2009 CIIM

Part of CIIM

Subcontests

(6)

1

CIIM 2009 Problem 5

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

, such that
i) For all

a \in \mathbb{R}

\epsilon > 0

\delta > 0

|x-a| < \delta \Rightarrow f(x) < f(a) + \epsilon.

b\in \mathbb{R}

\epsilon > 0

x,y \in \mathbb{R}

b - \epsilon < x < b < y < b + \epsilon

|f(x)-f(b)|< \epsilon

|f(y)-f(b)| < \epsilon.

f(a) < d < f(d)

c

a < c < b

b < c < a

f(c) = d

1

CIIM 2009 Problem 4

m

be a line in the plane and

M

m

. Find the locus of the focus of the parabolas with vertex

M

that are tangent to

m

1

CIIM 2009 Problem 6

\epsilon

n

-th root of the unity and suppose

z=p(\epsilon)

is a real number where

p

is some polinomial with integer coefficients. Prove there exists a polinomial

q

with integer coefficients such that

z=q(2\cos(2\pi/n))

1

CIIM 2009 Problem 3

r > n

be positive integers. A "good word" is an

n

\langle a_1,\dots, a_n \rangle

of distinct positive integers between 1 and

r

. A "play" consist of changing a integer

a_i

of a good word, in such a way that the resulting word is still a good word. The distance between two good words

A= \langle a_1,\dots, a_n \rangle

B = \langle b_1,\dots, b_n \rangle

is the minimun number of plays needed to obtain B from A. Find the maximun posible distance between two good words.

1

CIIM 2009 Problem 2

Determine if for all natural

n

n \times n

matrix of real entries such that its determinant is 0 and that changing any entry produce another matrix with nonzero determinant.

1

CIIM 2009 Problem 1

Prove that for any positive integer

n

\left(\frac{3+\sqrt{17}}{2}\right)^n+\left(\frac{3-\sqrt{17}}{2}\right)^n

is an odd integer.