Question 151
We operate on piles of cards placed at $n+1$ positions $A_1, A_2, \dots, A_n$ ($n \ge 3$) and $O$. In one operation, we can do either of the following:
(1) If there are at least three cards at $A_i$, we may take three cards from $A_i$ and place one at each of $A_{i-1}, A_{i+1}$ and $O$ (assume that $A_0 = A_n, A_{n+1} = A_1$);
(2) If there are at least $n$ cards at $O$, we may take $n$ cards from $O$ and place one at each of $A_1, A_2, \dots, A_n$. Prove that if the total number of cards is at least $n^2 + 3n + 1$, we can take some operations such that there are at least $n+1$ cards at each position.
(1) If there are at least three cards at $A_i$, we may take three cards from $A_i$ and place one at each of $A_{i-1}, A_{i+1}$ and $O$ (assume that $A_0 = A_n, A_{n+1} = A_1$);
(2) If there are at least $n$ cards at $O$, we may take $n$ cards from $O$ and place one at each of $A_1, A_2, \dots, A_n$. Prove that if the total number of cards is at least $n^2 + 3n + 1$, we can take some operations such that there are at least $n+1$ cards at each position.