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Problem 870

AMC 12 late, AIME early

Algebra Difficulty 4.6 Multiple choice AMC 10 B · United States

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 bin $B$ . The next $3$ 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, balls numbered $22, 23, \ldots, 28$ are deposited in bin $B$ at step $7$ of this process.) In which bin is ball $2024$ deposited?

Pick one

Official solution

Answer (D): After $n$ steps, a total of $1+2+3+\dots+n = \frac{1}{2}n(n+1)$ balls have been deposited. In particular, after step $n = 63$ , a total of $\frac{1}{2} \cdot 63 \cdot 64 = 2016$ balls have been deposited. The next batch of $64$ balls will include ball $2024$ . Because $64$ has remainder $4$ when divided by $5$ , ball $2024$ is deposited in bin $D$ .

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