I'm going to stick my neck out and say I believe the answer given on the site is wrong.
As you said, the probabilities that A-D do not win the contest are 2/3, 3/4, 4/5, and 5/6 respectively.
So the probability of winning is 1/3 for A, 1/4 for B, 1/5 for C and 1/6 for D.
Since only one person can win, these are mutually exclusive events.
So the probability that A, B, C, or D wins is 1/3 + 1/4 + 1/5 + 1/6 = 19/20.