Problem 1655

1. Initial Setup and Definitions:
- Let $B_1, B_2, \ldots, B_n$ be the boxes with cookies.
- Let $c_i$ be the number of cookies in box $B_i$.
- Alex records the distinct values of $c_i$ in a set $A$.

2. Serge's Process:
- Serge takes one cookie from each non-empty box and places them on the first plate.
- He repeats this process, placing cookies on subsequent plates until all boxes are empty.
- Let $P_1, P_2, \ldots, P_m$ be the plates used by Serge.

3. Recording the Number of Cookies on Each Plate:
- Let $p_i$ be the number of cookies on plate $P_i$.
- Serge records the distinct values of $p_i$ in a set $S$.

4. Visual Representation:
- Consider each box as a column in a vertical histogram where the height of the column represents the number of cookies.
- Draw horizontal lines at each integer height from 1 to the maximum number of cookies in any box.

5. Analysis of Serge's Process:
- Each horizontal line at height $y$ intersects the columns at points where the boxes have at least $y$ cookies.
- The number of intersections at height $y$ corresponds to the number of cookies on the $y$-th plate.

6. Distinct Heights and Plate Counts:
- If two boxes have the same height (same number of cookies), they contribute the same number of cookies to each plate until they are empty.
- Therefore, we can consider each distinct height as a unique contribution to the plates.

7. Equivalence of Records:
- The number of distinct heights in the histogram is equal to the number of distinct values in Alex's record $A$.
- The number of distinct plate counts is equal to the number of distinct values in Serge's record $S$.

8. Conclusion:
- Since each distinct height corresponds to a unique number of cookies on a plate, the number of distinct values in $A$ is equal to the number of distinct values in $S$.

$$\boxed{\text{Alex's record contains the same number of numbers as Serge's record.}}$$