5.103 Suppose there are 100 mutually hostile countries on Mars. To maintain peace, it is decided to form several alliances, with each alliance including at most 50 countries, and any two countries must belong to at least one alliance. Try to answer the following questions:
(1) What is the minimum number of alliances needed to meet the above requirements?
(2) If it is also required that the union of any two alliances does not exceed 80 countries, what would be the result?
[Solution] Since each alliance has at most 50 countries, in this alliance, each country has at most 49 allies. However, after forming alliances, each country has 99 allies, so each country must join at least 3 alliances. Therefore, the number of alliances formed by 100 countries is no less than $3 \times 100 \div 50 = 6$.

(1) Divide the 100 countries into 4 groups, with 25 countries in each group. Each pair of groups forms an alliance of 50 countries, resulting in $C_{4}^{2} = 6$ alliances. It is easy to verify that these 6 alliances meet the requirements, so the minimum number of alliances formed in this case is 6.
(2) Divide the 100 countries into 10 groups, with 10 countries in each group. Number these 10 groups from 1 to 10 and let the following sets of 5 groups form an alliance, resulting in 6 alliances:
$$
\begin{array}{l}
\{1,2,3,4,5\},\{1,2,6,7,8\},\{1,3,6,9,10\}, \\
\{2,4,7,9,10\},\{3,5,7,8,9\},\{4,5,6,8,10\}.
\end{array}
$$

It is easy to verify that these 6 alliances meet all the requirements, so the minimum number of alliances formed in this case is also 6.