MathDB

1

Limit distribution of a sum of sines

U

be a random variable that is uniformly distributed on the interval

[0,1]

, and let

S_n= 2\sum_{k=1}^n \sin(2kU\pi).

Show that, as

n\to \infty

, the limit distribution of

S_n

is the Cauchy distribution with density function

f(x)=\frac1{\pi(1+x^2)}

.

10

1

Convex sets assigned to a triangulation

To each vertex of a given triangulation of the two-dimensional sphere, we assign a convex subset of the plane. Assume that the three convex sets corresponding to the three vertices of any two-dimensional face of the triangulation have at least one point in common. Show that there exist four vertices such that the corresponding convex sets have at least one point in common.

9

1

Barycenters with a special weight function

Let

\rho:\mathbb{R}^n\to \mathbb{R}

,

\rho(\mathbf{x})=e^{-||\mathbf{x}||^2}

, and let

K\subset \mathbb{R}^n

be a convex body, i.e., a compact convex set with nonempty interior. Define the barycenter

\mathbf{s}_K

of the body

K

with respect to the weight function

\rho

by the usual formula

\mathbf{s}_K=\frac{\int_K\rho(\mathbf{x})\mathbf{x}d\mathbf{x}}{\int_K\rho(\mathbf{x})d\mathbf{x}}.

Prove that the translates of the body

K

have pairwise distinct barycenters with respect to

\rho

.

6

1

Direct sum of subspaces

Let

\rho:G\to GL(V)

be a representation of a finite

p

-group

G

over a field of characteristic

p

. Prove that if the restriction of the linear map

\sum_{g\in G} \rho(g)

to a finite dimensional subspace

W

of

V

is injective, then the subspace spanned by the subspaces

\rho(g)W

(g\in G)

is the direct sum of these subspaces.

1

Select a separating family

Let

n

be a positive integer. Let

\mathcal{F}

be a family of sets that contains more than half of all subsets of an

n

-element set

X

. Prove that from

\mathcal{F}

we can select

\lceil \log_2 n \rceil + 1

sets that form a separating family on

X

, i.e., for any two distinct elements of

X

there is a selected set containing exactly one of the two elements.
Moderator says: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=41&t=614827&hilit=Schweitzer+2014+separating

4

1

Zeta sum and big Oh

For a positive integer

n

, define

f(n)

to be the number of sequences

(a_1,a_2,\dots,a_k)

such that

a_1a_2\cdots a_k=n

where

a_i\geq 2

and

k\ge 0

is arbitrary. Also we define

f(1)=1

. Now let

\alpha>1

be the unique real number satisfying

\zeta(\alpha)=2

, i.e

\sum_{n=1}^{\infty}\frac{1}{n^\alpha}=2

Prove that
(a)

\sum_{j=1}^{n}f(j)=\mathcal{O}(n^\alpha)

(b) There is no real number

\beta<\alpha

such that

\sum_{j=1}^{n}f(j)=\mathcal{O}(n^\beta)

3

1

Colored triangles with disjoint interiors

We have

4n + 5

points on the plane, no three of them are collinear. The points are colored with two colors. Prove that from the points we can form

n

empty triangles (they have no colored points in their interiors) with pairwise disjoint interiors, such that all points occurring as vertices of the

n

triangles have the same color.

5

1

Divergent sum of inverses

Let

\alpha

be a non-real algebraic integer of degree two, and let

\mathbb{P}

be the set of irreducible elements of the ring \mathbb{Z}[ \alpha] . Prove that

\sum_{p\in \mathbb{P}}^{{}}\frac{1}{|p|^{2}}=\infty

2

1

Inequality on intervals

Let

k\geq 1

and let

I_{1},\dots, I_{k}

be non-degenerate subintervals of the interval

[0, 1]

. Prove that

\sum \frac{1}{\left | I_{i}\cup I_{j} \right |} \geq k^{2}

where the summation is over all pairs

(i, j)

of indices such that

I_i\cap I_j\neq \emptyset

.

8

1

Uniform approximation by polynomials...

Let

n\ge 1

be a fixed integer. Calculate the distance

\inf_{p,f}\, \max_{0\le x\le 1} |f(x)-p(x)|

, where

p

runs over polynomials of degree less than

n

with real coefficients and

f

runs over functions

f(x)= \sum_{k=n}^{\infty} c_k x^k

defined on the closed interval

[0,1]

, where

c_k \ge 0

and

\sum_{k=n}^{\infty} c_k=1

.

7

1

Does null Minkowski sum imply a function is linear?

Let

f : \mathbb{R} \to \mathbb{R}

be a continuous function and let

g : \mathbb{R} \to \mathbb{R}

be arbitrary. Suppose that the Minkowski sum of the graph of

f

and the graph of

g

(i.e., the set

\{( x+y; f(x)+g(y) ) \mid x, y \in \mathbb{R}\}

) has Lebesgue measure zero. Does it follow then that the function

f

is of the form

f(x) = ax + b

with suitable constants

a, b \in \mathbb{R}

?

2014 Miklós Schweitzer

Subcontests

Limit distribution of a sum of sines

Convex sets assigned to a triangulation

Barycenters with a special weight function

Direct sum of subspaces

Select a separating family

Zeta sum and big Oh

Colored triangles with disjoint interiors

Divergent sum of inverses

Inequality on intervals

Uniform approximation by polynomials...

Does null Minkowski sum imply a function is linear?