MathDB

1

Research flavoured combi

n > 3

be an integer. Let

\Omega

be the set of all triples of distinct elements of

\{1, 2, \ldots , n\}

. Let

m

denote the minimal number of colours which suffice to colour

\Omega

so that whenever

1\leq a<b<c<d \leq n

, the triples

\{a,b,c\}

and

\{b,c,d\}

have different colours. Prove that

\frac{1}{100}\log\log n \leq m \leq100\log \log n

.

3

1

Generating functions and recursions smelling from 1000 km

Let

p

be a prime number. A flea is staying at point

0

of the real line. At each minute, the flea has three possibilities: to stay at its position, or to move by

1

to the left or to the right. After

p-1

minutes, it wants to be at

0

again. Denote by

f(p)

the number of its strategies to do this (for example,

f(3) = 3

: it may either stay at

0

for the entire time, or go to the left and then to the right, or go to the right and then to the left). Find

f(p)

modulo

p

.

2

1

A good test on how well do you know matrix theory

For a positive integer

n

determine all

n\times n

real matrices

A

which have only real eigenvalues and such that there exists an integer

k\geq n

with

A + A^k = A^T

.

1

Welcome back to Blagoevgrad with a walk in the park

Let

f: [0,1] \to (0, \infty)

be an integrable function such that

f(x)f(1-x) = 1

for all

x\in [0,1]

. Prove that

\int_0^1f(x)dx \geq 1

.

8

1

Fun combinatorial geo

Let

n, k \geq 3

be integers, and let

S

be a circle. Let

n

blue points and

k

red points be chosen uniformly and independently at random on the circle

S

. Denote by

F

the intersection of the convex hull of the red points and the convex hull of the blue points. Let

m

be the number of vertices of the convex polygon

F

(in particular,

m=0

when

F

is empty). Find the expected value of

m

.

7

1

Idempotent theoretical problem

Let

A_1, \ldots, A_k

be

n\times n

idempotent complex matrices such that

A_iA_j = -A_iA_j

for all

1 \leq i < j \leq k

. Prove that at least one of the matrices has rank not exceeding

\frac{n}{k}

.

5

1

Red and Blue Counting

We colour all the sides and diagonals of a regular polygon

P

with

43

vertices either red or blue in such a way that every vertex is an endpoint of

20

red segments and

22

blue segments. A triangle formed by vertices of

P

is called monochromatic if all of its sides have the same colour. Suppose that there are

2022

blue monochromatic triangles. How many red monochromatic triangles are there?

6

1

Striking notorious similarity with ICMC 2021 Round 2

Let

p \geq 3

be a prime number. Prove that there is a permutation

(x_1,\ldots, x_{p-1})

of

(1,2,\ldots,p-1)

such that

x_1x_2 + x_2x_3 + \cdots + x_{p-2}x_{p-1} \equiv 2 \pmod p

.

2022 IMC

Subcontests

Research flavoured combi

Generating functions and recursions smelling from 1000 km

A good test on how well do you know matrix theory

Welcome back to Blagoevgrad with a walk in the park

Fun combinatorial geo

Idempotent theoretical problem

Red and Blue Counting

Striking notorious similarity with ICMC 2021 Round 2