Assume that there is no two boys and two girls that each boy danced with both girls.
Let the girls are G1, G2, ...G20, and they danced with n1, n2, ...n20 boys.
We have that
n1 + n2 + ... + n20 = 99*2 = 198
For a girl, say G1, we can select 2 boys from the n1 boys who danced with G1. There are C(n1,2) = n1*(n1-1)/2 such pair of boys. Similar, for girl G2, we have C(n2,2) pairs of boy who danced with G2. We say, the boy-pairs of G1 and G2 has no overlap, otherwise we will have boy B1 and B2, they both danced with G1 and G2, contradiction.
Boy-pairs of each girl should have no overlap. So, the sum should be not greater than C(20,2)=190
It's to say:
but the left side is
(n1^2 + n2^2 + ... + n20^2 - n1 - n2 - ... -n20)/2
it's greater or equal than
( ((n1+...+n20)/20)^2 * 20 - (n1+...+n20) )/2
= ( (198/20)^2 * 20 - 198 ) / 2