A Pre - RMO 2012 Question

Number Theory Level 4

Let S n = n 2 + 20 n + 12 S_n = n^2 + 20n + 12 , where n n is a positive integer.

What is the sum of all possible values of n n , for which S n S_n is a perfect square?


The answer is 16.

Abc Xyz by abc xyz , 18, India

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2 solutions

Aditya Kumar
May 30, 2016

n^2 < (n^2+20n+12) < (n+10)^2 Therefore n can only take the values 13 and 3 . The sum of these two gives 16 as the right ans

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please elaborate, how n can take only 13 and 3

Ayush Jaiswal - 5 years ago

We may bound S n as follows: (n+3)^2 < S n < (n+10)^2 . again S_n can't be equal to (n+i)^2 if i is odd. Checking the remaining perfect squares in between the possible values comes out to be 3 and 13, which gives the required sum: 16

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