2023 IMC

Part of IMC

Subcontests

(10)

1

The denominator is big

For every positive integer

n

f(n)

g(n)

be the minimal positive integers such that

1+\frac{1}{1!}+\frac{1}{2!}+\dots +\frac{1}{n!}=\frac{f(n)}{g(n)}.

Determine whether there exists a positive integer

n

g(n)>n^{0.999n}

1

Two convex bodies with disjoint projections

We say that a real number

V

is good if there exist two closed convex subsets

X

Y

of the unit cube in

\mathbb{R}^3

V

each, such that for each of the three coordinate planes (that is, the planes spanned by any two of the three coordinate axes), the projections of

X

Y

onto that plane are disjoint.
Find

\sup \{V\mid V\ \text{is good}\}

1

Inequality on a tree

T

n

vertices; that is, a connected simple graph on

n

vertices that contains no cycle. For every pair

u

v

of vertices, let

d(u,v)

denote the distance between

u

v

, that is, the number of edges in the shortest path in

T

u

v

.
Consider the sums W(T)=\sum_{\substack{\{u,v\}\subseteq V(T)\\ u\neq v}}d(u,v) \text{and} H(T)=\sum_{\substack{\{u,v\}\subseteq V(T)\\ u\neq v}}\frac{1}{d(u,v)} Prove that

W(T)\cdot H(T)\geq \frac{(n-1)^3(n+2)}{4}.

1

Necessary values of f'-f

V

be the set of all continuous functions

f\colon [0,1]\to \mathbb{R}

, differentiable on

(0,1)

, with the property that

f(0)=0

f(1)=1

. Determine all

\alpha \in \mathbb{R}

such that for every

f\in V

, there exists some

\xi \in (0,1)

f(\xi)+\alpha = f'(\xi)

1

Ivan now prefers matrices

Ivan writes the matrix

\begin{pmatrix} 2 & 3\\ 2 & 4\end{pmatrix}

on the board. Then he performs the following operation on the matrix several times:
1. he chooses a row or column of the matrix, and
2. he multiplies or divides the chosen row or column entry-wise by the other row or column, respectively.
Can Ivan end up with the matrix

\begin{pmatrix} 2 & 4\\ 2 & 3\end{pmatrix}

after finitely many steps?

1

Preferred permutations

Fix positive integers

n

k

2 \le k \le n

M

n

fruits. A permutation is a sequence

x=(x_1,x_2,\ldots,x_n)

\{x_1,\ldots,x_n\}=M

. Ivan prefers some (at least one) of these permutations. He realized that for every preferred permutation

x

k

i_1 < i_2 < \ldots < i_k

with the following property: for every

1 \le j < k

x_{i_j}

x_{i_{j+1}}

, he obtains another preferred permutation. \\ Prove that he prefers at least

k!

1

Residues of a polynomial modulo p

p

be a prime number and let

k

be a positive integer. Suppose that the numbers

a_i=i^k+i

i=0,1, \ldots,p-1

form a complete residue system modulo

p

. What is the set of possible remainders of

a_2

upon division by

p

1

Polynomial equation

Find all polynomials

P

in two variables with real coefficients satisfying the identity

P(x,y)P(z,t)=P(xz-yt,xt+yz).

1

Matrix equations

A

B

C

n \times n

matrices with complex entries satisfying

A^2=B^2=C^2 \text{ and } B^3 = ABC + 2I.

A^6=I

1

Differential FE

Find all functions

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

that have a continuous second derivative and for which the equality

f(7x+1)=49f(x)

x \in \mathbb{R}