OSN 2002

Prove that $n^4 - n^2$ is divisible by $12$ for all integers $n > 1$ .

Five dice are rolled. The product of the faces are then computed. Which result has a larger probability of occurring; $180$ or $144$ ?

Find all solutions (real and complex) for $x,y,z$ , given that: $$x+y+z = 6 x^2+y^2+z^2 = 12 x^3+y^3+z^3 = 24$$

Given a triangle $ABC$ where $AC > BC$ , $D$ is located on the circumcircle of $ABC$ such that $D$ is the midpoint of the arc $AB$ that contains $C$ . $E$ is a point on $AC$ such that $DE$ is perpendicular to $AC$ . Prove that $AE = EC + CB$ .

Find all primes $p$ such that $4p^2+1$ and $6p^2+1$ are both primes.

Let $ABCD$ be a rhombus where $\angle DAB = 60^\circ$ , and $P$ be the intersection between $AC$ and $BD$ . Let $Q,R,S$ be three points on the boundary of $ABCD$ such that $PQRS$ is a rhombus. Prove that exactly one of $Q,R,S$ lies on one of $A,B,C,D$ .