Question 29
Let $n, k$ be positive integers satisfying $n \ge k$.
There is a group consisting of $n$ people. Each person of this group belongs to one and only one of $k$ clubs, called club $C_1, C_2, \dots, C_k$. Each club has at least one member belonging to it. Prove that it is possible to distribute $n^2$ pieces of cake to these $n$ people in such a way to satisfy all of the following conditions:
* Every one receives at least 1 piece of cake.
* For each $i$ $1 \le i \le k$, every member of the club $C_i$ receives $a_i$ pieces of cake.
* If $1 \le i \lt j \le k$, then $a_i \gt a_j$ is satisfied.
There is a group consisting of $n$ people. Each person of this group belongs to one and only one of $k$ clubs, called club $C_1, C_2, \dots, C_k$. Each club has at least one member belonging to it. Prove that it is possible to distribute $n^2$ pieces of cake to these $n$ people in such a way to satisfy all of the following conditions:
* Every one receives at least 1 piece of cake.
* For each $i$ $1 \le i \le k$, every member of the club $C_i$ receives $a_i$ pieces of cake.
* If $1 \le i \lt j \le k$, then $a_i \gt a_j$ is satisfied.