Let \( n \geq 4 \) be an integer. Show that at a party of \( n \) people, it is possible for each person to have greeted exactly three other people if and only if \( n \) is even.

Find all integers \( n > 1 \) such that all prime divisors of \( n^6 - 1 \) divide \( (n^2 - 1)(n^3 - 1) \).

Let \( \triangle ABC \) be an acute triangle with orthocenter \( H \), and let \( M \) be the midpoint of \( BC \). Let \( P \) be the foot of the perpendicular from \( H \) to \( AM \). Prove that \( AM \cdot MP = BM^2 \).

In a country, there are \( n \) cities. Each pair of cities is connected either by a paved road or a dirt road. It is known that there exists a pair of cities such that it is impossible to travel between them using only paved roads. Show that, in this case, it is possible to travel between any two cities using only dirt roads.

Find all triples of integers \( (x, y, z) \) such that \[ x - yz = 11 \]\[ xz + y = 13 \]

The points inside a circle \( \Gamma \) are painted with \( n \geq 1 \) colors. A color is said to be dense in a circle \( \Omega \) if every circle contained within \( \Omega \) has points of that color in its interior. Prove that there exists at least one color that is dense in some circle contained within \( \Gamma \).