PRMO 2012 question 5

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5. Let $S_n = n^2 + 20n + 12$, $n$ a positive integer. What is the sum of all possible values of $n$ for which $S_n$ is a perfect square?
[PRE-PRMO–2012]

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Solution

Answer: 3 + 13 = 16

Let $n^2 + 20n + 12 = u^2$

$$(n + 10)^2 - 88 = u^2$$ $$(n + 10 - u)(n + 10 + u) = 88$$

As

$$88 = 88 \times 1 = 44 \times 2 = 22 \times 4 = 11 \times 8$$

Using each of the pair of factors of 88, we can get different values of $n$ and $u$.
Of all the values, we get $n = 3$ and $n = 13$ as positive integer solutions.
Thus, sum of all possible values of $n$ is $3 + 13 = 16$

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