2018 CIIM

Part of CIIM

Subcontests

(6)

1

CIIM 2018 Problem 6

\{x_n\}

be a sequence of real numbers in the interval

[0,1)

. Prove that there exists a sequence

1 < n_1 < n_2 < n_3 < \cdots

of positive integers such that the following limit exists

\lim_{i,j \to \infty} x_{n_i+n_j}.

That is, there exists a real number

L

such that for every

\epsilon > 0,

there exists a positive integer

N

i,j > N

|x_{n_i+n_j}-L| < \epsilon.

1

CIIM 2018 Problem 5

Consider the transformation

T(x,y,z) = (\sin y + \sin z - \sin x,\sin z + \sin x - \sin y,\sin x +\sin y -\sin z).

Determine all the points

(x,y,z) \in [0,1]^3

T^n(x,y,z) \in [0,1]^3,

n \geq 1

1

CIIM 2018 Problem 4

\alpha < 0 < \beta

and consider the polynomial

f(x) = x(x-\alpha)(x-\beta)

S

be the set of real numbers

s

f(x) - s

has three different real roots. For

s\in S

p(x)

the product of the smallest and largest root of

f(x)-s

. Determine the smallest possible value that

p(s)

s\in S

1

CIIM 2018 Problem 3

m

be an integer and

\mathbb{Z}_m

the set of integer modulo

m

. An equivalence relation is defined in

\mathbb{Z}_m

x \sim y

if there exists a natural

t

y \equiv 2^tx \, (\bmod m)

. Find al values of

m

such that the number of equivalent classes is even.

1

CIIM 2018 Problem 2

p(x)

q(x)

non constant real polynomials of degree at most

n

n > 1

). Show that there exists a non zero polynomial

F(x,y)

in two variables with real coefficients of degree at most

2n-2,

F(p(t),q(t)) = 0

t\in \mathbb{R}

1

CIIM 2018 Problem 1

Show that there exists a

2 \times 2

matrix of order 6 with rational entries, such that the sum of its entries is 2018. Note: The order of a matrix (if it exists) is the smallest positive integer

n

A^n = I

I

is the identity matrix.