Suppose that in a unit sphere in Euclidean space, there are $2m$ points $x_1, x_2, ..., x_{2m}.$ Prove that it's possible to partition them into two sets of $m$ points in such a way that the centers of mass of these sets are at a distance of at most $\frac{2}{\sqrt{m}}$ from one another.

Let $ \gamma: [0,1]\rightarrow [0,1]\times [0,1]$ be a mapping such that for each $ s,t\in [0,1]$ \[ |\gamma(s) \minus{} \gamma(t)|\leq M|s \minus{} t|^\alpha \] in which $ \alpha,M$ are fixed numbers. Prove that if $ \gamma$ is surjective, then $ \alpha\leq\frac12$

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]

Let $P_1 , P_2 , \ldots$ be a sequence of distinct points which is dense in the interval $(0,1)$. The points $P_1 , \ldots , P_{n-1}$ decompose the interval into $n$ parts, and $P_n$ decomposes one of these into two parts. Let $a_n$ and $b_n$ be the length of these two intervals. Prove that $$\sum_{n=1}^{\infty} a_n b_n (a_n +b_n) =1 \slash 3.$$

Let ( M , g ) be a Riemannian manifold. Extend the metric tensor g to the set of tangents TM with the following specification: if $a,b\in T_v TM \, (v\in T_p M)$, then $$\tilde g_v (a, b): = g_p (\dot {\alpha} (0), \dot {\beta} (0) ) + g_p (D _ {\alpha} X(0) , D_{\beta} Y(0) )$$ where $\alpha, \beta$ are curves in M such that $\alpha(0) = \beta(0) = p$. X and Y are vector fields along $\alpha,\beta$ respectively, with the condition $\dot X (0) = a,\dot Y(0) = b$. $D _{\alpha}$ and $D _{\beta}$ are the operators of the covariant derivative along the corresponding curves according to the Levi-Civita connection. Is the eigenfunction from the Riemannian manifold (M,g) to the Riemannian manifold $(TM, \tilde g)$ harmonic?

Prove that for a $2$-dimensional Riemannian manifold there is a metric linear connection with zero curvature if and only if the Gaussian curvature of the Riemannian manifold can be written as the divergence of a vector field.

A ring is the area between two circles with the same center, and width of a ring is the difference between the radii of two circles. [img]http://i18.tinypic.com/6cdmvi8.png[/img] a) Can we put uncountable disjoint rings of width 1(not necessarily same) in the space such that each two of them can not be separated. [img]http://i19.tinypic.com/4qgx30j.png[/img] b) What's the answer if 1 is replaced with 0?

Let $U$ denote the set $\{ f\in C[0, 1] \colon |f(x)|\leq 1\, \mathrm{for}\,\mathrm{all}\, x\in [0, 1]\}$. Prove that there is no topology on $C[0, 1]$ that, together with the linear structure of $C[0,1]$, makes $C[0,1]$ into a topological vector space in which the set $U$ is compact. (Assume that topological vector spaces are Hausdorff) [V. Totik]

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.

[b]3.[/b] Let $A$ be a subset of n-dimensional space containing at least one inner point and suppose that, for every point pair $x, y \in A$, the subset $A$ contains the mid point of the line segment beteween $x$ and $y$. Show that $A$ consists of a convex open set and of some of its boundary points. [b](St. 1)[/b]

Let $M$ be the set of real numbers of the form $\frac{m+n}{\sqrt{m^2+n^2}}$, where $m$ and $n$ are positive integers. Prove that for every pair $x \in M, y \in M$ with $x < y$, there exists an element $z \in M$ such that $x < z < y.$

Let $G_0, G_1,\ldots$ be infinite open subsets of a Hausdorff space. Prove that there exist some infinite pairwise disjoint open sets $V_0,V_1,\ldots$ and some indices $n_0<n_1<\cdots$ such that $V_i\subseteq G_{n_i}$ for every $i\geqslant 0.$

Suppose that the $ T_3$-space $ X$ has no isolated points and that in $ X$ any family of pairwise disjoint, nonempty, open sets is countable. Prove that $ X$ can be covered by at most continuum many nowhere-dense sets. [i]I. Juhasz[/i]

Let $M$ be the set of real numbers of the form $\frac{m+n}{\sqrt{m^2+n^2}}$, where $m$ and $n$ are positive integers. Prove that for every pair $x \in M, y \in M$ with $x < y$, there exists an element $z \in M$ such that $x < z < y.$

Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous function. A point $x$ is called a [i]shadow[/i] point if there exists a point $y\in \mathbb{R}$ with $y>x$ such that $f(y)>f(x).$ Let $a<b$ be real numbers and suppose that $\bullet$ all the points of the open interval $I=(a,b)$ are shadow points; $\bullet$ $a$ and $b$ are not shadow points. Prove that a) $f(x)\leq f(b)$ for all $a<x<b;$ b) $f(a)=f(b).$ [i]Proposed by José Luis Díaz-Barrero, Barcelona[/i]

Let $f:[-1,1]\to\mathbb{R}$ be a continuous function such that (i) $f(x)=\frac{2-x^2}{2}f\left(\frac{x^2}{2-x^2}\right)$ for every $x$ in $[-1,1],$ (ii) $ f(0)=1,$ and (iii) $\lim_{x\to 1^-}\frac{f(x)}{\sqrt{1-x}}$ exists and is finite. Prove that $f$ is unique, and express $f(x)$ in closed form.

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function with the following property: for all $\alpha \in \mathbb{R}_{>0}$, the sequence $(a_n)_{n \in \mathbb{N}}$ defined as $a_n = f(n\alpha)$ satisfies $\lim_{n \to \infty} a_n = 0$. Is it necessarily true that $\lim_{x \to +\infty} f(x) = 0$?

Let $(X,d)$ be a nonempty connected metric space such that the limit of every convergent sequence, is a term of that sequence. Prove that $X$ has exactly one element.

Let $A=(a_{ij})$ be a $5 \times 5$ matrix with $a_{ij}=\min\{i,j\}$. Suppose $f:\mathbb{R}^5 \to \mathbb{R}^5$ is a smooth map such that $f(\Sigma) \subset \Sigma$, where $\Sigma=\{x \in \mathbb{R}^5: xAx^T=1\}$. Denote by $f^{(n)}$ te $n$-th iterate of $f$. Prove that there does not exist $N \ge 1$ such that \[\inf_{x \in \Sigma} \| f^{(n)}(x)-x\|>0, \forall n \ge N.\]

Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous function and let $g : \mathbb{R} \to \mathbb{R}$ be arbitrary. Suppose that the Minkowski sum of the graph of $f$ and the graph of $g$ (i.e., the set $\{( x+y; f(x)+g(y) ) \mid x, y \in \mathbb{R}\}$) has Lebesgue measure zero. Does it follow then that the function $f$ is of the form $f(x) = ax + b$ with suitable constants $a, b \in \mathbb{R}$ ?

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.

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.

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 $A$ be a closed subset of $\mathbb{R}^{n}$ and let $B$ be the set of all those points $b \in \mathbb{R}^{n}$ for which there exists exactly one point $a_{0}\in A $ such that $|a_{0}-b|= \inf_{a\in A}|a-b|$. Prove that $B$ is dense in $\mathbb{R}^{n}$; that is, the closure of $B$ is $\mathbb{R}^{n}$

Let $(X,d)$ be a nonempty connected metric space such that the limit of every convergent sequence, is a term of that sequence. Prove that $X$ has exactly one element.