The diagonals $ AC$ and $ BD$ of a convex quadrilateral $ ABCD$ intersect at point $ M$. The bisector of $ \angle ACD$ meets the ray $ BA$ at $ K$. Given that $ MA \cdot MC +MA \cdot CD = MB \cdot MD$, prove that $ \angle BKC = \angle CDB$.

Let $0\le a\le b\le c$ be real numbers. Prove that \[(a+3b)(b+4c)(c+2a)\ge 60abc \]

Find all pairs $ (p, q)$ of prime numbers such that $ p$ divides $ 5^q + 1$ and $ q$ divides $ 5^p + 1$.

We are given $2001$ balloons and a positive integer $k$. Each balloon has been blown up to a certain size (not necessarily the same for each balloon). In each step it is allowed to choose at most $k$ balloons and equalize their sizes to their arithmetic mean. Determine the smallest value of $k$ such that, whatever the initial sizes are, it is possible to make all the balloons have equal size after a finite number of steps.