Question

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

Answer 1

maximum product occurs when a=b=50
That makes the product 2500
The larger a gets, the smaller b must get, so try
a=40 b=60: ab=2400
a=30 b=70: ab=2100
a=20 b=80: ab=1600
and narrow in on it.

Or, algebraically,

a(100-a) >= 2000
a^2 - 100a + 2000 <= 0
10(5-√5) <= a <= 10(5+√5)
27.6 <= a <= 72.3

so, 28 < a <= 72

check:
28*72 = 2016
27*73 = 1971