DMOPC '18 Contest 4 P1 - Dr. Henri and Differential Photometry - Olympiads Online Judge

DMOPC '18 Contest 4 P1 - Dr. Henri and Differential Photometry

Points: 3 (partial)

Time limit: 2.0s

Memory limit: 64M

Author:

Problem type

Dr. Henri is looking through his telescope at the MRD Observatory. He is observing a certain star $Y$ $Y$ and wants to find its magnitude (a measure of brightness), $m_Y$ $m_{Y}$ . The magnitude of a star can be any real number.

Dr. Henri is using a device called a differential photometer to measure magnitude. Although this device is very precise, it cannot directly measure the magnitude of a star; it can only measure the difference in magnitudes between two stars.

Fortunately, Dr. Henri knows the magnitude $m_X$ $m_{X}$ of a certain star $X$ $X$ . He decides to find $m_Y$ $m_{Y}$ by constructing a sequence of $n + 1$ $n + 1$ stars beginning with $X$ $X$ and ending with $Y$ $Y$ . Then, for each star $i$ $i$ on the list (except $Y$ $Y$ ), he records the difference $d_i = m_{i + 1} - m_i$ $d_{i} = m_{i + 1} - m_{i}$ between the magnitudes of the stars $i + 1$ $i + 1$ and $i$ $i$ , for a total of $n$ $n$ observations. He can then calculate a value for $m_Y$ $m_{Y}$ from this sequence.

Dr. Henri knows that he must take multiple measurements in order to ensure accuracy, so he constructs $K$ $K$ such sequences. Sequence $i$ $i$ consists of $n_i$ $n_{i}$ observations, and the value of $m_Y$ $m_{Y}$ calculated from $i$ $i$ is denoted as $m_{Yi}$ $m_{Y i}$ . Of course, due to natural error in measuring, the $m_{Yi}$ $m_{Y i}$ 's calculated from each sequence may not be exactly the same. So Dr. Henri will use the mean of the $m_{Yi}$ $m_{Y i}$ 's, $\frac{m_{Y1} + m_{Y2} + \dots + m_{YK}}{K}$ $\frac{m_{Y 1} + m_{Y 2} + \dots + m_{Y K}}{K}$ , as the final $m_Y$ $m_{Y}$ , which he denotes $m_{Yf}$ $m_{Y f}$ .

Given $K$ $K$ sequences of observations, please help Dr. Henri find $m_{Yf}$ $m_{Y f}$ .

Constraints

$2 \le K \le 1\,000$ $2 \leq K \leq 1 000$
$1 \le n_i \le 1\,000$ $1 \leq n_{i} \leq 1 000$
$-100.0 \le m_X, d_i \le 100.0$ $- 100.0 \leq m_{X}, d_{i} \leq 100.0$

Input Specification

The first line of input will contain one integer, $K$ $K$ .
The second line will contain one real number, $m_X$ $m_{X}$ .
The next $K$ $K$ lines will contain one integer $n_i$ $n_{i}$ , followed by $n_i$ $n_{i}$ space-separated real numbers $d_{i1}, d_{i2}, \dots, d_{in_i}$ $d_{i 1}, d_{i 2}, \dots, d_{i n_{i}}$ , the observations from the $i$ $i$ -th list.

Output Specification

A single line containing one real number, $m_{Yf}$ $m_{Y f}$ . Your answer will be judged as correct if it has an absolute error of no more than $10^{-3}$ $10^{- 3}$ .

Sample Input

Copy

3
-1.46
2 4.53 1.20
3 4.77 -1.45 2.35
1 5.69

Sample Output

Copy

4.236667

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