2012 CIIM

Part of CIIM

Subcontests

(6)

1

CIIM 2012 Problem 5

D=\{0,1,\dots,9\}

. A direction function for

D

f:D \times D \to \{0,1\}.

r\in [0,1]

is compatible with

f

if it can be written in the form

r = \sum_{j=1}^{\infty} \frac{d_j}{10^j}

d_j \in D

f(d_j,d_{j+1})=1

for every positive integer

j

.
Determine the least integer

k

such that for any direction fuction

f

k

compatible reals with

f

then there are infinite reals compatible with

f

1

CIIM 2012 Problem 4

f(x) = \frac{\sin(x)}{x}

\lim_{T\to\infty}\frac{1}{T}\int_0^T\sqrt{1+f'(x)^2}dx.

1

CIIM 2012 Problem 1

For each positive integer

n

A_n

n \times n

matrix such that its

a_{ij}

entry is equal to

{i+j-2 \choose j-1}

1\leq i,j \leq n.

Find the determinant of

A_n

1

CIIM 2012 Problem 6

n \geq 2

p(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0

a polynomial with real coefficients. Show that if there exists a positive integer

k

(x-1)^{k+1}

p(x)

\sum_{j=0}^{n-1}|a_j| > 1 +\frac{2k^2}{n}.

1

CIIM 2012 Problem 2

A\subset \mathbb{Z}

is "padre" if whenever

x,y \in A

x\leq y

2y -x \in A

. Prove that if

A

0,a,b \in A

0< a < b

d = g.c.d(a,b)

a+b-3d, a+b-2d \in A.

1

CIIM 2012 Problem 3

a,b,c,

the lengths of the sides of a triangle. Prove that

\sqrt{\frac{(3a+b)(3b+a)}{(2a+c)(2b+c)}} + \sqrt{\frac{(3b+c)(3c+b)}{(2b+a)(2c+a)}} + \sqrt{\frac{(3c+a)(3a+c)}{(2c+b)(2a+b)}} \geq 4.