05/22/2020 (2011: National Team Round, Problem 3)

Q: If x and y are positive integers such that 5x + 3y = 100, what is the greatest possible value of xy?

A: 160

We know that x and y are positive integers, so we can find the point closest to the x-axis and the point closest to the y-axis that fulfills the requirements. We can list out all the points and find out what xy equals:

(17, 5) --> 17 * 5 = 85

(14, 10) --> 14 * 10 = 140

(11, 15) --> 11 * 15 = 150

(8, 20) --> 8 * 20 = 160

(5, 25) --> 5 * 25 = 125

(2, 30) --> 2 * 30 = 60

The greatest value out of 85, 140, 150, 160, 125, and 60 is 160