let $ \mathcal{H}$ be the class of all graphs with at most $ 2^{\aleph_0}$ vertices not containing a complete subgraph of size $ \aleph_1$. Show that there is no graph $ H \in \mathcal{H}$ such that every graph in $ \mathcal{H}$ is a subgraph of $ H$.
[i]F. Galvin[/i]
Let $ f$ and $ g$ be polynomials with rational coefficients, and let $ F$ and $ G$ denote the sets of values of $ f$ and $ g$ at rational numbers. Prove that $ F \equal{} G$ holds if and only if $ f(x) \equal{} g(ax \plus{} b)$ for some suitable rational numbers $ a\not \equal{} 0$ and
$ b$.
[i]E. Fried[/i]
For every natural number $ r$, the set of $ r$-tuples of natural numbers is partitioned into finitely many classes. Show that if $ f(r)$ is a function such that $ f(r)\geq 1$ and $ \lim _{r\rightarrow \infty} f(r)\equal{}\plus{}\infty$, then there exists an infinite set of natural numbers that, for all $ r$, contains $ r$-triples from at most $ f(r)$ classes. Show that if $ f(r) \not \rightarrow \plus{}\infty$, then there is a family of partitions such that no such infinite set exists.
[i]P. Erdos, A. Hajnal[/i]