System of Equations

Quickly solve this system of equations:

x - y = 5
x^2 - y^2 = 35

Objects

There are 50 distinct objects lying on a table. I can choose any of the objects, and any of the objects can choose me. How many ways are there to pick the objects?

Counting Divisors

If n is the perfect cube of a natural number and has 28 divisors, 2 of which are prime, how many divisors does the cube root of n have?
(Source: Art of Problem Solving: Introduction to Number Theory)

Three Difficult Problems

13. There were a total of 1,050,697 crimes in the US way back in 1997. Newark, NJ housed 0.56% of these crimes. In 1998, Newark City officials started a safety program to lower their crime rate. During the year of 1999, Newark was home to only 5,540 crimes. A program is considered successful if it reduces its…well, whatever it’s testing…by at least 5%. Was this program successful?

14. Jimmy, a novice salesperson, earns a base pay of $200 plus 5% commission on his weekly sales. His goal commission is $100. Right now, Jimmy totals $1200 in sales. How much does Jimmy need to raise his total sales each week in order to reach his goal in two weeks? (Example: if your answer was $200, he would make $200 more every week, leading to $1600 at the end)

15. Frank needs to raise $2000 in two years in order to pay off his mortgage. However, every year he pays $500 back to the government for taxes. At what percent does the interest need to be in order to achieve his goal? Assume he starts at $1000.