UNCA High School Programming Competition 2018


2018-04-14 16:00 UTC

UNCA High School Programming Competition 2018


2018-04-14 20:00 UTC
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Problem D
Studying For Exams

Photo by CollegeDegrees360 cc by-sa 2.0
It is exam time! You have, of course, been spending too much time participating in various programming contests and have not done much studying. Now you have $N$ subjects to study for, but only a limited amount of time before the final exams. You have to decide how much time to allocate to studying each subject, so that your average grade over all $N$ subjects is maximized.

As a seasoned programming contest competitor, you recognize immediately that you can determine the optimal allocation with a computer program. Of course, you have decided to ignore the amount of time you spend solving this problem (i.e. procrastinating).

You have a total of $T$ hours that you can split among different subjects. For each subject $i$, the expected grade with $t$ hours of studying is given by the function $f_ i(t) = a_ i t^2 + b_ i t + c_ i$, satisfying the following properties:

  • $f_ i(0) \geq 0$;

  • $f_ i(T) \leq 100$;

  • $a_ i < 0$;

  • $f_ i(t)$ is a non-decreasing function in the interval $[0,T]$.

You may allocate any fraction of an hour to a subject, not just whole hours. What is the maximum average grade you can obtain over all $n$ subjects?


The first line of each input contains the integers $N$ ($1 \leq N \le 10$) and $T$ ($1 \leq T \le 240$) separated by a space. This is followed by $N$ lines, each containing the three parameters $a_ i$, $b_ i$, and $c_ i$ describing the function $f_ i(t)$. The three parameters are separated by a space, and are given as real numbers with $4$ decimal places. Their absolute values are no more than $100$.


Output in a single line the maximum average grade you can obtain. Answers within $0.01$ of the correct answer will be accepted.

Sample Input 1 Sample Output 1
2 96
-0.0080 1.5417 25.0000
-0.0080 1.5417 25.0000
Sample Input 2 Sample Output 2
3 34
-0.0657 4.4706 23.0000
-0.0562 3.8235 34.0000
-0.0493 3.3529 42.0000