20212  There are 2004 small boxes on the table, each containing a small ball. It is known that some of the balls are white, and the total number of white balls is even. You are allowed to point to any two boxes and ask: "Do at least one of them contain a white ball?" How many times do you need to ask, at a minimum, to be able to determine two boxes that both contain white balls?
Parse at least 4005 questions.
We will number the boxes (and the balls inside them) from 1 to 2004. We will also refer to the questions by the numbers of the box pairs. We will call all non-white balls black balls.

We will prove that asking 4005 questions can find two white balls. Ask the questions $(1,2),(1,3), \cdots$, $(1,2004),(2,3),(2,4), \cdots,(2,2001)$. If all the answers are affirmative, then the balls in boxes 1 and 2 are both white (in fact, if the ball in box 1 is black, then at least one more black ball can be found, which forms a black ball pair with the ball in box 1). If at least one of the answers is negative (let's assume it is a question involving box 1), then the ball in box 1 is black. In this case, the other ball in the questions involving box 1 that gave an affirmative answer must be white. Thus, we have found all the white balls, and there are at least two.

Now we will prove that fewer than 4005 questions cannot ensure finding two white balls. Assume there exists a strategy that can ensure finding two white balls with fewer questions. We can give affirmative answers to all the questions. At this point, after asking at most 4004 questions, we must point out two boxes (let's say the first and second boxes) and affirm that they contain white balls. At this time, there must be a question of the form $(1, n)$ or $(2, n)$ that has not been asked, for example, the question $(1, k)$, otherwise, 4005 questions would be needed. If the balls in the first and the $k$-th boxes are black, and the rest of the boxes contain white balls, then the answers we gave are correct, but the first ball pointed out by the player is black, which leads to a contradiction.