Question number 18. A chess player played 40 matches and scored 25 points. How many more matches did he win than lose? And here we know that a win counts as one point, a draw counts as one half of a point, and the loss counts as zero points. So let's consider two possibilities. Either there are no draws, so the 25 points correspond to 25 wins. And if there are 25 wins out of 40 games, then we must have 15 losses and that corresponds to zero points. So adding these together, we have 25 points, and adding these together, we have 40 matches. And the difference between them, if we subtract, is 10 matches. Okay, so we should select C as an answer, but we are not sure that would be a correct answer. We have to consider the possibility that there are draws. And so we note here that draws have to come in pairs. Because otherwise, with for example three draws, we would see 1.5 points and we scored exactly an integer number of points. And so if we have 20 wins, that would correspond to 20 points, and then we would have to have some draws. So 5 points would be earned if we draw 10 times. Now we see that that's exactly a total of 30 matches, so we need 10 losses here for zero points. And again, we have the right number of points, and the difference here between wins and losses is again 10 matches. And now we have to argue that this is the only possibility. See, if we increase the number of draws by 2, we would have one more point, but then we would have to decrease the number of wins by 1, and we would still have 41 matches instead of 40. And likewise, if we decrease the number of draws by 2, we would have 38 matches but 24 points, and increasing that number of points by 1 would give us 21 wins, 8 draws, and then a total of 39 matches for that number of points, which is not possible. So the answer has to be C, that's the only possibility for both draws and no draws in the final score. So the answer here is 10.

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