Giving out Star Stickers

Let the scores of the students be $s_1, s_2, \ldots s_{23}$ . If everyone did better than the class mean, then $s_j > \frac {\sum_{i=1}^{23} s_j} {23}.$

If we were to add up these $n$ inequalities, we get $\sum_{j=1}^{23} s_j > \sum_{i=1}^{23} s_i$ , which is a contradiction since both of these terms are equal. Hence, not all 23 of the students can receive a star sticker.

By considering $s_1=0$ , $s_2 = s_3 = \ldots s_{23} = 100$ , we see that all students except for the first did better than the class mean. Hence, it is possible for 22 students to receive a star sticker. Thus, the maximum number of students who can receive a star sticker is 22.