For every positive integer $n$, $P(n)$ is defined as follows: For each prime divisor $p$ of $n$ is considered the largest integer $k$ such that $p^k\le n$ and all the $p^k$ are added. For example, for $n=100=2^2 \cdot 5^2$, as $2^6<100<2^7$ and $5^2<100<5^3$, it turns out that $P(100)=2^6+5^2=89$ Prove that there are infinitely many positive integers $n$ such that $P(n)>n$..

Determine all positive integers $n$ such that numbers from $1$ to $n$ can be sorted in some order $x_1,x_2,...,x_n$ with the property that the number $x_1+x_2+...+x_k$ is divisible by $k$, for all $1\le k\le n$., that is $1$ is divides $x_1$, $2$ divides $x_1+x_2$, $3$ divides $x_1+x_2+x_3$, and so on until $n$ divides $x_1+x_2+...+x_n$.

Given a square $ABCD$, let us consider an equilateral triangle $KLM$, whose vertices $K$, $L$ and $M$ belong to the sides $AB$, $BC$ and $CD$ respectively. Find the locus of the midpoints of the sides $KL$ for all possible equilateral triangles $KLM$. Note: The set of points that satisfy a property is called a locus.

We consider a square board of $1000\times 1000$ with $1000000$ squares $1\times 1$ . A piece placed on a square threatens all squares on the board that are inside a $19\times 19$ square. with a center in the square where the piece is placed, and with sides parallel to those of the board, except for the squares in the same row and those in the same column. Determine the maximum number of pieces that can be placed on the board so that no two pieces threaten each other.

Find all pairs of positive integers $x,y$ such that $$x^3+y^3=4(x^2y+xy^2-5).$$

For every positive integer $n$, we consider the polynomial of real coefficients, of $2n+1$ terms, $$P(x)=a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+...+a_1x+a_0$$where all coefficients are real numbers satisfying $100 \le a_i \le 101$ for $0 \le i \le 2n$. Find the smallest possible value of $n$ such that the polynomial can have at least one real root.