Nine friends bought three cakes, each cut into eight slices. All of them ate cake and no piece was left. Knowing that each of them only ate whole slices of the cake, we can be sure that:

(a) someone ate four slices;

(b) one of them ate only one slice;

(c) everyone ate at least two slices;

(d) some ate two slices and the others ate three slices;

(e) one of them ate, at least, three slices.

![](https://cdn.mathpix.com/cropped/2024_05_01_666a1aec7e447b0a6cbeg-004.jpg?height=287&width=517&top_left_y=953&top_left_x=1318)
The correct option is (e).

We have a total of $8 \times 3=24$ slices of cake that were eaten. Since everyone ate cake, initially each of the nine friends ate one slice, leaving $24-9=$ 15 slices to be eaten by nine people. If all nine friends had eaten less than two of these 15 slices, we could write the number 15 as a sum of nine terms, each being 0 (those who did not eat any of the 15 slices) or 1 (those who ate one of the 15 slices), which is clearly not possible. Therefore, someone must have eaten at least two of these 15 slices. Since everyone had already eaten one slice initially, we conclude that someone ate at least 3 slices.