OBM 2012

In a culturing of bacteria, there are two species of them: red and blue bacteria.

When two red bacteria meet, they transform into one blue bacterium. When two blue bacteria meet, they transform into four red bacteria. When a red and a blue bacteria meet, they transform into three red bacteria.

Find, in function of the amount of blue bacteria and the red bacteria initially in the culturing, all possible amounts of bacteria, and for every possible amount, the possible amounts of red and blue bacteria.

$ABC$ is a non-isosceles triangle. $T_A$ is the tangency point of incircle of $ABC$ in the side $BC$ (define $T_B$ , $T_C$ analogously). $I_A$ is the ex-center relative to the side BC (define $I_B$ , $I_C$ analogously). $X_A$ is the mid-point of $I_BI_C$ (define $X_B$ , $X_C$ analogously).

Show that $X_AT_A$ , $X_BT_B$ , $X_CT_C$ meet in a common point, colinear with the incenter and circumcenter of $ABC$ .

Find the least non-negative integer $n$ such that exists a non-negative integer $k$ such that the last 2012 decimal digits of $n^k$ are all $1$ 's.

There exists some integers $n,a_1,a_2,\ldots,a_{2012}$ such that

$$n^2=\sum_{1 \leq i \leq 2012}{{a_i}^{p_i}}$$

where $p_i$ is the i-th prime ( $p_1=2,p_2=3,p_3=5,p_4=7,\ldots$ ) and $a_i>1$ for all $i$ ?

In how many ways we can paint a $N \times N$ chessboard using 4 colours such that squares with a common side are painted with distinct colors and every $2 \times 2$ square (formed with 4 squares in consecutive lines and columns) is painted with the four colors?

Find all surjective functions $f\colon (0,+\infty) \to (0,+\infty)$ such that $2x f(f(x)) = f(x)(x+f(f(x)))$ for all $x>0$ .