SMO 2011 Junior Round 2

Suppose $a,b,c,d>0$ and $x=\sqrt{a^2+b^2}$, $y=\sqrt{c^2+d^2}$. Prove that $$xy\ge ac+bd.$$

Two circles $\Gamma_1,\Gamma_2$ with radii $r_1,r_2$, respectively, touch internally at the point $P$. A tangent parallel to the diameter through $P$ touches $\Gamma_1$ at $R$ and intersects $\Gamma_2$ at $M$ and $N$. Prove that $PR$ bisects $\angle MPN$.

Let $S_1,S_2,\ldots,S_{2011}$ be nonempty sets of consecutive integers such that any $2$ of them have a common element. Prove that there is an integer that belongs to every $S_i$, $i=1,\ldots,2011$. (For example, $\{2,3,4,5\}$ is a set of consecutive integers while $\{2,3,5\}$ is not.)

Any positive integer $n$ can be written in the form $n=2^aq$, where $a\ge0$ and $q$ is odd. We call $q$ the odd part of $n$. Define the sequence $a_0,a_1,\ldots$ as follows: $a_0=2^{2011}-1$ and for $m\ge0$, $a_{m+1}$ is the odd part of $3a_m+1$. Find $a_{2011}$.

Initially the number $10$ is written on the board. In each subsequent move, you can either (i) erase the number $1$ and replace it with a $10$, or (ii) erase the number $10$ and replace it with a $1$ and a $25$, or (iii) erase a $25$ and replace it with two $10$s. After some time, you notice that there are exactly one hundred copies of $1$ on the board. What is the least possible sum of all the numbers on the board at that moment?