Problem D
Marathon
Erik wants to run a marathon. Most of all, he wants to win the race. To plan his training, he has looked up how the other contestants performed in previous races and made a model to predict his chances of winning. The finishing time for each contestant is distributed uniformly at random in an interval $[a_ i, b_ i]$. What is the largest finishing time Erik can have while still having a $50\% $ change of winning?
Input
The first line contains an integer $1 \leq N \leq 10^5$, the number of other contestants. Then follows $N$ lines, each with two floating point values $0 \leq a_ i \leq 10^5$ and $a_ i \le b_ i \leq 10^5$ with at exactly one decimal place, the start and end time in seconds for their finishing time.
Output
A single real number, the largest finishing time in seconds that Erik needs to have a $50\% $ chance of winning. The answer must be with a relative or absolute error of at most $10^{-6}$.
Scoring
|
Group |
Points |
Limits |
|
$1$ |
$20$ |
No intervals overlap. |
|
$2$ |
$80$ |
No further restrictions. |
| Sample Input 1 | Sample Output 1 |
|---|---|
1 1.0 3.0 |
2.0 |
| Sample Input 2 | Sample Output 2 |
|---|---|
3 0.0 10.0 3.5 6.7 2.2 4.5 |
2.883937700856755 |
