I have some problems relating to the pigeonhole principle. Here they are:
Prove that at a party where there are at least two people, there are two people who know the same number of other people there.
The other one is,
Let n1, n2, ... , nt be positive integers. Show that if n1+n2+...+nt - t + 1 objects are placed into t boxes, then for some i, i = 1, 2, ... , t, the ith box contains at least ni objects.
Now, I was given the hint that the pigeonhole principle can help me here, but I don't know how to apply it. Are there any other hints anyone could think of to help me prove this? Thanks.