PE-MO 2011

We say that a positive integer is irregular if said number is not a multiple of none of its digits. For example, $203$ is irregular because $203$ is not a multiple of $2$ , it is not multiple of $0$ and is not a multiple of $3$ . Consider a set consisting of $n$ consecutive positive integers. If all the numbers in that set are irregular, determine the largest possible value of $n$ .

If $\alpha, \beta, \gamma$ are angles whose measures in radians belong to the interval $\left[0, \frac{\pi}{2}\right]$ such that: $$\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 1$$ calculate the minimum possible value of $\cos \alpha + \cos \beta + \cos \gamma$ .

Let $ABC$ be a right triangle, right in $B$ . Inner bisectors are drawn $CM$ and $AN$ that intersect in $I$ . Then, the $AMIP$ and $CNIQ$ parallelograms are constructed. Let $U$ and $V$ are the midpoints of the segments $AC$ and $PQ$ , respectively. Prove that $UV$ is perpendicular to $AC$ .