1. It is known that the free term $a_{0}$ of the polynomial $P(x)$ with integer coefficients is less than 100 in modulus, and $P(20)=P(16)=2016$. Find $a_{0}$.
Solution. We can write $P(x)-2016=(x-16)(x-20) Q(x)$, where $Q(x)$ is a polynomial with integer coefficients. The constant term of the right side is $320 k$, where $k$ is an integer. Thus, $a_{0}=2016-320 k$. The condition is satisfied only by the value $k=6, a_{0}=96$.

Answer. $a_{0}=96$.