Question

Find all pairs of integers such that when they are added they equal 100, and when they are multiplied they are at least 2000.

Answer 1

the the two integers be x and y
x+y = 100
y = 100-x

product = xy
= x(100-x)
100x - x^2 ≥ 2000
x^2 - 100x + 2000≤0

so we are looking for all those integer values of x where the parabola y = x^2 - 100x + 2000 lie below the x-axis, so we have to find the x-intercepts or the zeros of the function

I will use completing the square
x^2 - 100x = -2000
x^2 - 100x + 2500 = -2000+2500
(x-50)^2 = 500
x-50 = ± 22.36
x = 50-22.36 or x = 50+22.3
for integers
x = 28 or x = 72
then y = 100-28=72 or y = 28
so we want all the values of 28 ≤ x ≤ 72

so
x -- y--xy
28 72 2016
29 71 2059
30 72 ...
...
71 29 2059 ----> notice the x+y = 100 and xy > 2000
72 28 2016

so the number of such pairs = 72-28 +1= 45