Let $f(n)$ be the product of all factors of $n$. Find all natural numbers $n$ such that $f(n)$ is not a perfect power of $n$.

In ZS Chess, an Ivanight attacks like a knight, except that if the attacked square is out of range, it goes through the edge and comes out from the other side of the board, and attacks that square instead. The ZS chessboard is an $8 \times 8$ board, where cells are coloured with $n$ distinct colours, where $n$ is a natural number, such that a Ivanight placed on any square attacks $ 8 $ squares that consist of all $n$ colours, and the colours appear equally many times in those $ 8 $ squares. For which values of $n$ does such a ZS chess board exist?

There is a complete graph $G$ with $4027$ vertices drawn on the whiteboard. Ivan paints all the edges by red or blue colour. Find all ordered pairs $(r, b)$ such that Ivan can paint the edges so that every vertex is connected to exactly $r$ red edges and $b$ blue edges.

One day, Ivan was imprisoned by an evil king. The evil king said : "If you can correctly determine the polynomial that I'm thinking of, you'll be free. If after $2014$ tries, you can't guess it, you'll be executed." Ivan answered : "Are there any clues?" The evil king replied : "I can tell you that the polynomial has real coefficients and is monic. Furthermore, all roots are positive real numbers." That night, a kind wizard, told him the polynomial. The conversation was heard by the king who was visiting Ivan. He killed the wizard. The next day, Ivan forgot the polynomial, except that the coefficients of $x^{2013}$ is $2014$, and that the constant term is $1$. Can Ivan guarantee freedom? And if so, in how many tries? (Assume that Ivan is very unlucky so any random guess fails.)

Given $\triangle ABC$ with circumcircle $\Gamma$ and circumcentre $O$, let $X$ be a point on $\Gamma$. Let $XC_1$, $XB_1$ to be feet of perpendiculars from $X$ to lines $AB$ and $AC$. Define $\omega_C$ as the circle with centre the midpoint of $AB$ and passing through $C_1$ . Define $\omega_B$ similarly. Prove that $\omega_B$ and $\omega_C$ has a common point on $XO$.