2022 AMC 12A Problems/Problem 16
A is a positive integer that can be expressed in the form , for some positive integer . The three smallest triangular numbers that are also perfect squares are , , and . What is the sum of the digits of the fourth smallest triangular number that is also a perfect square?
We have . If is a perfect square, then it can be written as , where is a positive integer.
Thus, . Rearranging, we get , a Pell equation. So must be a truncation of the continued fraction for :
Therefore, , so the answer is .
- Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Edited by wzs26843545602
Solution 2 (Bash)
As mentioned above, . If is a perfect square, one of two things must occur when the fraction is split into a product. Either and must both be squares, or and must both be squares, and thus the search for the next perfect square triangular number can be narrowed down by testing values of that are close to or are perfect squares. After some work, we reach , less than , and . This product is a perfect square, and thus the sum of the digits of the fourth smallest perfect square triangular number is therefore .
According to the problem, we want to find integer such , after expanding, we have , we call , the equation becomes , obviously is the elementary solution for this pell equation, thus the forth smallest solution set , which indicates leads to
If is a square, then is also a square. We can prove this quite simply:
Therefore, is a square. Note that . We can easily check all smaller possibilities using a bit of casework, and they don't work. Our solution is thus , and so the answer is .
We want to find integer and such that
We use the formula and get
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
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