1B. In the plane, two sets of parallel lines $p_{1}, p_{2}, \ldots, p_{13}$ and $q_{1}, q_{2}, \ldots, q_{7}$ are given such that the lines from the first set intersect with the lines from the second set. How many parallelograms are determined by these lines?
Solution. Each pair of lines from the first set and each pair from the second set determine exactly one parallelogram, and vice versa. A pair from the first set can be chosen in $\binom{13}{2}$ ways, and from the second set in $\binom{7}{2}$ ways, from which it follows that there are $\binom{13}{2}\binom{7}{2}=1638$.

4B Prove that, for every natural number $n$

$$
\cos \frac{\pi}{2^{n+1}}=\frac{1}{2} \underbrace{\sqrt{2+\sqrt{2+\cdots+\sqrt{2}}}}_{n-\text { roots }}
$$