1. Find the sum of the coefficients of the terms with odd powers of $x$ in the polynomial

$$
p(x)=\left(x^{2}+2 x+2\right)^{1982}+\left(x^{2}-3 x-3\right)^{1982}
$$
Solution. If $f(x)=a_{0}+a_{1} x+a_{2} x^{2}+\ldots+a_{n} x^{n}$ is a polynomial of degree $n$, then

$$
f(1)=a_{0}+a_{1}+a_{2}+\ldots+a_{n}
$$

$$
f(-1)=a_{0}-a_{1}+a_{2}-\ldots+(-1)^{n} a_{n}
$$

thus, $\frac{1}{2}(f(1)+f(-1))$ is the sum of the coefficients of the even powers of $x$, and $\frac{1}{2}(f(1)-f(-1))$ is the sum of the coefficients of the odd powers of $x$.

According to this, the sum of the coefficients of the odd powers of $x$ of the polynomial $p(x)$ will be:

$$
\frac{1}{2}[p(1)-p(-1)]=\frac{1}{2}\left(5^{1982}+5^{1982}-1-1\right)=5^{1982}-1
$$