MathDB

1

IMC 2021 P8: Maximum number of vectors such that for any 3, 2 are orthogonal

n

be a positive integer. At most how many distinct unit vectors can be selected in

\mathbb{R}^n

such that from any three of them, at least two are orthogonal?

7

1

IMC 2021 P7: Maximum in the boundary multiplied by monic polynomial

Let

D \subseteq \mathbb{C}

be an open set containing the closed unit disk

\{z : |z| \leq 1\}

. Let

f : D \rightarrow \mathbb{C}

be a holomorphic function, and let

p(z)

be a monic polynomial. Prove that

|f(0)| \leq \max_{|z|=1} |f(z)p(z)|

6

1

IMC 2021 P6: No injective homomorphism from GL_2(Z/pZ) to S_p

For a prime number

p

, let

GL_2(\mathbb{Z}/p\mathbb{Z})

be the group of invertible

2 \times 2

matrices of residues modulo

p

, and let

S_p

be the symmetric group (the group of all permutations) on

p

elements. Show that there is no injective group homomorphism

\phi : GL_2(\mathbb{Z}/p\mathbb{Z}) \rightarrow S_p

.

5

1

IMC 2021 P5: 2021B=A^m+B^2

Let

A

be a real

n \times n

matrix and suppose that for every positive integer

m

there exists a real symmetric matrix

B

such that

2021B = A^m+B^2.

Prove that

|\text{det} A| \leq 1

.

4

1

IMC 2021 first day , problem 4

Let

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

be a function. Suppose that for every

\varepsilon >0

, there exists a function

g:\mathbb{R}\to (0,\infty)

such that for every pair

(x,y)

of real numbers, if

|x-y|<\text{min}\{g(x),g(y)\}

, then

|f(x)-f(y)|<\varepsilon

Prove that

f

is pointwise limit of a squence of continuous

\mathbb{R}\to \mathbb{R}

functions i.e., there is a squence

h_1,h_2,...,

of continuous

\mathbb{R}\to \mathbb{R}

such that

\lim_{n\to \infty}h_n(x)=f(x)

for every

x\in \mathbb{R}

3

1

IMC 2021, first day , problem 3

We say that a positive real number

d

is

good

if there exists an infinite squence

a_1,a_2,a_3,...\in (0,d)

such that for each

n

, the points

a_1,a_2,...,a_n

partition the interval

[0,d]

into segments of length at most

\frac{1}{n}

each . Find

\text{sup}\{d| d \text{is good}\}

.

2

1

IMC 2021, first day , problem 2

Let

n

and

k

be fixed positive integers , and

a

be arbitrary nonnegative integer . Choose a random

k

-element subset

X

of

\{1,2,...,k+a\}

uniformly (i.e., all k-element subsets are chosen with the same probability) and, independently of

X

, choose random n-elements subset

Y

of

\{1,2,..,k+a+n\}

uniformly. Prove that the probability

P\left( \text{min}(Y)>\text{max}(X)\right)

does not depend on

a

.

1

IMC 2021, first day, problem 1

Let

A

be a real

n\times n

matrix such that

A^3=0

a) prove that there is unique real

n\times n

matrix

X

that satisfied the equation

X+AX+XA^2=A

b) Express

X

in terms of

A

2021 IMC

Subcontests

IMC 2021 P8: Maximum number of vectors such that for any 3, 2 are orthogonal

IMC 2021 P7: Maximum in the boundary multiplied by monic polynomial

IMC 2021 P6: No injective homomorphism from GL_2(Z/pZ) to S_p

IMC 2021 P5: 2021B=A^m+B^2

IMC 2021 first day , problem 4

IMC 2021, first day , problem 3

IMC 2021, first day , problem 2

IMC 2021, first day, problem 1