RMO 2009

Show that $3^{2008} + 4^{2009}$ can be written as product of two positive integers each of which is larger than $2009^{182}$ .

A convex polygon is such that the distance between any two vertices does not exceed $1$ . $(i)$ Prove that the distance between any two points on the boundary of the polygon does not exceed $1$ . $(ii)$ If $X$ and $Y$ are two distinct points inside the polygon, prove that there exists a point $Z$ on the boundary of the polygon such that $XZ + YZ\le1$ .

In a book with page numbers from $1$ to $100$ some pages are torn off. The sum of the numbers on the remaining pages is $4949$ . How many pages are torn off?