[Hints] 2013 APMO Problem 2

2013 APMO 2. Determine all positive integers n for which \displaystyle \frac{n^2+1}{[\sqrt{n}]^2 + 2} is an integer. Here [r] denotes the greatest integer less than or equal to r.

Hint 1: Let k := [\sqrt{n}]. Then n = k^2 + \alpha, where \alpha = 0, 1, \dots 2k. Think of the problem in terms of k instead.

 

 

 

 

 

 

 

 

 

Hint 2: After rewriting the fraction in the question in terms of k and \alpha, perform long division so that the degree of k is lower in the numerator than in the denominator.

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