Hanoi Open Mathematics Competition 2010 Junior

Compare the numbers
$$P=\underbrace{888\ldots888}_{2010\text{ digits}}\times\underbrace{333\ldots333}_{2010\text{ digits}}$$
and
$$Q=\underbrace{444\ldots444}_{2010\text{ digits}}\times\underbrace{666\ldots667}_{2010\text{ digits}}.$$

The number of integers $n$ from the set $\{2000,2001,\ldots,2010\}$ such that $2^{2n}+2^n+5$ is divisible by $7$ is:

The last five digits of the number $M=5^{2010}$ are:

How many real numbers $a\in(1,9)$ are there such that $a-\dfrac1a$ is an integer?

Each box in a $2\times2$ table can be colored black or white. How many different colorings of the table are there?

The greatest integer less than $(2+\sqrt3)^5$ is:

Determine all positive integers $a$ such that the equation $2x^2-30x+a=0$ has two prime roots, i.e. both roots are prime numbers.

If $n$ and $n^3+2n^2+2n+4$ are both perfect squares, find $n$.

Given a triangle $ABC$, let $D,M,N$ belong to $BC,AB,AC$, respectively. Suppose that $MD\parallel AC$ and $ND\parallel AB$. If $S_{\triangle BMD}=9\text{ cm}^2$ and $S_{\triangle DNC}=25\text{ cm}^2$, compute $S_{\triangle AMN}$.

Find the maximum value of $$M=\frac{x}{2x+y}+\frac{y}{2y+z}+\frac{z}{2z+x},\qquad x,y,z>0.$$