For positive integer $k$, let $p(k)$ denote the greatest odd divisor of $k$. Prove that for every positive integer $n$, $$\frac{2n}{3} < \frac{p(1)}{1}+ \frac{p(2)}{2}+... +\frac{ p(n)}{n}<\frac{2(n + 1)}{3}$$

A box contains answer $4032$ scripts out of which exactly half have odd number of marks. We choose 2 scripts randomly and, if the scores on both of them are odd number, we add one mark to one of them, put the script back in the box and keep the other script outside. If both scripts have even scores, we put back one of the scripts and keep the other outside. If there is one script with even score and the other with odd score, we put back the script with the odd score and keep the other script outside. After following this procedure a number of times, there are 3 scripts left among which there is at least one script each with odd and even scores. Find, with proof, the number of scripts with odd scores among the three left.

We consider the real sequence ($x_n$) defined by $x_0=0, x_1=1$ and $x_{n+2}=3x_{n+1}-2 x_{n}$ for $n=0,1,2,...$ We define the sequence ($y_n$) by $y_n=x^2_n+2^{n+2}$ for every nonnegative integer $n$. Prove that for every $n>0, y_n$ is the square of an odd integer.

Prove that if $p$ is an odd prime, then $p^2(p^2 -1999)$ is divisible by $6$ but not by $12$.