1998 Miklós Schweitzer

Part of Miklós Schweitzer

Subcontests

(10)

1

probability

\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

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

1

graph partition

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.

1

analysis

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)

\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

).
[hide=not sure about this part]For what numbers

1 \leq p <\infty

is it true that whenever H is 1-periodic, positive measure, the sequence

a_n( H )

belongs to the space

l_p

1

fourier transform on Zp

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}\qquad(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

1

analysis

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}

\forall d\geq 2

q_d = d

1

first countability of space and subspace

X is a compact T2 space such that every subspace of cardinality

\aleph_1

is first countable. Prove that X is first countable.

1

analysis

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

D_2

be the outside of

D_1

. Suppose that the function

u_1( z )

D_1

and continuous on its closure,

u_2(z)

D_2

\infty

) and continuous on its closure, and

u_1(z) = u_2(z)

at the common boundary of the domains

D_1

D_2

. Prove that if

u_1( x )\geq 0

-1 < x <1

u_2 ( x )\geq 0

x>1

x<-1

1

quotient perimeter area

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".

1

continuum sets

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.

1

simple arc in domain

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

q\in v_q

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

R^3

space, the statement does not hold.