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]

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]

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]

In a non-obtuse triangle $ABC$, prove that \[ \frac{\sin A \sin B}{\sin C} + \frac{\sin B \sin C}{\sin A} + \frac{\sin C \sin A}{ \sin B} \ge \frac 52. \][i]Proposed by Ryan Alweiss[/i]

Prove that if X is a compact $T_2$ space, and X has density d(X), then $X^3$ contains a discrete subspace of cardinality $d(X)$. note: $d(X)$ is the smallest cardinality of a dense subspace of X.

Construct a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having the following properties: $ \text{(i)} f $ is not monotonic on any real interval. $ \text{(ii)} f $ has Darboux property (intermediate value property) on any real interval. $ \text{(iii)} f(x)\leqslant f\left( x+1/n \right) ,\quad \forall x\in\mathbb{R} ,\quad \forall n\in\mathbb{N} $ [i]Alexandru Cioba[/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]

Let X be a $\kappa$ weighted compact $T_2$ space. Prove that for every $\omega\leq\lambda<\kappa$, X has a continuous image of a $T_2$ space of weight $\lambda$. (The weight of a space X is the smallest infinite cardinality of a base of X.)

Let $M$ be a compact orientable $2n$-manifold with boundary, where $n \ge 2$. Suppose that $H_0(M;\mathbb{Q}) \cong \mathbb{Q}$ and $H_i(M;\mathbb{Q})=0$ for $i>0$. Prove that the order of $H_{n-1}(\partial M; \mathbb{Z})$ is a square number.

Let $a \neq b a,b \in \mathbb{R}$ such that $(x^2+20ax+10b)(x^2+20bx+10a)=0$ has no roots for $x$. Prove that $20(b-a)$ is not an integer.

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

Let $K$ be a closed subset of the closed unit ball in $\mathbb{R}^3$. Suppose there exists a family of chords $\Omega$ of the unit sphere $S^2$, with the following property: for every $X,Y\in S^2$, there exist $X',Y'\in S^2$, as close to $X$ and $Y$ correspondingly, as we want, such that $X'Y'\in \Omega$ and $X'Y'$ is disjoint from $K$. Verify that there exists a set $H\subset S^2$, such that $H$ is dense in the unit sphere $S^2$, and the chords connecting any two points of $H$ are disjoint from $K$. EDIT: The statement fixed. See post #4

Let $M$ be a connected, compact $C^{\infty}$-differentiable manifold, and denote the vector space of smooth real functions on $M$ by $C^{\infty}(M)$. Let the subspace $V\le C^{\infty}(M)$ be invariant under $C^{\infty}$-diffeomorphisms of $M$, that is, let $f\circ h\in V$ for every $f\in V$ and for every $C^{\infty}$-diffeomorphism $h\colon M\rightarrow M$. Prove that if $V$ is different from the subspaces $\{ 0\}$ and $C^{\infty}(M)$ then $V$ only contains the constant functions.

We say that the real-valued function $ f(x)$ defined on the interval $ (0,1)$ is approximately continuous on $ (0,1)$ if for any $ x_0 \in (0,1)$ and $ \varepsilon >0$ the point $ x_0$ is a point of interior density $ 1$ of the set \[ H\equal{} \{x : \;|f(x)\minus{}f(x_0)|< \varepsilon \ \}.\] Let $ F \subset (0,1)$ be a countable closed set, and $ g(x)$ a real-valued function defined on $ F$. Prove the existence of an approximately continuous function $ f(x)$ defined on $ (0,1)$ such that \[ f(x)\equal{}g(x) \;\textrm{for all}\ \;x \in F\ .\] [i]M. Laczkovich, Gy. Petruska[/i]

Is it possible to select a non-degenerate segment from each line of the plane such that any two selected segments are disjoint?

Calculate the integral of the Gaussian curvature of the surface \[z^4+(x^2+y^2-1)(2x^2+3y^2-1)=0\]

Show that there is an absolute constant $c < 1$ with the following property: whenever $\mathcal P$ is a polygon with area $1$ in the plane, one can translate it by a distance of $\frac{1}{100}$ in some direction to obtain a polygon $\mathcal Q$, for which the intersection of the interiors of $\mathcal P$ and $\mathcal Q$ has total area at most $c$. [i]Linus Hamilton[/i]

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.

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 $ n \geq 2$ be an integer, and let $ X$ be a connected Hausdorff space such that every point of $ X$ has a neighborhood homeomorphic to the Euclidean space $ \mathbb{R}^n$. Suppose that any discrete (not necessarily closed ) subspace $ D$ of $ X$ can be covered by a family of pairwise disjoint, open sets of $ X$ so that each of these open sets contains precisely one element of $ D$. Prove that $ X$ is a union of at most $ \aleph_1$ compact subspaces. [i]Z. Balogh[/i]

Prove that these three statements are equivalent: (a) For every continuous function $f:S^n \to \mathbb R^n$, there exists an $x\in S^n$ such that $f(x)=f(-x)$. (b) There is no antipodal mapping $f:S^n \to S^{n-1}$. (c) For every covering of $S^n$ with closed sets $A_0,\dots,A_n$, there exists an index $i$ such that $A_i\cap -A_i\neq \emptyset$.

We define a relation between subsets of $\mathbb R ^n$. $A \sim B\Longleftrightarrow$ we can partition $A,B$ in sets $A_1,\dots,A_n$ and $B_1,\dots,B_n$(i.e $\displaystyle A=\bigcup_{i=1} ^n A_i,\ B=\bigcup_{i=1} ^n B_i, A_i\cap A_j=\emptyset,\ B_i\cap B_j=\emptyset$) and $A_i\simeq B_i$. Say the the following sets have the relation $\sim$ or not ? a) Natural numbers and composite numbers. b) Rational numbers and rational numbers with finite digits in base 10. c) $\{x\in\mathbb Q|x<\sqrt 2\}$ and $\{x\in\mathbb Q|x<\sqrt 3\}$ d) $A=\{(x,y)\in\mathbb R^2|x^2+y^2<1\}$ and $A\setminus \{(0,0)\}$

Let $f:[0,\infty)\to\mathbb R$ be a continuous function s.t. $\lim_{x\to\infty}\frac {f(x)}x=0$. Let $(x_n)_n$ be a sequence of positive real numbers s.t. $\left(\frac{x_n}n\right)_n$ is bounded. Prove that $\lim_{n\to\infty}\frac{f(x_n)}n=0$. [i]Dorin Andrica, Eugen Paltanea[/i]

Let $D\subseteq \mathbb{C}$ be a compact set with at least two elements and consider the space $\Omega=\bigtimes_{i=1}^{\infty} D$ with the product topology. For any sequence $(d_n)_{n=0}^{\infty} \in \Omega$ let $f_{(d_n)}(z)=\sum_{n=0}^{\infty}d_nz^n$, and for each point $\zeta \in \mathbb{C}$ with $|\zeta|=1$ we define $S=S(\zeta,(d_n))$ to be the set of complex numbers $w$ for which there exists a sequence $(z_k)$ such that $|z_k|<1$, $z_k \to \zeta$, and $f_{d_n}(z_k) \to w$. Prove that on a residual set of $\Omega$, the set $S$ does not depend on the choice of $\zeta$.

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