MathDB

1

Criterion equivalent to A^n \le B^n for all n

A

and

B

be positive self-adjoint operators on a complex Hilbert space

H

. Prove that

\limsup_{n \to \infty} \|A^n x\|^{1/n} \le \limsup_{n \to \infty} \|B^n x\|^{1/n}

holds for every

x \in H

if and only if

A^n \le B^n

for each positive integer

n

.

9

1

ELMO 2019/6

8

1

f(x+t) - f(x) is locally integrable for all t

Let

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

be a measurable function such that

f(x+t) - f(x)

is locally integrable for every

t

as a function of

x

. Prove that

f

is locally integrable.

7

1

Roots of P' are in {z: |P(z)| = 1}

Given a polynomial

P

, assume that

L = \{z \in \mathbb{C}: |P(z)| = 1\}

is a Jordan curve. Show that the zeros of

P'

are in the interior of

L

.

6

1

Nonlinear recurrence approaches a-1

Let

d

be a positive integer and

1 < a \le (d+2)/(d+1)

. For given

x_0, x_1,\dots, x_d \in (0, a-1)

, let

x_{k+1} = x_k (a - x_{k-d})

,

k \ge d

. Prove that

\lim_{k \to \infty} x_k = a-1

.

5

1

Inequality related to defining convex body from halfspaces

Let

S \subset \mathbb{R}^d

be a convex compact body with nonempty interior. Show that there is an

\alpha > 0

such that if

S = \cap_{i \in I} H_i

, where

I

is an index set and

(H_i)_{i \in I}

are halfspaces, then for any

P \in \mathbb{R}^d

, there is an

i \in I

for which

\mathrm{dist}(P, H_i) \ge \alpha \, \mathrm{dist}(P, S)

.

4

1

m*n matrices of 1..mn where only 1 is the smallest in its row/column

An

n \times m

matrix is nice if it contains every integer from

1

to

mn

exactly once and

1

is the only entry which is the smallest both in its row and in its column. Prove that the number of

n \times m

nice matrices is

(nm)!n!m!/(n+m-1)!

.

3

1

All four gcds are at least c*sqrt(n)

Prove that there are infinitely many integers

m

,

n

, such that

1 < m < n

, and the greatest common divisors

(m, n)

,

(m, n+1)

,

(m+1, n)

and

(m+1, n+1)

are all greater than

\sqrt{n}/999

.

2

1

Criterion for a ring to be simple

Let

R

be a noncommutative finite ring with multiplicative identity element

1

. Show that if the subring generated by

I \cup \{1\}

is

R

for each nonzero ideal

I

then

R

is simple.

1

Sigma-compact subspaces of Hausdorff spaces

Prove that if every subspace of a Hausdorff space

X

is

\sigma

-compact, then

X

is countable.

2019 Miklós Schweitzer

Subcontests

Criterion equivalent to A^n \le B^n for all n

ELMO 2019/6

f(x+t) - f(x) is locally integrable for all t

Roots of P' are in {z: |P(z)| = 1}

Nonlinear recurrence approaches a-1

Inequality related to defining convex body from halfspaces

m*n matrices of 1..mn where only 1 is the smallest in its row/column

All four gcds are at least c*sqrt(n)

Criterion for a ring to be simple

Sigma-compact subspaces of Hausdorff spaces

2019 Mikl&oacute;s Schweitzer

Subcontests

Criterion equivalent to A^n \le B^n for all n

ELMO 2019/6

f(x+t) - f(x) is locally integrable for all t

Roots of P' are in {z: |P(z)| = 1}

Nonlinear recurrence approaches a-1

Inequality related to defining convex body from halfspaces

m*n matrices of 1..mn where only 1 is the smallest in its row/column

All four gcds are at least c*sqrt(n)

Criterion for a ring to be simple

Sigma-compact subspaces of Hausdorff spaces

2019 Miklós Schweitzer