2025/26 AHSMC Part 1 Problem 12

Problem 12 of 16

A triangular number is any integer of the form $n(n+1)/2$ where $n$ is a positive integer. The number of positive integers $k$ for which $k+1$ is a perfect square and $2k+30$ is a triangular number is

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Solution

We need $k+1 = m^2$ and $2k+30 = n(n+1)/2$ for positive integers $m,n$ . This rearranges to $(2n+1)^2 - (4m)^2 = 225 = 15^2,$ so the positive factor pairs of $225$ with first factor smaller than the second and of the same parity are $(1,225)$ , $(3,75)$ , $(5,45)$ , $(9,25)$ . These yield $(k,m,n) = (783,28,56),\ (80,9,19),\ (24,5,12),\ (3,2,8).$ Four values of $k$ . The answer is (D).

University of Alberta

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