Hanoi Open Mathematics Competition 2010 Senior

The number of integers $n\in[2000,2010]$ such that $2^{2n}+2^n+5$ is divisible by $7$ is:

The last five digits of the number $5^{2010}$ 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?

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.

Let $a,b$ be the roots of $x^2-px+q=0$ and let $c,d$ be the roots of $x^2-rx+s=0$, where $p,q,r,s$ are positive real numbers. Suppose that
$$M=\frac{2(abc+bcd+cda+dab)}{p^2+q^2+r^2+s^2}$$
is an integer. Determine $a,b,c,d$.

Let $P$ be the common point of the three internal angle bisectors of a triangle $ABC$. The line passing through $P$ and perpendicular to $CP$ intersects $AC$ and $BC$ at $M$ and $N$, respectively. If $AP=3\text{ cm}$ and $BP=4\text{ cm}$, compute $\dfrac{AM}{BN}$.

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

Let $x,y$ be positive integers such that $3x^2+x=4y^2+y$. Prove that $x-y$ is a perfect integer.

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