[b]p1.[/b] A bus leaves San Mateo with $n$ fairies on board. When it stops in San Francisco, each fairy gets off, but for each fairy that gets off, $n$ fairies get on. Next it stops in Oakland where $6$ times as many fairies get off as there were in San Mateo. Finally the bus arrives at Berkeley, where the remaining $391$ fairies get off. How many fairies were on the bus in San Mateo?


[b]p2.[/b] Let $a$ and $b$ be two real solutions to the equation $x^2 + 8x - 209 = 0$. Find $\frac{ab}{a+b}$ . Express your answer as a decimal or a fraction in lowest terms.


[b]p3.[/b] Let $a$, $b$, and $c$ be positive integers such that the least common multiple of $a$ and $b$ is $25$ and the least common multiple of $b$ and $c$ is $27$. Find $abc$.


[b]p4.[/b] It takes Justin $15$ minutes to finish the Speed Test alone, and it takes James $30$ minutes to finish the Speed Test alone. If Justin works alone on the Speed Test for $3$ minutes, then how many minutes will it take Justin and James to finish the rest of the test working together? Assume each problem on the Speed Test takes the same amount of time.


[b]p5.[/b] Angela has $128$ coins. $127$ of them have the same weight, but the one remaining coin is heavier than the others. Angela has a balance that she can use to compare the weight of two collections of coins against each other (that is, the balance will not tell Angela the weight of a collection of coins, but it will say which of two collections is heavier). What is the minumum number of weighings Angela must perform to guarantee she can determine which coin is heavier?



PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
To solve the given quadratic equation \(x^2 + 8x - 209 = 0\) and find \(\frac{ab}{a+b}\), we can use Vieta's formulas. Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots.

1. **Identify the roots using Vieta's formulas:**
   - For a quadratic equation \(ax^2 + bx + c = 0\), the sum of the roots \(a + b\) is given by \(-\frac{b}{a}\).
   - The product of the roots \(ab\) is given by \(\frac{c}{a}\).

   Here, \(a = 1\), \(b = 8\), and \(c = -209\).

2. **Calculate the sum and product of the roots:**
   - The sum of the roots \(a + b\) is:
     \[
     a + b = -\frac{8}{1} = -8
     \]
   - The product of the roots \(ab\) is:
     \[
     ab = \frac{-209}{1} = -209
     \]

3. **Find \(\frac{ab}{a+b}\):**
   - Substitute the values obtained from Vieta's formulas:
     \[
     \frac{ab}{a+b} = \frac{-209}{-8} = \frac{209}{8}
     \]

4. **Simplify the fraction:**
   - The fraction \(\frac{209}{8}\) is already in its simplest form.

The final answer is \(\boxed{\frac{209}{8}}\).