Problem:
A triangular number 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?
Answer Choices:
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Solution:
The triangular number . Let , where is a positive integer. Because and are relatively prime, there must exist nonnegative integers and such that and , or and . Hence , which is equivalent to . That is, it suffices to find the least odd perfect square greater than 49 that differs by 1 from twice the square of some integer.
Note that none of , or meets this condition. However, differs by 1 from 288 , which is . Therefore the required triangular number is
and the sum of its digits is .
Note: Euler showed in 1730 that there are infinitely many triangular numbers that are perfect squares. The fifth smallest one is . See sequence A001110 in the On-Line Encyclopedia of Integer Sequences.
The problems on this page are the property of the MAA's American Mathematics Competitions