Question 1176058
<br>
The average of the remaining (n-4) integers is 89.5625 = 89 9/16.<br>
That means n-4 is a multiple of 16.<br>
The sum of the remaining (n-4) numbers is approximately 90(n-4), and four integers greater than 100 were removed from the list.  So the average of the n integers is slightly more than 90.  Since the list is the consecutive integers starting with 1, that means the number of numbers in the list, n, is about 180.<br>
n-4 is a multiple of 16, so n is 4 more than a multiple of 16.  The numbers close to 180 that are 4 more than a multiple of 16 are 164, 180, and 196.  So it is almost certain that n is 180.<br>
The sum of the first n=180 positive integers is (180)(181)/2 = 16290.<br>
The sum of the n-4=176 integers remaining in the list after p, q, r, and s are removed is 176(89.5625) = 15763.<br>
So the sum of p, q, r, and s is 16290-15763 = 527.<br>
p, q, r, and s are all greater than 100; at least three of them are consecutive; and their sum is 527.  The possibilities are....<br>
101, 102, 103, 221
102, 103, 104, 218
103, 104, 105, 215
...
129, 130, 131, 137
130, 131, 132, 134<br>
That is 30 different combinations, with 30 different values of s.<br>
ANSWER: The number of possible values of s is 30.<br>