Find all pairs of integers such that when they are added they equal 100, and when they are multiplied they are at least 2000.
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
To find all pairs of integers that satisfy both conditions, we need to analyze the problem step by step.
First, let's consider the condition that the sum of the integers equals 100.
We can represent this condition mathematically as: x + y = 100, where x and y are the two integers.
Next, let's consider the condition that the product of the integers is at least 2000.
We can represent this condition as: x * y ≥ 2000.
To find the pairs of integers that satisfy both conditions, we can iterate through all possible values of x and y, and check if they meet both conditions. Let's start by assuming x ranges from 1 to 100 (inclusive), and y ranges from 1 to 100 (inclusive) as well.
Here's a Python implementation to find the pairs of integers:
```python
pairs = []
for x in range(1, 101):
for y in range(1, 101):
if x + y == 100 and x * y >= 2000:
pairs.append((x, y))
print(pairs)
```
The code above will iterate through all possible combinations of x and y values from 1 to 100. It checks if the sum of the integers is equal to 100 and if the product is at least 2000. If both conditions are satisfied, it adds the pair (x, y) to the `pairs` list.
After iterating through all values, the `pairs` list will contain all the pairs of integers that meet the given conditions.
Please note that there might be multiple pairs that satisfy the conditions, so the result will not be unique.