KMO 2010

On a circular table are sitting $2n$ people, equally spaced in between. $m$ cookies are given to these people, and they give cookies to their neighbors according to the following rule.

(i) One may give cookies only to people adjacent to himself. (ii) In order to give a cookie to one's neighbor, one must eat a cookie.

Select arbitrarily a person $A$ sitting on the table. Find the minimum value $m$ such that there is a strategy in which $A$ can eventually receive a cookie, independent of the distribution of cookies at the beginning.

An arbitrary prime $p$ is given. If an integer sequence $(n_1 , n_2 , \cdots , n_k )$ satisfying the conditions

• For all $i= 1, 2, \cdots , k$ , $n_i \geq \frac{p+1}{2}$
• For all $i= 1, 2, \cdots , k$ , $p^{n_i} - 1$ is divisible by $n_{i+1}$ , and $\frac{p^{n_i} - 1}{n_{i+1}}$ is coprime to $n_{i+1}$ . Let $n_{k+1} = n_1$ . exists not for $k=1$ , but exists for some $k \geq 2$ , then call the prime a good prime. Prove that a prime is good iff it is not $2$ .