9.3 On the table, there are 2004 small boxes, 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 they contain at least one white ball?" How many times do you need to ask, at a minimum, to determine two boxes that both contain white balls?
Solving requires 4005 questions. We will sequentially number the boxes (and the balls placed inside them) from 1 to 2004. We will refer to the pairs of boxes as the problem. We will also call all non-white balls black balls.

We will prove that 4005 questions are sufficient to find two white balls. We sequentially ask the questions (1, 2), (1, 3), ..., (1, 2004), (2, 3), (2, 4), ..., (2, 2004). If all the answers are affirmative, then balls 1 and 2 are white (Fact 1: If ball 1 is black, then at least one more black ball can be found, which, together with ball 1, forms a black pair). If at least one answer is negative (let's assume it is one of the questions containing ball 1), then ball 1 is black. In this case, the other balls in the questions containing ball 1 that gave affirmative answers must be white, thus we have found all the white balls, and there are at least two.

Now we prove that at least 4005 questions are necessary. Assume there exists a strategy that allows finding two white balls with fewer questions. We can give consistent answers to all the questions. In this case, after asking at most 4004 questions, we must point out two boxes (for definiteness, let's say the first and the second boxes) and assert that they contain white balls. At this point, there must be at least one 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. Suppose the first and the k-th boxes contain black balls, and the rest contain white balls. Then the answers we gave are correct, but the first ball indicated by the player is black, which is a contradiction.