M. Murraikin

In a racing tournament, there are 12 stages and $n$ participants. After each stage, all participants receive points $a_{k}$ depending on the place $k$ they occupy (the numbers $a_{k}$ are natural, and $a_{1}>a_{2}>\ldots>a_{n}$). For what smallest $n$ can the tournament organizer choose the numbers $a_{1}, \ldots, a_{n}$ so that after the penultimate stage, in any possible distribution of places, at least two participants have a chance to take the first place.
Evaluation. Suppose there are no more than 12 participants.

Let one of the participants (call him $A$) win all 11 stages, and each of the remaining participants at least once took the last place.

Then participant $A$ after 12 stages will score no less than $11 a_{1} + a_{n}$ points, while each of the remaining participants will score no more than $a_{n} + 10 a_{2} + a_{1}$ points, which is less than

$11 a_{1} + a_{n}$.

Therefore, only $A$ can take the first place. Hence, to meet the condition, there must be at least 13 participants.

Example. Suppose there are 13 participants, then the organizer can choose the numbers $a_{1}=1011, a_{2}=1010, \ldots, a_{12}=1000, a_{13}=1$.

Consider the participant who scored the most points after 11 stages (call him $A$). Then among the remaining 12 participants, there will be at least one who did not take the last place on any stage (call him $B$). Note that $A$ scored no more than $1011 \cdot 11 = 11121$ points, while $B$ scored no less than $1000 \cdot 11 = 11000$ points. That is, after 11 stages, $A$ is ahead of $B$ by no more than 121 points.

Obviously, after 11 stages, $A$ has a chance to take the first place. However, if on the 12th stage $A$ takes the last place, then $B$ will score at least 999 points more than $A$ on the 12th stage and overtake him.

## Answer

For $n=13$.