1. Prove that every tournament contains a Hamiltonian path.
(Tournament map: a directed graph such that for every pair of distinct vertices
u and v, there is either an edge from u to v or v to u, but never both.)
(I can prove this easily by double induction, but I heard there is a classic
proof by strong induction. How?)
2. Given Bertrand's Theorem (there always exists a prime p such that n<p<2n
for every n that is a positive integer), prove that all integers greater
than 6 can be written as the sum of one or more distinct primes.