Found problems: 79
1971 Miklós Schweitzer, 11
Let $ C$ be a simple arc with monotone curvature such that $ C$ is congruent to its evolute. Show that under appropriate differentiability conditions, $ C$ is a part of a cycloid or a logarithmic spiral with polar equation $ r\equal{}ae^{\vartheta}$.
[i]J. Szenthe[/i]
2011 Pre-Preparation Course Examination, 2
by using the formula $\pi cot(\pi z)=\frac{1}{z}+\sum_{n=1}^{\infty}\frac{2z}{z^2-n^2}$ calculate values of $\zeta(2k)$ on terms of bernoli numbers and powers of $\pi$.
1968 Miklós Schweitzer, 7
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]
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]
2009 Miklós Schweitzer, 10
Let $ U\subset\mathbb R^n$ be an open set, and let $ L: U\times\mathbb R^n\to\mathbb R$ be a continuous, in its second variable first order positive homogeneous, positive over $ U\times (\mathbb R^n\setminus\{0\})$ and of $ C^2$-class Langrange function, such that for all $ p\in U$ the Gauss-curvature of the hyper surface
\[ \{ v\in\mathbb R^n \mid L(p,v) \equal{} 1 \}\]
is nowhere zero. Determine the extremals of $ L$ if it satisfies the following system
\[ \sum_{k \equal{} 1}^n y^k\partial_k\partial_{n \plus{} i}L \equal{} \sum_{k \equal{} 1}^n y^k\partial_i\partial_{n \plus{} k} L \qquad (i\in\{1,\dots,n\})\]
of partial differetial equations, where $ y^k(u,v) : \equal{} v^k$ for $ (u,v)\in U\times\mathbb R^k$, $ v \equal{} (v^1,\dots,v^k)$.
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]
1972 Miklós Schweitzer, 2
Let $ \leq$ be a reflexive, antisymmetric relation on a finite set $ A$. Show that this relation can be extended to an appropriate finite superset $ B$ of $ A$ such that $ \leq$ on $ B$ remains reflexive, antisymmetric, and any two elements of $ B$ have a least upper bound as well as a greatest lower bound. (The relation $ \leq$ is extended to $ B$ if for $ x,y \in A , x \leq y$ holds in $ A$ if and only if it holds in $ B$.)
[i]E. Freid[/i]
1969 Miklós Schweitzer, 7
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]
MIPT student olimpiad autumn 2024, 1
$F$* is the multiplicative group of the field $F$.
$F$* is of finitely generated.
Is it true that $F$* is cyclic?
Additional question: (wasn’t at the olympiad)
$K$* is the multiplicative group of the field $K$.
$L \subseteq $$K$* is a finitely generated subgroup.
Is it true that $L$ is cyclic?
2011 Pre-Preparation Course Examination, 4
represent a way to calculate $\sum_{k=0}^{\infty}\frac{(-1)^k}{(2k+1)^3}=1-\frac{1}{3^3}+\frac{1}{5^3}-\frac{1}{7^3}+...$.
1964 Miklós Schweitzer, 2
Let $ p$ be a prime and let \[ l_k(x,y)\equal{}a_kx\plus{}b_ky \;(k\equal{}1,2,...,p^2)\ .\] be homogeneous linear polynomials with integral coefficients. Suppose that for every pair $ (\xi,\eta)$ of integers, not both divisible by $ p$, the values $ l_k(\xi,\eta), \;1\leq k\leq p^2 $, represent every residue class $ \textrm{mod} \;p$ exactly $ p$ times. Prove that the set of pairs $ \{(a_k,b_k): 1\leq k \leq p^2 \}$ is identical $ \textrm{mod} \;p$ with the set $ \{(m,n): 0\leq m,n \leq p\minus{}1 \}.$
1972 Miklós Schweitzer, 8
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]
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]
1950 Miklós Schweitzer, 10
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.
1974 Miklós Schweitzer, 9
Let $ A$ be a closed and bounded set in the plane, and let $ C$ denote the set of points at a unit distance from $ A$. Let $ p \in
C$, and assume that the intersection of $ A$ with the unit circle $ K$ centered at $ p$ can be covered by an arc shorter that a semicircle of $ K$. Prove that the intersection of $ C$ with a suitable neighborhood of $ p$ is a simple arc which $ p$ is not an endpoint.
[i]M. Bognar[/i]
1972 Miklós Schweitzer, 9
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]
1971 Miklós Schweitzer, 4
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]
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.
2013 Miklós Schweitzer, 11
[list]
(a) Consider an ellipse in the plane. Prove that there exists a Riemannian metric which is defined on the whole plane, and with respect to which the ellipse is a geodesic. Prove that the Gaussian curvature of any such Riemannian metric takes a positive value.
(b) Consider two nonintersecting, simple closed smooth curves in the plane. Prove that if there is a Riemmanian metric defined on the whole plane and the two curves are geodesics of that metric, then the Gaussian curvature of the metric vanishes somewhere.
[/list]
[i]Proposed by Tran Quoc Binh[/i]
1977 Miklós Schweitzer, 2
Construct on the real projective plane a continuous curve, consisting of simple points, which is not a straight line and is intersected in a single point by every tangent and every secant of a given conic.
[i]F. Karteszi[/i]
1982 Miklós Schweitzer, 7
Let $ V$ be a bounded, closed, convex set in $ \mathbb{R}^n$, and denote by $ r$ the radius of its circumscribed sphere (that is, the radius of the smallest sphere that contains $ V$). Show that $ r$ is the only real number with the following property: for any finite number of points in $ V$, there exists a point in $ V$ such that the arithmetic mean of its distances from the other points is equal to $ r$.
[i]Gy. Szekeres[/i]
1978 Miklós Schweitzer, 7
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]
1965 Miklós Schweitzer, 5
Let $ A\equal{}A_1A_2A_3A_4$ be a tetrahedron, and suppose that for each $ j \not\equal{} k, [A_j,A_{jk}]$ is a segment of length $ \rho$ extending from $ A_j$ in the direction of $ A_k$. Let $ p_j$ be the intersection line of the planes $ [A_{jk}A_{jl}A_{jm}]$ and $ [A_kA_lA_m]$. Show that there are infinitely many straight lines that intersect the straight lines $ p_1,p_2,p_3,p_4$ simultaneously.
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]
1962 Miklós Schweitzer, 1
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]