2021 AIME I Problems/Problem 4
Problem
Find the number of ways identical coins can be separated into three nonempty piles so that there are fewer coins in the first pile than in the second pile and fewer coins in the second pile than in the third pile.
Solution 1
Suppose we have coin in the first pile. Then
all work for a total of
piles. Suppose we have
coins in the first pile, then
all work, for a total of
. Continuing this pattern until
coins in the first pile, we have the sum
.
Solution 2
Let the three piles have coins respectively. If we disregard order, then we just need to divide by
at the end.
We know . Since
are positive integers, there are
ways from Stars and Bars.
However, we must discard the cases where or
or
. The three cases are symmetric, so we just take the first case and multiply by 3. We have
for 32 solutions. Multiplying by 3, we will subtract 96 from our total.
But we undercounted where . This is first counted 1 time, then we subtract it 3 times, so we add it back twice. There is clearly only 1 way, for a total of 2.
Hence, the answer is
Solution 3
Let the piles have and
coins, with
. Then, let
, and
, such that each
. The sum is then
. This is simply the number of positive solutions to the equation
. Now, we take cases on
.
If , then
. Each value of
corresponds to a unique value of
, so there are
solutions in this case. Similarly, if
, then
, for a total of
solutions in this case. If
, then
, for a total of
solutions. In general, the number of solutions is just all the numbers that aren't a multiple of
, that are less than or equal to
.
We then add our cases to get as our answer.
See also
2021 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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