RU-ALLRUSSIAN 2019

An interstellar hotel has $100$ rooms with capacities $101,102,\ldots, 200$ people. These rooms are occupied by $n$ people in total. Now a VIP guest is about to arrive and the owner wants to provide him with a personal room. On that purpose, the owner wants to choose two rooms $A$ and $B$ and move all guests from $A$ to $B$ without exceeding its capacity. Determine the largest $n$ for which the owner can be sure that he can achieve his goal no matter what the initial distribution of the guests is.

Let $P(x)$ be a non-constant polynomial with integer coefficients and let $n$ be a positive integer. The sequence $a_0,a_1,\ldots$ is defined as follows: $a_0=n$ and $a_k=P(a_{k-1})$ for all positive integers $k.$ Assume that for every positive integer $b$ the sequence contains a $b$ th power of an integer greater than $1.$ Show that $P(x)$ is linear.