1. Given the polynomial

$$
P(x)=x^{2019}+2018 x^{2017}+2016 x+2015
$$

Find all integers $n$ such that

$$
P(n) \mid P(P(n)+1)
$$
1. For every integer $n$ we have:

$$
P(P(n)+1) \equiv P(1)=1+2018+2016+2015=6050 \quad(\bmod P(n))
$$

Therefore, to satisfy the given condition, it must hold that $P(n) \mid 6050$. However, note that for $n \geqslant 2$ we have $P(n)>2^{2019}>6050$, and for $n \leqslant-2$ we have $P(n)<-2^{2019}<-6050$. Thus, the only possibilities are $n \in\{-1,0,1\}$. We directly compute $P(-1)=-1-2018-2016+2015=-2020 \nmid 6050, P(0)=2015 \nmid 6050$ and $P(1)=6050$. Therefore, the only solution is $n=1$.