CIIM 2009

Prove that for any positive integer $n$ the number $\left(\frac{3+\sqrt{17}}{2}\right)^n+\left(\frac{3-\sqrt{17}}{2}\right)^n$ is an odd integer.

Determine if for all natural $n$ there is a $n \times n$ matrix of real entries such that its determinant is 0 and that changing any entry produce another matrix with nonzero determinant.

Let $r > n$ be positive integers. A "good word" is an $n$ -tuple $\langle a_1,\dots, a_n \rangle$ 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 $A= \langle a_1,\dots, a_n \rangle$ and $B = \langle b_1,\dots, b_n \rangle$ is the minimun number of plays needed to obtain B from A. Find the maximun posible distance between two good words.

Let $m$ be a line in the plane and $M$ a point not in $m$ . Find the locus of the focus of the parabolas with vertex $M$ that are tangent to $m$ .

Let $f:\mathbb{R} \to \mathbb{R}$ , such that

i) For all $a \in \mathbb{R}$ and all $\epsilon > 0$ , exists $\delta > 0$ such that $|x-a| < \delta \Rightarrow f(x) < f(a) + \epsilon.$ ii) For all $b\in \mathbb{R}$ and all $\epsilon > 0$ , exists $x,y \in \mathbb{R}$ with $b - \epsilon < x < b < y < b + \epsilon$ , such that $|f(x)-f(b)|< \epsilon$ and $|f(y)-f(b)| < \epsilon.$ Prove that if $f(a) < d < f(d)$ there exists $c$ with $a < c < b$ or $b < c < a$ such that $f(c) = d$ .

Let $\epsilon$ be an $n$ -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 $z=q(2\cos(2\pi/n))$ .