A cube is made by soldering twelve $ 3$-inch lengths of wire properly at the vertices of the cube. If a fly alights at one of the vertices and then walks along the edges, the greatest distance it could travel before coming to any vertex a second time, without retracing any distance, is: $ \textbf{(A)}\ 24\text{ in.}\qquad \textbf{(B)}\ 12\text{ in.}\qquad \textbf{(C)}\ 30\text{ in.}\qquad \textbf{(D)}\ 18\text{ in.}\qquad \textbf{(E)}\ 36\text{ in.}$

Show that if a connected, nowhere zero sectional curvature of Riemannian manifold, where symmetric (1,1)-tensor of the Levi-Civita connection covariant derivative vanishes, then the tensor is constant times the unit tensor. (translated by j___d)

Let $(X,d)$ be a metric space with $d:X\times X \to \mathbb{R}_{\geq 0}$. Suppose that $X$ is connected and compact. Prove that there exists an $\alpha \in \mathbb{R}_{\geq 0}$ with the following property: for any integer $n > 0$ and any $x_1,\dots,x_n \in X$, there exists $x\in X$ such that the average of the distances from $x_1,\dots,x_n$ to $x$ is $\alpha$ i.e. $$\frac{d(x,x_1)+d(x,x_2)+\cdots+d(x,x_n)}{n} = \alpha.$$

Prove that if every subspace of a Hausdorff space $X$ is $\sigma$-compact, then $X$ is countable.

Let $M = F_1\times\cdots\times F_k$ be the product of $k$ smooth, closed surfaces (2-dimensional, $C^\infty$, compact, connected, manifold without boundary), $s$ of which are non-orientable. Prove that $M$ can be embedded in $\mathbb{R}^{2k+s+1}$.

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]

Let $F$ be a set of filters on X so that if $ \sigma, \tau \in F$ , $\forall S \in\sigma$ , $\forall T\in\tau$ , we have $S \cap T\neq\emptyset$ , then $\sigma \cap \tau \in F$. We say that $F$ is compatible with a topology on X when $x \in X$ is a contact point of $A\subset X$ , if and only if , there is $\sigma \in F$ such that $x \in S$ and $S \cap A \neq\emptyset$ for all $S \in\sigma$ . When is there an $F$ compatible with the topology on X in which finite subsets of X and X are closed ? contact point is also known as adherent point.

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)$.

Let $A$ be a ring with $2^n+1$ elements, where $n$ is a positive integer and let \[ M = \{ k \in\mathbb{Z} \mid k \geq 2, \ x^k =x , \ \forall \ x\in A \} . \] Prove that the following statements are equivalent: a) $A$ is a field; b) $M$ is not empty and the smallest element in $M$ is $2^n+1$. [i]Marian Andronache[/i]

Let $ E$ be the set of all real functions on $ I\equal{}[0,1]$. Prove that one cannot define a topology on $ E$ in which $ f_n\rightarrow f$ holds if and only if $ f_n$ converges to $ f$ almost everywhere.

A map $ F : P(X) \rightarrow P(X)$, where $ P(X)$ denotes the set of all subsets of $ X$, is called a $ \textit{closure operation}$ on $ X$ if for arbitrary $ A,B \subset X$, the following conditions hold: (i) $ A \subset F(A);$ (ii) $ A \subset B \Rightarrow F(A) \subset F(B);$ (iii) $ F(F(A))\equal{}F(A)$. The cardinal number $ \min \{ |A| : \;A \subset X\ ,\;F(A)\equal{}X\ \}$ is called the $ \textit{density}$ of $ F$ and is denoted by $ d(F)$. A set $ H \subset X$ is called $ \textit{discrete}$ with respect to $ F$ if $ u \not \in F(H\minus{}\{ u \})$ holds for all $ u \in H$. Prove that if the density of the closure operation $ F$ is a singular cardinal number, then for any nonnegative integer $ n$, there exists a set of size $ n$ that is discrete with respect to $ F$. Show that the statement is not true when the existence of an infinite discrete subset is required, even if $ F$ is the closure operation of a topological space satisfying the $ T_1$ separation axiom. [i]A. Hajnal[/i]

Can a closed disk can be decomposed into a union of two congruent parts having no common point?

Let $(X,d)$ be a metric space and $f:X \to X$ be a function such that $\forall x,y\in X : d(f(x),f(y))=d(x,y)$. $\text{a})$ Prove that for all $x \in X$, $\lim_{n \rightarrow +\infty} \frac{d(x,f^n(x))}{n}$ exists, where $f^n(x)$ is $\underbrace{f(f(\cdots f(x)}_{n \text{times}} \cdots ))$. $\text{b})$ Prove that the amount of the limit does [b][u]not[/u][/b] depend on choosing $x$.

Let $ G$ be an infinite compact topological group with a Hausdorff topology. Prove that $ G$ contains an element $ g \not\equal{} 1$ such that the set of all powers of $ g$ is either everywhere dense in $ G$ or nowhere dense in $ G$. [i]J. Erdos[/i]

Let $M_n=\{(u,v) \in S^n \times S^n: u \cdot v=0\}$, where $n \ge 2$, and $u \cdot v$ is the Euclidean inner product of $u$ and $v$. Suppose that the topology of $M_n$ is induces from $S^n \times S^n$. (1) Prove that $M_n$ is a connected regular submanifold of $S^n \times S^n$. (2) $M_n$ is Lie Group if and only if $n=2$.

Show that one cannot find compact sets $A_1, A_2, A_3, \ldots$ in $\mathbb{R}$ such that (1) All elements of $A_n$ are rational. (2) Any compact set $K\subset \mathbb{R}$ which only contains rational numbers is contained in some $A_{m}$.

Two subsets $A$ and $B$ of the $(x,y)$-plane are said to be [i]equivalent[/i] if there exists a function $f: A\to B$ which is both one-to-one and onto. (i) Show that any two line segments in the plane are equivalent. (ii) Show that any two circles in the plane are equivalent.

We call the set $A\in \mathbb R^n$ CN if and only if for every continuous $f:A\to A$ there exists some $x\in A$ such that $f(x)=x$. a) Example: We know that $A = \{ x\in\mathbb R^n | |x|\leq 1 \}$ is CN. b) The circle is not CN. Which one of these sets are CN? 1) $A=\{x\in\mathbb R^3| |x|=1\}$ 2) The cross $\{(x,y)\in\mathbb R^2|xy=0,\ |x|+|y|\leq1\}$ 3) Graph of the function $f:[0,1]\to \mathbb R$ defined by \[f(x)=\sin\frac 1x\ \mbox{if}\ x\neq0,\ f(0)=0\]

Let $S$ be the Sierpiński triangle. What can we say about the Hausdorff dimension of the elevation sets $f^{-1}(y)$ for typical continuous real functions defined on $S$? (A property is satisfied for typical continuous real functions on $S$ if the set of functions not having this property is of the first Baire category in the metric space of continuous $S\rightarrow\mathbb{R}$ functions with the supremum norm.) (translated by Miklós Maróti)

Prove that if a function $f:R \to R$ is bounded and its graph is closed as subset of the $R^2$ plane, then the function f is continuous.

Suppose that the closed subset $K$ of the sphere $$S^2=\{ (x,y,z)\in \mathbb{R}^3\colon x^2+y^2+z^2=1 \}$$ is symmetric with respect to the origin and separates any two antipodal points in $S^2 \backslash K$. Prove that for any positive $\varepsilon$ there exists a homogeneous polynomial $P$ of odd degree such that the Hausdorff distance between $$Z(P)=\{ (x,y,z)\in S^2 \colon P(x,y,z)=0\}$$ and $K$ is less than $\varepsilon$.

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]

Show that there are infinitely many primes.

Let $A$ and $B$ be rectangles in the plane and $f : A \rightarrow B$ be a mapping which is uniform on the interior of $A$, maps the boundary of $A$ homeomorphically to the boundary of $B$ by mapping the sides of $A$ to corresponding sides in $B$. Prove that $f$ is an affine transformation.

Show that the unit sphere bundle of the $r$-fold direct sum of the tautological (universal) complex line bundle over the space $\mathbb{C}P^{\infty}$ is homotopically equivalent to $\mathbb{C}P^{r-1}$.