Safety in Numbers

A reality TV show has $N$ contestants. Each contestant gets a score from the judges, and the audience vote is added on top of it to form a final score.

Let $X$ be the sum of the judge scores of all contestants. If a contestant received $J$ points from the judges and a fraction $Y$ of the audience vote, where $0 \le Y \le 1$ (for example $0.3$), then that contestant's final score is $J + X \cdot Y$. The vote fractions of all contestants add up to exactly $1$.

The contestant with the lowest final score is eliminated. If two or more contestants share the lowest score, nobody is eliminated.

Once contestant $i$'s vote fraction is fixed at $Y$, the remaining fraction $1 - Y$ can be split among the other contestants in any way, with a nonnegative amount for each of them. Given the judge scores, find for every contestant the smallest $Y$, as a percentage, that keeps that contestant out of elimination no matter how the remaining votes are split.

The first line contains the number of test cases $T$. Each of the next $T$ lines holds one test case. A line starts with the number of contestants $N$, then a single space, then the judge scores $s_0, s_1, \dots, s_{N-1}$ separated by single spaces. The value $s_i$ is the score contestant $i$ received from the judges.

Limits

• $1 \le T \le 50$
• $2 \le N \le 200$
• $0 \le s_i \le 100$
• $s_i > 0$ for at least one $i$.

Print one line per test case. The line starts with Case #x: , then the real numbers $m_0, m_1, \dots, m_{N-1}$ separated by single spaces. Here $x$ is the test case number starting from $1$, and $m_i$ is the smallest percentage of the audience vote contestant $i$ must secure to be certain of avoiding elimination.

Always print $m_i$ with exactly six digits after the decimal point. Round the exact value at the seventh digit after the decimal point, rounding up when that digit is exactly $5$. A value of $0$ prints as 0.000000 and a value of $100$ prints as 100.000000.