CIIM 2010

Given two vectors $v = (v_1,\dots,v_n)$ and $w = (w_1\dots,w_n)$ in $\mathbb{R}^n$ , lets define $v*w$ as the matrix in which the element of row $i$ and column $j$ is $v_iw_j$ . Supose that $v$ and $w$ are linearly independent. Find the rank of the matrix $v*w - w*v.$

In one side of a hall there are $2N$ rooms numbered from 1 to $2N$ . In each room $i$ between 1 and $N$ there are $p_i$ beds. Is needed to move every one of this beds to the roms from $N+ 1$ to $2N$ , in such a way that for every $j$ between $N+1$ and $2N$ the room $j$ will have $p_j$ beds. Supose that each bed can be move once and the price of moving a bed from room $i$ to room $j$ is $(i-j)^2$ . Find a way to move every bed such that the total cost is minimize.

Note: The numbers $p_i$ are given and satisfy that $p_1 + p_2 + \cdots + p_N = p_{N+1} + p_{N+2} + \cdots+ p_{2N}.$

A set $X\subset \mathbb{R}$ has dimension zero if, for any $\epsilon > 0$ there exists a positive integer $k$ and intervals $I_1,I_2,...,I_k$ such that $X \subset I_1 \cup I_2 \cup \cdots \cup I_k$ with $\sum_{j=1}^k |I_j|^{\epsilon} < \epsilon$ .

Prove that there exist sets $X,Y \subset [0,1]$ both of dimension zero, such that $X+Y = [0,2].$

Let $f:[0,1] \to [0,1]$ a increasing continuous function, diferentiable in $(0,1)$ and with derivative smaller than 1 in every point. The sequence of sets $A_1,A_2,A_3,\dots$ is define as: $A_1 = f([0,1])$ , and for $n \geq 2, A_n = f(A_{n-1}).$ Prove that $\lim_{n\to+\infty} d(A_n) = 0$ , where $d(A)$ is the diameter of the set $A$ .

Note: The diameter of a set $X$ is define as $d(X) = \sup_{x,y\in X} |x-y|.$

Let $n,d$ be integers with $n,k > 1$ such that $g.c.d(n,d!) = 1$ . Prove that $n$ and $n+d$ are primes if and only if $$d!d((n-1)!+1) + n(d!-1) \equiv 0 \hspace{0.2cm} (\bmod n(n+d)).$$

A group is call locally cyclic if any finitely generated subgroup is cyclic. Prove that a locally cyclic group is isomorphic to one of its proper subgroups if and only if it's isomorphic to a proper subgroup of the rational numbers with the adition.