1984 AIME Problems/Problem 10
Mary told John her score on the American High School Mathematics Examination (AHSME), which was over . From this, John was able to determine the number of problems Mary solved correctly. If Mary's score had been any lower, but still over , John could not have determined this. What was Mary's score? (Recall that the AHSME consists of multiple choice problems and that one's score, , is computed by the formula , where is the number of correct answers and is the number of wrong answers. (Students are not penalized for problems left unanswered.)
Let Mary's score, number correct, and number wrong be respectively. Then
Therefore, Mary could not have left at least five blank; otherwise, 1 more correct and 4 more wrong would produce the same score. Similarly, Mary could not have answered at least four wrong (clearly Mary answered at least one right to have score above 80, or even 30.)
It follows that and , so and . So Mary scored at least 119. To see that no result other than 23 right/3 wrong produces 119, note that so . But if , then , which was the result given; otherwise and , but this implies at least 31 questions, a contradiction. This makes the minimum score .
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