SCHWEITZER 1999

Call a subset of the plane a circular set iff there exists a point such that for every ray starting from it, the ray intersects the subset once. show that the plane is a countable union of circular sets.

Let e>0. Prove that for a large enough natural n, there exist natural x,y,z st $n^2+x^2=y^2+z^2$ and $y,z\leq \frac{(1+e)n}{\sqrt{2}}$ .

Prove that for any finite graph G there is a constant c(G)>0 such that for every n-point graph that does not have an induced subgraph isomorphic to G, there are two disjoint sets of vertices, each with at least $n^{c(G)}$ elements, between which either all edges are connected, or none of the edges are.

A permutation f of the set of integers is called bounded if | x - f (x) | is bounded. Bounded permutations with permutation multiplication form a group W. Show that the additive group of rational numbers is not isomorphic to any subgroup of W.

Let $\alpha>-2$ , $n\in \mathbb{N}$ and $y_1,\cdots,y_n$ be the solutions to the system of equations: $\sum_{j=1}^n \frac{y_j}{j+k+\alpha}= \frac{1}{n+1+k+\alpha}$ , $k=1,\cdots,n$ Prove that $y_{j-1}y_{j+1}\leq y_j^2 \,\forall 1<j<n$

Show that for every real function f in 1-period $L^2(0, 1)$ there exist three functions $g_1, g_2, g_3$ with the same properties and constants $c_0, c_1, c_2, c_3$ satisfying $$f(x)=c_0+\sum_{i=1}^3(g_i(x+c_i)-g_i(x))$$

let $f:R\to R$ be a continuous function tf(t)>0 for $t\neq 0$ . Prove that there exists a non-zero differentiable function $y:[0,\infty)\to R$ such that $y'(t)=f(y(t-1))\,\forall t>1$ and the roots of y are bounded.

Let $P_1,...,P_n$ and $Q_1,...,Q_n$ be oppositely oriented convex polygons. Prove that there is a line passing through the n line segments $P_1Q_1,...,P_nQ_n$ .

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 $\{U_{n,1},...,U_{n,n}\}_{n=1}^\infty$ be iid rv, uniformly distributed over [0,1] , and for $\alpha\geq 1$ consider the sets $\{[n^\alpha U_{n,1}],...,[n^\alpha U_{n,n}]\}$ , where [·] denotes the whole part. Prove that the elements of the sets $H_n\cap(\cup_{m=n+1}^\infty H_m)$ form an almost surely bounded sequence if and only if $\alpha>3$ .