Olympiad problems

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.

2011 Pre-Preparation Course Examination, 2

Tags: advanced fields , function , geometric series

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, 4

Tags: advanced fields

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]

1973 Miklós Schweitzer, 7

Tags: advanced fields

Let us connect consecutive vertices of a regular heptagon inscribed in a unit circle by connected subsets (of the plane of the circle) of diameter less than $ 1$. Show that every continuum (in the plane of the circle) of diameter greater than $ 4$, containing the center of the circle, intersects one of these connected sets. [i]M. Bognar[/i]

1972 Miklós Schweitzer, 9

Tags: advanced fields , combinatorial geometry

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]

1972 Miklós Schweitzer, 7

Tags: advanced fields , function , inequalities

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]

1967 Miklós Schweitzer, 8

Tags: advanced fields

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]

1970 Miklós Schweitzer, 6

Tags: advanced fields , real analysis , topology

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]

1975 Miklós Schweitzer, 1

Tags: advanced fields , set theory , real analysis

Show that there exists a tournament $ (T,\rightarrow)$ of cardinality $ \aleph_1$ containing no transitive subtournament of size $ \aleph_1$. ( A structure $ (T,\rightarrow)$ is a $ \textit{tournament}$ if $ \rightarrow$ is a binary, irreflexive, asymmetric and trichotomic relation. The tournament $ (T,\rightarrow)$ is transitive if $ \rightarrow$ is transitive, that is, if it orders $ T$.) [i]A. Hajnal[/i]

1964 Miklós Schweitzer, 4

Tags: geometry , geometric transformation , reflection , advanced fields

Let $ A_1,A_2,...,A_n$ be the vertices of a closed convex $ n$-gon $ K$ numbered consecutively. Show that at least $ n\minus{}3$ vertices $ A_i$ have the property that the reflection of $ A_i$ with respect to the midpoint of $ A_{i\minus{}1}A_{i\plus{}1}$ is contained in $ K$. (Indices are meant $ \textrm{mod} \;n\ .$)

1962 Miklós Schweitzer, 4

Tags: advanced fields , floor function , modular arithmetic , quadratic , symmetry

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]

1970 Miklós Schweitzer, 5

Tags: advanced fields , topology

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]

2011 Pre-Preparation Course Examination, 1

Tags: advanced fields , function , geometry

[b]a)[/b] prove that the function $\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s}$ that is defined on the area $Re(s)>1$, is an analytic function. [b]b)[/b] prove that the function $\zeta(s)-\frac{1}{s-1}$ can be spanned to an analytic function over $\mathbb C$. [b]c)[/b] using the span of part [b]b[/b] show that $\zeta(1-n)=-\frac{B_n}{n}$ that $B_n$ is the $n$th bernoli number that is defined by generating function $\frac{t}{e^t-1}=\sum_{n=0}^{\infty}B_n\frac{t^n}{n!}$.

1978 Miklós Schweitzer, 8

Tags: advanced fields , inequalities

Let $ X_1, \ldots ,X_n$ be $ n$ points in the unit square ($ n>1$). Let $ r_i$ be the distance of $ X_i$ from the nearest point (other than $ X_i$). Prove that the inequality \[ r_1^2\plus{} \ldots \plus{}r_n^2 \leq 4.\] [i]L. Fejes-Toth, E. Szemeredi[/i]

1970 Miklós Schweitzer, 3

Tags: advanced fields , geometry

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]

1974 Miklós Schweitzer, 8

Tags: advanced fields , real analysis , function

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]

2014 Miklós Schweitzer, 9

Tags: advanced fields , function , integration , real analysis

Let $\rho:\mathbb{R}^n\to \mathbb{R}$, $\rho(\mathbf{x})=e^{-||\mathbf{x}||^2}$, and let $K\subset \mathbb{R}^n$ be a convex body, i.e., a compact convex set with nonempty interior. Define the barycenter $\mathbf{s}_K$ of the body $K$ with respect to the weight function $\rho$ by the usual formula \[\mathbf{s}_K=\frac{\int_K\rho(\mathbf{x})\mathbf{x}d\mathbf{x}}{\int_K\rho(\mathbf{x})d\mathbf{x}}.\] Prove that the translates of the body $K$ have pairwise distinct barycenters with respect to $\rho$.

2011 Pre-Preparation Course Examination, 4

Tags: advanced fields , geometry

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

Tags: advanced fields , rotation , geometry , geometric transformation

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.

MIPT student olimpiad autumn 2024, 1

Tags: advanced fields

$F$* 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?

1974 Miklós Schweitzer, 9

Tags: advanced fields

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]

1966 Miklós Schweitzer, 3

Tags: advanced fields , limit

Let $ f(n)$ denote the maximum possible number of right triangles determined by $ n$ coplanar points. Show that \[ \lim_{n\rightarrow \infty} \frac{f(n)}{n^2}\equal{}\infty \;\textrm{and}\ \lim_{n\rightarrow \infty}\frac{f(n)}{n^3}\equal{}0 .\] [i]P. Erdos[/i]

1977 Miklós Schweitzer, 2

Tags: advanced fields

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]

2007 VJIMC, Problem 1

Tags: advanced fields

Can the set of positive rationals be split into two nonempty disjoint subsets $\mathbb Q_1$ and $\mathbb Q_2$, such that both are closed under addition, i.e. $p+q\in\mathbb Q_k$ for every $p,q\in\mathbb Q_k$, $k=1,2$? Can it be done when addition is exchanged for multiplication, i.e. $p\cdot q\in\mathbb Q_k$ for every $p,q\in\mathbb Q_k$, $k=1,2$?

1972 Miklós Schweitzer, 2

Tags: advanced fields

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]