SCHWEITZER 1998

Can there be a continuum set of continuum sets such that (i) the intersection of any two is finite, and (ii) every set that intersects all sets intersects any in an infinite set? note: a continuum set is a set that can be put into a 1-to-1 bijection with the reals.

For any polynomial f, denote by $P_f$ the number of integers n for which f(n) is a (positive) prime number. Let $q_d = max P_f$ , where f runs over all polynomials with integer coefficients with degree d and reducible over $\mathbb{Q}$ . Prove that $\forall d\geq 2$ , $q_d = d$ .

Let p be a prime and $f: Z_p \to C$ a complex valued function defined on a cyclic group of order p. Define the Fourier transform of f by the formula: $$\hat f (k) = \sum_{l = 0}^{p-1} f (l) e^{i2\pi kl / p} (k \in Z_p)$$ Show that if the combined number of zeros of f and $\hat f$ is at least p, then f is identically zero.

related: https://artofproblemsolving.com/community/c7h22594

For any measurable set $H \subset R$ , we define the sequence $a_n(H)$ by the formula: $$a_n(H) = \lambda \bigg([0,1] \setminus \bigcup_{k = n}^{2n} (H + \log_2 k) \bigg)$$ where $\lambda$ denotes the Lebesgue measure and $\log_2$ denotes the binary logarithm. Prove that there is a measurable, 1-periodic, positive measure set $H \subset R$ , such that the sequence $a_n( H )$ does not belong to any space $l_p$ ( $1 \leq p < \infty$ ).

Let $K_1$ be an open disk in the complex plane whose boundary passes through the points -1 and +1, and let $K_2$ be the mirror image of $K_1$ across the real axis. Also, let $D_1 = K_1 \cap K_2$ , and let $D_2$ be the outside of $D_1$ . Suppose that the function $u_1( z )$ is harmonic on $D_1$ and continuous on its closure, $u_2(z)$ harmonic on $D_2$ (including $\infty$ ) and continuous on its closure, and $u_1(z) = u_2(z)$ at the common boundary of the domains $D_1$ and $D_2$ . Prove that if $u_1( x )\geq 0$ for all $-1 < x <1$ , then $u_2 ( x )\geq 0$ for all $x>1$ and $x<-1$ .

Let U be the union of a finite number (not necessarily connected and not necessarily disjoint) of closed unit squares lying in the plane. Can the quotient of the perimeter and area of U be arbitrarily large?

@below: i think "single" means "connected".

Let P be a set of 4n points in the plane such that none of the three points are collinear. Prove that if n is large enough, then the following two statements are equivalent. (i) P can be divided into n four-element subsets such that each subset forms the vertices of a convex quadrilateral. (ii) P can not be split into two sets A and B, each with an odd number of elements, so that each convex quadrilateral whose vertices are in P has an even number of vertices in A and B.

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

Let G be a domain (connected open set) in the $R^2$ plane whose boundary is locally connected. Prove that for every point q of the boundary of G there exists a simple arc $v_q$ in which $q\in v_q$ and $v_q\setminus\{q\}\subset G$ .

other questions: (i) Show that local connectedness cannot be replaced by connectedness. (ii) Show that if we replace the $R^2$ plane with $R^3$ space, the statement does not hold.

Let $\xi_1 , \xi_2 , ...$ be a series of independent, zero-expected-value random variables for which $\lim_{n\to\infty} E(\xi_n ^ 2) = 0$ , and $S_n = \sum_{j = 1}^n \xi_j$ . Denote by I(A) the indicator function of event A. Prove that $$\frac{1}{\log n} \sum_{k = 1}^n \frac1k I\bigg(\max_{1\leq j\leq k} |S_j|>\sqrt k\bigg) \to 0$$ with probability 1 if $n\to\infty$ .