3.4. Five students Ana, Bojan, Vesna, Goran, and Dejan took a test consisting of five questions. For the first two questions, the possible answers were $\alpha, \beta$, and $\gamma$, while for the last three questions, the possible answers were YES and NO. The students gave the following answers to these five questions:

|  | 1 | 2 | 3 | 4 | 5 |
| :--- | :---: | :---: | :---: | :---: | :---: |
| ANA | $\alpha$ | $\alpha$ | YES | YES | YES |
| BOJAN | $\beta$ | $\beta$ | YES | NO | YES |
| VESNA | $\alpha$ | $\beta$ | YES | YES | NO |
| GORAN | $\beta$ | $\gamma$ | YES | YES | NO |
| DEJAN | $\gamma$ | $\alpha$ | NO | YES | YES |

If no two students had the same number of correct answers, determine which student performed the best on the test.
3.4. 1) Let some student have all 5 correct answers. Every 2 students have one identical YES-NO question, so if someone has all 5 correct answers, then each of the others will have at least one correct answer. Since everyone has a different number of correct answers, the total number of correct answers is $1+2+3+4+5=15$. Since the maximum number of correct answers is $2(\alpha$ or $\beta)+2(\alpha$ or $\beta)+4($ YES $)+4$ (YES) $+3($ YES $)=15$, only Ana's answers can all be correct, but then Bojan and Goran have 2 correct answers each, and Vesna and Dejan each have 3.

2) If no one has all 5 correct answers, the total number of correct answers is $0+1+2+3+4=10$. Neither Ana, nor Vesna, nor Goran can have all wrong answers because then the correct answers to questions 3 and 4 would be NO (3/NO, 4/NO). The maximum number of correct answers would be $2(\alpha$ or $\beta)+$ $2(\alpha$ or $\beta)+1($ NO $)+1($ NO $)+3$ (YES) $=9<10$. If Bojan had all wrong answers, they would be: (3/NO, 4/YES, 5/NO). Vesna, Goran, and Dejan would each have at least 2 correct answers, and Ana would have only one correct answer (4/YES). The correct answers would be (1/ $\gamma, 2 / \gamma$), and the maximum number of correct answers would be $1+1+1+4+2=9<10$. Therefore, Dejan has all wrong answers. Thus, the correct answers are (3/YES, 4/NO, 5/NO). So, Bojan, Vesna, and Goran each have at least 2 correct answers. From this, it follows that Ana has only one correct answer (3/YES), which means (1/ $\beta$). Since the total number of correct answers is 10, it must be (2/ $\gamma$). Then the best

is Goran with 4 correct answers.