SCHWEITZER 2017

Can one divide a square into finitely many triangles such that no two triangles share a side? (The triangles have pairwise disjoint interiors and their union is the square.)

Prove that a field $K$ can be ordered if and only if every $A\in M_n(K)$ symmetric matrix can be diagonalized over the algebraic closure of $K$ . (In other words, for all $n\in\mathbb{N}$ and all $A\in M_n(K)$ , there exists an $S\in GL_n(\overline{K})$ for which $S^{-1}AS$ is diagonal.)

For every algebraic integer $\alpha$ define its positive degree $\text{deg}^+(\alpha)$ to be the minimal $k\in\mathbb{N}$ for which there exists a $k\times k$ matrix with non-negative integer entries with eigenvalue $\alpha$ . Prove that for any $n\in\mathbb{N}$ , every algebraic integer $\alpha$ with degree $n$ satisfies $\text{deg}^+(\alpha)\le 2n$ .

Let $K$ be a number field which is neither $\mathbb{Q}$ nor a quadratic imaginary extension of $\mathbb{Q}$ . Denote by $\mathcal{L}(K)$ the set of integers $n\ge 3$ for which we can find units $\varepsilon_1,\ldots,\varepsilon_n\in K$ for which $$\varepsilon_1+\dots+\varepsilon_n=0,$$ but $\sum_{i\in I}\varepsilon_i\neq 0$ for any nonempty proper subset $I$ of $\{1,2,\dots,n\}$ . Prove that $\mathcal{L}(K)$ is infinite, and that its smallest element can be bounded from above by a function of the degree and discriminant of $K$ . Further, show that for infinitely many $K$ , $\mathcal{L}(K)$ contains infinitely many even and infinitely many odd elements.

For every non-constant polynomial $p$ , let $H_p=\big\{z\in \mathbb{C} \big| |p(z)|=1\big\}$ . Prove that if $H_p=H_q$ for some polynomials $p,q$ , then there exists a polynomial $r$ such that $p=r^m$ and $q=\xi\cdot r^n$ for some positive integers $m,n$ and constant $|\xi|=1$ .

Let $I$ and $J$ be intervals. Let $\varphi,\psi:I\to\mathbb{R}$ be strictly increasing continuous functions and let $\Phi,\Psi:J\to\mathbb{R}$ be continuous functions. Suppose that $\varphi(x)+\psi(x)=x$ and $\Phi(u)+\Psi(u)=u$ holds for all $x\in I$ and $u\in J$ . Show that if $f:I\to J$ is a continuous solution of the functional inequality $$f\big(\varphi(x)+\psi(y)\big)\le \Phi\big(f(x)\big)+\Psi\big(f(y)\big) (x,y\in I),$$ then $\Phi\circ f\circ \varphi^{-1}$ and $\Psi\circ f\circ \psi^{-1}$ are convex functions.

Characterize all increasing sequences $(s_n)$ of positive reals for which there exists a set $A\subset \mathbb{R}$ with positive measure such that $\lambda(A\cap I)<\frac{s_n}{n}$ holds for every interval $I$ with length $1/n$ , where $\lambda$ denotes the Lebesgue measure.

Let the base $2$ representation of $x\in[0;1)$ be $x=\sum_{i=0}^\infty \frac{x_i}{2^{i+1}}$ . (If $x$ is dyadically rational, i.e. $x\in\left\{\frac{k}{2^n}\,: k,n\in\mathbb{Z}\right\}$ , then we choose the finite representation.) Define function $f_n:[0;1)\to\mathbb{Z}$ by $$f_n(x)=\sum_{j=0}^{n-1}(-1)^{\sum_{i=0}^j x_i}.$$ Does there exist a function $\varphi:[0;\infty)\to[0;\infty)$ such that $\lim_{x\to\infty} \varphi(x)=\infty$ and $$\sup_{n\in\mathbb{N}}\int_0^1 \varphi(|f_n(x)|)\mathrm{d}x<\infty ?$$

Let $N$ be a normed linear space with a dense linear subspace $M$ . Prove that if $L_1,\ldots,L_m$ are continuous linear functionals on $N$ , then for all $x\in N$ there exists a sequence $(y_n)$ in $M$ converging to $x$ satisfying $L_j(y_n)=L_j(x)$ for all $j=1,\ldots,m$ and $n\in \mathbb{N}$ .

Let $X_1,X_2,\ldots$ be independent and identically distributed random variables with distribution $\mathbb{P}(X_1=0)=\mathbb{P}(X_1=1)=\frac12$ . Let $Y_1$ , $Y_2$ , $Y_3$ , and $Y_4$ be independent, identically distributed random variables, where $Y_1:=\sum_{k=1}^\infty \frac{X_k}{16^k}$ . Decide whether the random variables $Y_1+2Y_2+4Y_3+8Y_4$ and $Y_1+4Y_3$ are absolutely continuous.