Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.5 AIME, harder Prove it Slovenia

At a contest students were presented with 24 multiple choice questions. Contestants who gave no answer or gave more than one answer were awarded 0 points for that question. For the correct answer they received 1 point and for a wrong answer $\frac{1}{4}$ of a point was deducted. If a student received 13 points, at most how many questions did he answer correctly?

Solution

Let $P$ be the number of questions for which the contestant was awarded 1 point and let $N$ denote the number of those for which $\frac{1}{4}$ of a point was deducted. Then $P - \frac{1}{4}N = 13$ or $N = 4P - 52$ . Since there were 24 questions, we get $P + N \le 24$ or $5P - 52 \le 24$ . So, $5P \le 76$ and we have $P \le 15$ . The greatest possible value of $P$ is 15 and it is attained when $N = 8$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.