Auto: : Agakhanov N. $\mathbf{H}$.

Find the free term of the polynomial $P(x)$ with integer coefficients, if it is known that it is less than a thousand in absolute value, and $P(19)=P(94)=1994$.
Let $a_{0}$ be the constant term of the polynomial $P(x)$. Then $P(x)=x Q(x)+a_{0}$, where $Q(x)$ is a polynomial with integer coefficients. Therefore, $P(19)=19 n+a_{0}$, and $P(94)=94 m+a_{0}$, where $m$ and $n$ are integers. From the condition, it follows that $19 n=94 m$, hence $n=94 k, m=19 k$. Thus, $19 \cdot 94 k+a_{0}=1994$, from which

$a_{0}=1994-1786 k$. From the condition $\left|a_{0}\right|<1000$ it follows that $k=1$, and $a_{0}=208$.

## Answer

208.