MathDB

1

Affine transformation with rectangles in the plane

A

and

B

be rectangles in the plane and

f : A \rightarrow B

be a mapping which is uniform on the interior of

A

, maps the boundary of

A

homeomorphically to the boundary of

B

by mapping the sides of

A

to corresponding sides in

B

. Prove that

f

is an affine transformation.

2.2

1

A wicked game of expectations... and integral probability functions

Petya and Vasya are playing the following game. Petya chooses a non-negative random value

\xi

with expectation

\mathbb{E} [\xi ] = 1

, after which Vasya chooses his own value

\eta

with expectation

\mathbb{E} [\eta ] = 1

without reference to the value of

\xi

. For which maximal value

p

can Petya choose a value

\xi

in such a way that for any choice of Vasya's

\eta

, the inequality

\mathbb{P}[\eta \geq \xi ] \leq p

holds?

1.5 & 2.5

1

Scary triginequalinometry on the reals

Prove the inequality

\sum _{k = 1} ^n (x_k - x_{k-1})^2 \geq 4 \sin ^2 \frac{\pi}{2n} \cdot \sum ^n _{k = 0} x_k ^2

for any sequence of real numbers

x_0, x_1, ..., x_n

for which

x_0 = x_n = 0.

1.4

1

Subsets and their centers of mass in a unit sphere

Suppose that in a unit sphere in Euclidean space, there are

2m

points

x_1, x_2, ..., x_{2m}.

Prove that it's possible to partition them into two sets of

m

points in such a way that the centers of mass of these sets are at a distance of at most

\frac{2}{\sqrt{m}}

from one another.

1.3

1

Rank preservation after interchanging positives and negatives in a matrix

Given a natural number

n

, for what maximal value

k

it is possible to construct a matrix of size

k \times n

consisting only of elements

\pm 1

in such a way that for any interchange of a

+1

with a

-1

or vice versa, its rank is equal to

k

?

1.2

1

An irrational, strictly-increasing, real-valued function

Does there exist a strictly increasing function

f: \mathbb{R} \rightarrow \mathbb{R}

that takes on only irrational values?

1.1 & 2.1

1

A sequence of homothetic cubes

In

\mathbb{R}^3

, let there be a cube

Q

and a sequence of other cubes, all of which are homothetic to

Q

with coefficients of homothety that are each smaller than

1

. Prove that if this sequence of homothetic cubes completely fills

Q

, the sum of their coefficients of homothety is not less than

4

.

MIPT Undergraduate Contest 2019

Subcontests

Affine transformation with rectangles in the plane

A wicked game of expectations... and integral probability functions

Scary triginequalinometry on the reals

Subsets and their centers of mass in a unit sphere

Rank preservation after interchanging positives and negatives in a matrix

An irrational, strictly-increasing, real-valued function

A sequence of homothetic cubes