Here's an alternative that gets us there directly with algebra: it's quicker and requires a lot less guessing and checking.

Suppose we have w women and m men. Since (Sum of Scores) = Average * (# of Scores), we know that

(Sum of all scores) = 80*(m + w)

Let's also say that the average female score is a and the average male score is b. Then we have

w*a + m*b = 80*(m + w)

or

w*a + m*b = 80m + 80w

or

w*(a - 80) = m*(80 - b)

We want to know if a > 85.

S1:

b < 75

That means that m * (80 - b) > m * (80 - 75), or m * (80 - b) > 5m.

Since w*(a - 80) = m * (80 - b), we must also have

w*(a - 80) > 5m

Since we're looking to find a, let's rearrange everything in terms of a.

w*a - 80w > 5m

w*a > (5m + 80w)

a > (5m + 80w)/w

a > 5*(m/w) + 80

So close!

S2::

(m/w) > 1

By itself this is useless.

S1 + S2

We have

a > 5*(m/w) + 80
and
(m/w) > 1

so

a > 5*1 + 80

or

a > 85

Touchdown!