2023 CIIM

Part of CIIM

Subcontests

(6)

1

Find the constants of a limit

n

be a positive integer. We define

f(n)

as the number of finite sequences

(a_1, a_2, \ldots , a_k)

of positive integers such that

a_1 < a_2 < a_3 < \cdots < a_k

a_1+a_2^2+a_3^3+\cdots + a_k^k \leq n.

Determine the positive constants

\alpha

C

\lim\limits_{n\rightarrow \infty} \frac{f(n)}{n^\alpha}=C.

1

Find all polynomial with modulus 1 root

Given a positive integer

k > 1

, find all positive integers

n

such that the polynomial

P(z) = z^n + \sum_{j=0}^{2^k-2} z^j = 1 +z +z^2 + \cdots +z^{2^k-2} + z^n

has a complex root

w

|w| = 1

1

Find the limsup!

For a positive integer

n

\sigma(n)

denotes the sum of the positive divisors of

n

\limsup\limits_{n\rightarrow \infty} \frac{\sigma(n^{2023})}{(\sigma(n))^{2023}}

Note: Given a sequence (

a_n

) of real numbers, we say that

\limsup\limits_{n\rightarrow \infty} a_n = +\infty

a_n

) is not upper bounded, and, otherwise,

\limsup\limits_{n\rightarrow \infty} a_n

is the smallest constant

C

such that, for every real

K > C

, there is a positive integer

N

a_n < K

n > N

1

Sequence of symmetric matrices

3 \times 3

symmetric real matrix

A

f(A)

3 \times 3

matrix with the same eigenvectors of

A

A

has eigenvalues

a

b

c

f(A)

has eigenvalues

b+c

c+a

a+b

(in that order). We define a sequence of symmetric real

3\times3

A_0, A_1, A_2, \ldots

A_{n+1} = f(A_n)

n \geq 0

. If the matrix

A_0

has no zero entries, determine the maximum number of indices

j \geq 0

for which the matrix

A_j

has any null entries.

1

what is the first number we cannot make with some dice

k

dice at his disposal, each with

6

blank sides. On each side of each of these dice, the toymaker must draw one of the digits

0, 1, 2, \ldots , 9

.
Determine (in terms of

k

) the largest integer

n

such that the toymaker can draw digits on the

k

dice such that, for any positive integer

r \leq n

, it is possible to choose some of the

k

dice and form with them the decimal representation of

r

.
Note: The digits 6 and 9 are distinguishable: they appear as 6 and 9.

1

Series of integrals

Determine all the pairs of positive real numbers

(a, b)

a < b

such that the following series

\sum_{k=1}^{\infty} \int_a^b\{x\}^k dx =\int_a^b\{x\} dx + \int_a^b\{x\}^2 dx + \int_a^b\{x\}^3 dx + \cdots

is convergent and determine its value in function of

a

b

\{x\} = x - \lfloor x \rfloor

denotes the fractional part of

x