2021 AIME I Problems/Problem 4
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.
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
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
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.
We make an equation for how this is true. a+b+c = 66, where a is less than b, is less than c. We don't have a clear solution, so we'll try complimentary counting. First, let's find where a doesn't have to be less than b, which doesn't have to be less than c. We have =2080 by stars and bars. Now we need to subtract off the cases where it doesn't satisfy the condition.
We first by starting out with a = b. We can write that as 2b + c = 66. We can find there are 32 integer solutions to this equation. There are 32 solutions for b=c, and a = c by symmetry. We also need to subtract 2 from the 96, because we counted 22, 22, 22 3 times. We then have to divide by 6 because there are 3! ways to order the a, b, and c. Therefore, we have =
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