Progetto Olimpiadi della Matematica - GARA di FEBBRAIO

Solution:

The answer is (B). Let $r$ be the number of correct answers given by Camilla; by difference, the number of wrong answers is $90 - r$, and the total score is $5r + (-1)(90 - r) = 6r - 90 = 6(r - 15)$. It follows that Camilla's score is a multiple of 6, and therefore it cannot be equal to $116 = 2^{2} \cdot 29$, since this number is not divisible by 3. All the other scores are possible: Camilla can total $-78, 204, 318$ and $402$ points by giving respectively 2, 49, 68 and 82 correct answers; indeed, Camilla's score can be any multiple of 6 between $6 \cdot (0 - 15) = -90$ and $6 \cdot (90 - 15) = 450$.