document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1176058>Question 1176058</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Yann writes down the first n consecutive positive integers, 1, 2, 3, 4, . . . , n − 1, n. He removes four different integers p, q, r, s from the list. At least three of p, q, r, s are consecutive and 100 < p < q < r < s. The average of the integers remaining in the list is 89.5625. The number of possible values of s is<BR>\n" );
document.write( "answer options: 30,32,28,24,26 </SPAN>\n" );
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document.write( "The average of the remaining (n-4) integers is 89.5625 = 89 9/16.<br><BR>\n" );
document.write( "That means n-4 is a multiple of 16.<br><BR>\n" );
document.write( "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><BR>\n" );
document.write( "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><BR>\n" );
document.write( "The sum of the first n=180 positive integers is (180)(181)/2 = 16290.<br><BR>\n" );
document.write( "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><BR>\n" );
document.write( "So the sum of p, q, r, and s is 16290-15763 = 527.<br><BR>\n" );
document.write( "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><BR>\n" );
document.write( "101, 102, 103, 221<BR>\n" );
document.write( "102, 103, 104, 218<BR>\n" );
document.write( "103, 104, 105, 215<BR>\n" );
document.write( "...<BR>\n" );
document.write( "129, 130, 131, 137<BR>\n" );
document.write( "130, 131, 132, 134<br><BR>\n" );
document.write( "That is 30 different combinations, with 30 different values of s.<br><BR>\n" );
document.write( "ANSWER: The number of possible values of s is 30.<br><BR>\n" );
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