OMCC 2008

Find the least positive integer $N$ such that the sum of its digits is 100 and the sum of the digits of $2N$ is 110.

Let $ABCD$ be a convex cuadrilateral inscribed in a circumference centered at $O$ such that $AC$ is a diameter. Pararellograms $DAOE$ and $BCOF$ are constructed. Show that if $E$ and $F$ lie on the circumference then $ABCD$ is a rectangle.

There are 2008 bags numbered from 1 to 2008, with 2008 frogs in each one of them. Two people play in turns. A play consists in selecting a bag and taking out of it any number of frongs (at least one), leaving $x$ frogs in it ( $x\geq 0$ ). After each play, from each bag with a number higher than the selected one and having more than $x$ frogs, some frogs scape until there are $x$ frogs in the bag. The player that takes out the last frog from bag number 1 looses. Find and explain a winning strategy.

Five girls have a little store that opens from Monday through Friday. Since two people are always enough for taking care of it, they decide to do a work plan for the week, specifying who will work each day, and fulfilling the following conditions:

a) Each girl will work exactly two days a week b) The 5 assigned couples for the week must be different In how many ways can the girls do the work plan?

Find a polynomial $p\left(x\right)$ with real coefficients such that

$\left(x+10\right)p\left(2x\right)=\left(8x-32\right)p\left(x+6\right)$

for all real $x$ and $p\left(1\right)=210$ .

Let $ABC$ be an acute triangle. Take points $P$ and $Q$ inside $AB$ and $AC$ , respectively, such that $BPQC$ is cyclic. The circumcircle of $ABQ$ intersects $BC$ again in $S$ and the circumcircle of $APC$ intersects $BC$ again in $R$ , $PR$ and $QS$ intersect again in $L$ . Prove that the intersection of $AL$ and $BC$ does not depend on the selection of $P$ and $Q$ .