2014 CIIM

Part of CIIM

Subcontests

(6)

1

CIIM 2014 Problem 6

\{x_n\}

be a sequence with

x_n \in [0,1]

n

. Prove that there exists

C > 0

such that for every positive integer

r

m \geq 1

n > m + r

(n-m)|x_n-x_m| \leq C

. b) Prove that for every

C > 0

, there exists a sequence

\{x_n\}

x_n \in [0,1]

n

r

m \geq 1

n > m+r

(n-m)|x_n-x_m| > C.

1

CIIM 2014 Problem 5

A analityc function

f:\mathbb{C}\to\mathbb{C}

is call interesting if

f(z)

is real along the parabola

Re (z) = (Im (z))^2

.
a) Find an example of a non constant interesting function. b) Show that every interesting function

f

f'(-3/4) = 0.

1

CIIM 2014 Problem 4

\{a_i\}

be a strictly increasing sequence of positive integers. Define the sequence

\{s_k\}

s_k = \sum_{i=1}^{k}\frac{1}{[a_i,a_{i+1}]},

[a_i,a_{i+1}]

is the least commun multiple of

a_i

a_{i+1}

. Show that the sequence

\{s_k\}

1

CIIM 2014 Problem 3

n\geq2

\mathcal{A}

be a family of subsets of the set

\{1,2,\dots,n\}

such that, for any

A_1,A_2,A_3,A_4 \in \mathcal{A}

, it holds that

|A_1 \cup A_2 \cup A_3 \cup A_4| \leq n -2

|\mathcal{A}| \leq 2^{n-2}.

1

CIIM 2013 Problem 2

n

be an integer and

p

a prime greater than 2. Show that:

(p-1)^nn!|(p^n-1)(p^n-p)(p^n-p^2)\cdots(p^n-p^{n-1}).

1

CIIM 2014 Problem 1

g:[2013,2014]\to\mathbb{R}

a function that satisfy the following two conditions:
i)

g(2013)=g(2014) = 0,

a,b \in [2013,2014]

g\left(\frac{a+b}{2}\right) \leq g(a) + g(b).

g

has zeros in any open subinterval

(c,d) \subset[2013,2014].