2024 AMC 10B Problems/Problem 4
- The following problem is from both the 2024 AMC 10B #4 and 2024 AMC 12B #4, so both problems redirect to this page.
Contents
[hide]Problem
Balls numbered 1, 2, 3, ... 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, ..., 28 are deposited in bin B at step 7 of this process.) In which bin is ball 2024 deposited?
Solution 1
Consider the triangular array of numbers: .
The numbers in a row congruent to will be in bucket A. Similarly, the numbers in a row congruent to will be in buckets B, C, D, and E respectively. Note that the row ends with the triangle number, .
We must find values of , that make close to .
Trying we find that . Since, will be the last ball in row , ball will be in row . Since , ball will be placed in bucket .
~numerophile
Video Solution 1 by Pi Academy (Fast and Easy ⚡🚀)
https://youtu.be/DIl3rLQQkQQ?feature=shared
~ Pi Academy
Solution 2
~Kathan