2015 CIIM

Part of CIIM

Subcontests

(6)

1

CIIM 2015 Problem 6

Show that there exists a real

C > 1

that satisfy the following property: if

n > 1

a_0 < a_1 < \cdots < a_n

are positive integers such that

\frac{1}{a_0},\frac{1}{a_1},\dots,\frac{1}{a_n}

are in arithmetic progression, then

a_0 > C^n.

1

CIIM 2015 Problem 5

n

people seated on a circular table that have seats numerated from 1 to

n

k

be a fix integer with

2 \leq k \leq n

. The people can change their seats. There are two types of moves permitted:
1. Each person moves to the next seat clockwise. 2. Only the ones in seats 1 and

k

exchange their seats.
Determine, in function of

n

k

, the number of possible configurations of people in the table that can be attain by using a sequence of permitted moves.

1

CIIM 2015 Problem 4

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

a continuos function and

\alpha

a real number such that

\lim_{x\to\infty}f(x) = \lim_{x\to-\infty}f(x) = \alpha.

Prove that for any

r > 0,

x,y \in \mathbb{R}

y-x = r

f(x) = f(y).

1

CIIM 2015 Problem 3

Consider the matrices A = \left(\begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right) \\ \mbox{ and } \\ B = \left(\begin{matrix} 1 & 0 \\ 2 & 1 \end{matrix}\right). Let

k\geq 1

an integer. Prove that for any nonzero

i_1,i_2,\dots,i_{k-1},j_1,j_2,\dots,j_k

and any integers

i_0,i_k

A^{i_0}B^{j_1}A^{i_1}B^{j_2}\cdots A^{i_{k-1}}B^{i_k}A^{i_k} \not = I.

1

CIIM 2015 Problem 2

Find all polynomials

P(x)

with real coefficients that satisfy the identity

P(x^3-2)=P(x)^3-2,

for every real number

x

1

CIIM 2015 Problem 1

Find the real number

a

such that the integral

\int_a^{a+8}e^{-x}e^{-x^2}dx

attain its maximum.