CMO 2006

For positive integers $a_1,a_2 ,\ldots,a_{2006}$ such that $\frac{a_1}{a_2},\frac{a_2}{a_3},\ldots,\frac{a_{2005}}{a_{2006}}$ are pairwise distinct, find the minimum possible amount of distinct positive integers in the set $\{a_1,a_2,...,a_{2006}\}$ .

Positive integers $k, m, n$ satisfy $mn=k^2+k+3$ , prove that at least one of the equations $x^2+11y^2=4m$ and $x^2+11y^2=4n$ has an odd solution.

In a right angled-triangle $ABC$ , $\angle{ACB} = 90^o$ . Its incircle $O$ meets $BC$ , $AC$ , $AB$ at $D$ , $E$ , $F$ respectively. $AD$ cuts $O$ at $P$ . If $\angle{BPC} = 90^o$ , prove $AE + AP = PD$ .

Suppose $X$ is a set with $|X| = 56$ . Find the minimum value of $n$ , so that for any 15 subsets of $X$ , if the cardinality of the union of any 7 of them is greater or equal to $n$ , then there exists 3 of them whose intersection is nonempty.