Prove that there are $2$ students , who have the same result in this test .

Question

$30$ students pass a test containing $16$ questions . If a student answers a question correctly in less than or in $1$ minute , he gets $10$ points . If he answers correctly in more than 1 minute , he gets $5$ points . If he answers incorrectly , he gets $0$ points . After the competition , the results are this:

• More than half of answers were correct and answered in less than a minute.
• The number of correct answers answered in more than $1$ minute, and the number of incorrect answers are equal .

Prove that there are $2$ students , who have the same result in this test .

I am a really stuck here, please help me

Answer 1

There are $480$ answers in total. Let $a$ be the number of answers that score $10$ points. Then the numbers of answers that score $5$ and $0$ points are both $(480 - a)/2$.

This gives a total score of $10a + 5\cdot (480 - a)/2 = 1200 + 7.5a$. Since we know that $a > 240$, the total score is $> 3000$.

A student can have a maximum score of $160$ and the score is always a multiple of $5$. If all students have different scores, then the total score is at most $160 + 155 + \dots + 15 = 2625$, which is smaller than $3000$. Contradiction.

Answer 2

First suppose each of the $30$ students has a different score. We know each score is a multiple of $5$, and each student's score must lie between $0$ and $160$. There are only $33$ different possible scores in this interval. Let the total of the scores across all $30$ students be $N$. If we omit the $3$ lowest scores $0,5,10$ then we can find an upper limit for $N$ if all students have different scores:

$\displaystyle N \le 15 + 20 + 25 + \dots + 155 + 160 = \sum_{k=1}^{30} (10 + 5k)$

Now suppose there are $m$ correct answers in more than a minute and also $m$ incorrect answers. The total number of answers from all $30$ students is $30 \times 16 = 480$, so there must be $480-2m$ correct answers in less than a minute. We know this is greater than $240$, so we have $0 \le m \lt 120$.

We have $m$ answers scoring $5$ points and $480-2m$ answers scoring $10$ points, so the total points scored across all students is

$N = 5m + 10(480-2m) = 4800 - 15m$

and since $m < 120$ we have $N > 4800 - 15 \times 120 = 3000$. So the total score across all $30$ students is greater than $3000$ points.

So it is only possible for each student to have a different score if

$\displaystyle 3000 < \sum_{k=1}^{30} (10 + 5k)$