Difference between revisions of "2024 AMC 10B Problems/Problem 4"
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Revision as of 14:42, 14 November 2024
- The following problem is from both the 2024 AMC 10B #4 and 2024 AMC 12B #4, so both problems redirect to this page.
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