Question

Balls numbered $1, 2, 3, \ldots$ are deposited in $5$ bins, labeled $A, B, C, D,$ and $E,$ using the following procedure. Ball $1$ is deposited in bin $A,$ and balls $2$ and $3$ are deposited in $B.$ The next three balls are deposited in bin $C,$ the next $4$ in bin $D,$ and so on, cycling back to bin $A$ after balls are deposited in bin $E.$ (For example, $22, 23, \ldots, 28$ are deposited in bin $B$ at step $7$ of this process.) In which bin is ball $2024$ deposited?

$A$

$B$

$C$

$D$

$E$

Solution:

Step $k$ deposits $k$ balls, so after step $k$ a total of $\dfrac{k(k+1)}{2}$ balls have been placed. Since $\dfrac{63 \cdot 64}{2} = 2016$ and $\dfrac{64 \cdot 65}{2} = 2080,$ ball $2024$ falls in step $64.$

The steps cycle through the bins $A, B, C, D, E,$ so step $k$ uses position $(k-1) \bmod 5.$ Here $(64 - 1) \bmod 5 = 63 \bmod 5 = 3,$ which is the fourth bin, $D.$

Thus, the correct answer is D.

2024 AMC 12B Problem 4

Solution:

Problem 4 in Other Years