05/31/2020 (2011: National Sprint Round, Problem 4)

Q: If you subtract 100 from the square of an integer n, the result is 50 more than five times the integer n.What is the greatest possible value of n?

A: 15

This is a simple algebra problem, where you set up the equation and solve:

n² - 100 = 50 + 5n 

Put everything on one side

n² - 5n - 150 = 0

Use the Quadratic Formula

(-(-5) ± √( (-5)²- 4(1)(-150))) ÷ (2) =

(5 ± √(25 + 600)) ÷ (2) =

(5 ± 25) ÷ (2) 

We are trying to find the greatest value so we use addition only

(30) ÷ (2) =

15