Let $S_0=\{z\in\mathbb C:|z|=1,z\ne-1\}$ and $f(z)=\frac{\operatorname{Im}z}{1+\operatorname{Re}z}$. Prove that $f$ is a bijection between $S_0$ and $\mathbb R$. Find $f^{-1}$.

Prove that the sets $$ \{ \left( x_1,x_2,x_3,x_4 \right)\in\mathbb{R}^4|x_1^2+x_2^2+x_3^2=x_4^2 \} , $$ $$ \{ \left( x_1,x_2,x_3,x_4 \right)\in\mathbb{R}^4|x_1^2+x_2^2=x_3^2+x_4^2 \}, $$ are not homeomorphic on the Euclidean topology induced on them.

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 $F_1, F_2, ...$ be Borel-measurable sets on the plane whose union is the whole plane. Prove that there is a natural number n and circle S for which the set $S \cap F_n$ is dense in S. Also show that the statement is not necessarily true if we omit the condition for the measurability of sets $F_j$.

$11$ people are sitting around a circle table, orderly (means that the distance between two adjacent persons is equal to others) and $11$ cards with numbers $1$ to $11$ are given to them. Some may have no card and some may have more than $1$ card. In each round, one [and only one] can give one of his cards with number $ i $ to his adjacent person if after and before the round, the locations of the cards with numbers $ i-1,i,i+1 $ don’t make an acute-angled triangle. (Card with number $0$ means the card with number $11$ and card with number $12$ means the card with number $1$!) Suppose that the cards are given to the persons regularly clockwise. (Mean that the number of the cards in the clockwise direction is increasing.) Prove that the cards can’t be gathered at one person.

Does the circle T = R / Z have a self-homeomorphism $\phi$ that is singular (that is, its derivative is almost everywhere 0), but the mapping $f:T \to T$ , $f(x) = \phi^{-1} (2\phi(x))$ is absolutely continuous?

Does there exist a nowhere dense, nonempty compact set $C \subset [0,1]$ such that \[ \liminf_{h \to 0^+} \frac{\lambda(C \cap (x, x+h))}{h} > 0 \quad \text{or} \quad \liminf_{h \to 0^+} \frac{\lambda(C \cap (x-h, x))}{h} > 0 \] holds for every point $x \in C$, where $\lambda(A)$ denotes the Lebesgue measure of $A$?

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]

Let $H(D)$ denote the space of functions holomorphic on the disc $D=\{ z\colon |z|<1 \}$, endowed with the topology of uniform convergence on each compact subset of $D$. If $f(z)=\sum_{n=0}^{\infty} a_nz^n$, then we shall denote $S_n(f,z)=\sum_{k=0}^n a_kz^k$. A function $f\in H(D)$ is called [i]universal[/i] if, for every continuous function $g\colon\partial D\rightarrow \mathbb{C}$ and for every $\varepsilon >0$, there are partial sums $S_{n(j)}(f,z)$ approximating $g$ uniformly on the arc $\{ e^{it} \colon 0\le t\le 2\pi - \varepsilon\}$. Prove that the set of universal functions contains a dense $G_{\delta}$ subset of $H(D)$.

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

The Lindelöf number $L(X)$ of a topological space $X$ is the least infinite cardinal $\lambda$ with the property that every open covering of $X$ has a subcovering of cardinality at most $\lambda$. Prove that if evert non-countably infinite subset of a first countable space $X$ has a point of condensation, then $L(X)=\sup L(A)$, where $A$ runs over the separable closed subspaces of $X$. (A point of condensation of a subset $H\subseteq X$ is a point $x\in X$ such that any neighbourhood of $x$ intersects $H$ in a non-countably infinite set.)

Construct an uncountable Hausdorff space in which the complement of the closure of any nonempty, open set is countable. [i]A. Hajnal, I. Juhasz[/i]

In $\mathbb{R}^3$, let there be a cube $Q$ and a sequence of other cubes, all of which are homothetic to $Q$ with coefficients of homothety that are each smaller than $1$. Prove that if this sequence of homothetic cubes completely fills $Q$, the sum of their coefficients of homothety is not less than $4$.

Let $ W$ be a dense, open subset of the real line $ \mathbb{R}$. Show that the following two statements are equivalent: (1) Every function $ f : \mathbb{R} \rightarrow \mathbb{R}$ continuous at all points of $ \mathbb{R} \setminus W$ and nondecreasing on every open interval contained in $ W$ is nondecreasing on the whole $ \mathbb{R}$. (2) $ \mathbb{R} \setminus W$ is countable. [i]E. Gesztelyi[/i]

Let $(M,g)$ be an $n$-dimensional complete Riemannian manifold with $n \ge 2$. Suppose $M$ is connected and $\text{Ric} \ge (n-1)g$, where $\text{Ric}$ is the Ricci tensor of $(M,g)$. Denote by $\text{d}g$ the Riemannian measure of $(M,g)$ and by $d(x,y)$ the geodesic distance between $x$ and $y$. Prove that \[\int_{M \times M} \cos d(x,y) \text{d}g(x)\text{d}g(y) \ge 0.\] Moreover, equality holds if and only if $(M,g)$ is isometric to the unit round sphere $S^n$.

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.

Let X be a dense set homeomorphic to $\mathbb R^n$ in the compact Hausdorff space Y. Prove that for $n\geq 2$ , $Y \setminus X$ is connected, and for n=1 it consists of at most two components.

Suppose that $ K$ is a compact Hausdorff space and $ K\equal{} \cup_{n\equal{}0}^{\infty}A_n$, where $ A_n$ is metrizable and $ A_n \subset A_m$ for $ n<m$. Prove that $ K$ is metrizable. [i]Z. Balogh[/i]

Let $ D \subset \mathbb {R} ^ {2} $ be a finite Lebesgue measure of a connected open set and $ u: D \rightarrow \mathbb {R} $ a harmonic function. Show that it is either a constant $ u $ or for almost every $ p \in D $ $$ f ^ {\prime} (t) = (\operatorname {grad} u) (f (t)), \quad f (0) = p $$has no initial value problem(differentiable everywhere) solution to $ f:[0,\infty) \rightarrow D $.

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]

Let $X$ be a regular topological space and let $S$ be a countably compact dense subspace in $X$. (The countably compact property means that every infinite subset of $S$ has an accumulation point in $S$.) Show that $S$ is also $G_\delta$-dense in $X$, i.e., $S$ intersects all nonempty $G_\delta$ sets.

X is a compact T2 space such that every subspace of cardinality $\aleph_1$ is first countable. Prove that X is first countable.

Let $S$ be a nonempty closed set in the euclidean plane for which there is a closed disk $D$ containing $S$ such that $D$ is a subset of every closed disk that contains $S$. Prove that every point inside $D$ is the midpoint of a segment joining two points of $S.$

Suppose that $X$ is a compact metric space and $T: X\rightarrow X$ is a continous function. Prove that $T$ has a returning point. It means there is a strictly increasing sequence $n_i$ such that $\lim_{k\rightarrow \infty} T^{n_k}(x_0)=x_0$ for some $x_0$.

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]