KZ-MO 2006 Problem 3

The racing tournament has $12$ stages and $n$ participants. After each stage, all participants, depending on the occupied place $k$ , receive points $a_k$ (the numbers $a_k$ are natural and $a_1> a_2> \dots> a_n$ ). For what is the smallest $n$ the tournament organizer can choose the numbers $a_1$ , $\dots$ , $a_n$ so that after the penultimate stage for any possible distribution of places at least two participants had a chance to take first place.