SMO 2009 Junior Round 2

In $\triangle ABC$, $\angle A=2\angle B$. Let $a,b,c$ be the lengths of its sides $BC,CA,AB$, respectively. Prove that $$a^2=b(b+c).$$

The set of $2000$-digit integers is divided into two sets: the set $M$ consisting of all integers that can be represented as the product of two $1000$-digit integers, and the set $N$ containing the other integers. Which of the sets $M$ and $N$ contains more elements?

Suppose $\overline{a_1a_2\ldots a_{2009}}$ is a $2009$-digit integer such that for each $i=1,2,\ldots,2007$, the $2$-digit integer $\overline{a_i a_{i+1}}$ contains $3$ distinct prime factors. Find $a_{2008}$.

(Note: $\overline{xyz\ldots}$ denotes an integer whose digits are $x,y,z,\ldots$.)

Let $S$ be the set of integers that can be written in the form $50m+3n$ where $m$ and $n$ are non-negative integers. For example, $3$, $50$ and $53$ are all in $S$. Find the sum of all positive integers not in $S$.

Let $a,b$ be positive real numbers satisfying $a+b=1$. Show that if $x_1,x_2,\ldots,x_5$ are positive real numbers such that $x_1x_2\cdots x_5=1$, then $$(ax_1+b)(ax_2+b)\cdots(ax_5+b)\ge1.$$