USC invited each South Carolina high school to send up to 39 students
to watch a football game. A section which has 199 seats in each row is
reserved for those students. What is the least number of rows needed to
guarantee that if 2006 students show up, then all students from the same
high school can be seated in the same row? Pick one:	A) 11	B) 12	C) 13	D) 14	E) 15
The answer is 12 rows. If 59 schools send 34 students each, then we can sit at   most 5 groups of students in the same row, so we will need 12 rows. Next, 12 rows are   sufficient. Assume that this is not the case. Suppose the groups of students are seated   like this: first the largest group, then the second largest group, then the third largest   group, etc. Suppose we run out of space - there are not enough seats in any row to   seat together the next group. Suppose the first group that can not be seated together   is the kth group and it consists of N students. Then k 61 since any row fits at least   5 groups. Also, N 2006/k 2006/61 &lt; 33 (all groups already seated are no smaller   than the kth group). So, N 32. Since there is not enough space in any of the 12 rows   to seat the kth group, then there must be at least 168 students seated in each of the 12   rows. Now, 12 × 168 = 2016 &gt; 2006 a contradiction. So, 12 rows are sufficient.   correct answer 12
<result>B</result>
USC invited each South Carolina high school to send up to 39 students
to watch a football game. A section which has 199 seats in each row is
reserved for those students. What is the least number of rows needed to
guarantee that if 2006 students show up, then all students from the same
high school can be seated in the same row?