KURSCHAK 2005

Let $N>1$ and let $a_1,a_2,\dots,a_N$ be nonnegative reals with sum at most $500$ . Prove that there exist integers $k\ge 1$ and $1=n_0<n_1<\dots<n_k=N$ such that $$\sum_{i=1}^k n_ia_{n_{i-1}}<2005.$$

A and B play tennis. The player to first win at least four points and at least two more than the other player wins. We know that A gets a point each time with probability $p\le \frac12$ , independent of the game so far. Prove that the probability that A wins is at most $2p^2$ .

We build a tower of $2\times 1$ dominoes in the following way. First, we place $55$ dominoes on the table such that they cover a $10\times 11$ rectangle; this is the first story of the tower. We then build every new level with $55$ domioes above the exact same $10\times 11$ rectangle. The tower is called stable if for every non-lattice point of the $10\times 11$ rectangle, we can find a domino that has an inner point above it. How many stories is the lowest stable tower?