60 students attend the one or both of the following courses: algebra and calculus. On algebra they are divided into 10 teams of equal size. On calculus they are divided into different 10 teams of equal size. Prove that there is a set S of 10 students containing exactly 1 person from every algebra team and exactly 1 person from every calculus team.
I have to prove this applying concepts of graph theory, I'd appreciate if any of you could prove it.
You can, indeed, see the problem as a graph problem.
Consider a bipartite graph, with vertices are $a_1, ..., a_{10}, c_1, ..., c_{10}$. Each vertex represents a team (either a algebra or a calculus team).
Now, you put an between $a_i$ and $c_j$ if and only if there is a student who belongs to the $i$-th algebra team, and to the $j$-th calculus one.
Your graph has $60$ edges, one per student, representing the student's teams.
The problem can now be reformulated as "Does this graph admit a perfect matching ?" Indeed, you would like to pick students (=edges), such that you "cover" all the teams both in algebra and calculus (="perfect"), and you want a single student per team (=matching).
If you still need help to solve the problem, I can try to give some hints, but at least you can use some graph theory now !