There is a complicated formula that would determine how much extra funds Oprah would have to give them so that the bonus + tax given to the recipient would result in zero tax to the recipient. It goes something like this:

Bonus $5,000 X 20% tax = 1,000
Tax on $1,000 X 20% = 200.00
Tax on $200 X 20% = 40.00
Tax on $40 X 20% = 8
Tax on $8 X 20% = 1.60
And on and on until you get to zero.

You may think it's complicated, but I would argue it's very simple. Using your 20% rate, the grossed-up amount is simply 5000/(1-.20) = $6250.

Ira