26.2.11 ** 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 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 a box that contains a white ball?
Parse: At least 2003 questions are needed.
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 by asking 2003 questions, we can find a white ball. We ask the questions (1, 2), (1, 3), ..., (1, 2004). If all the answers are affirmative, then ball 1 is white (in fact, if ball 1 were black, then there would be at least one more black ball that forms a black pair with ball 1). If at least one of the answers is negative (let's assume it is a question involving ball 1), then ball 1 is black. In this case, the other ball in the questions involving ball 1 that gave an affirmative answer must be white, thus we can even find all the white balls.

Now we prove that fewer than 2003 questions cannot suffice. Assume there exists a strategy that can ensure finding a white ball with fewer questions. We can give affirmative answers to all the questions. In this case, at most after 2002 questions, we can point to a box (for definiteness, let's say the first box) and assert that it contains a white ball. At this point, there must be a question of the form (1, n) that has not been asked, for example, question (1, 2), because there are 2003 such questions. If we place black balls in the first and second boxes and white balls in all the other boxes, then the answers we gave are correct, but the ball the player points to is black, which is a contradiction.