Question

A certain list consists of several different integers. is the product of all the integers in the list positive? 1). the product of the greatest and the smallest of the integers in the list are positive. 2). there is even number of integers in the list.

Answer 1

Let's take the worst case scenario. Suppose the list consists of 
-2, - 4, - 5, - 8

The product of the first and the last integers is positive
-2 * -8 = 16 so that checks out.
There's an even number in the list so the result will be positive.

Not enough variety. Suppose the list is
1, - 2, - 4 , -5
The largest is 1 the smallest - 5. The result disobeys the first condition of the problem.  

Any other list is trivial.

Answer: yes if those two conditions are met.