Olympiad problems

2009 CIIM, Problem 2

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.

2009 CIIM, Problem 1

Tags: 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.

2009 CIIM, Problem 6

Tags: CIIM 2009 , CIIM , undergraduate

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

2009 CIIM, Problem 5

Tags: CIIM 2009 , CIIM , Undegraduate

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 $f(c) = d$.

2009 CIIM, Problem 4