Maths Olympiad Prep

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, 2016

Algebra Difficulty 4.3 AIME Prove it Baltic Way

Prove that
1+1n2+1(n+1)2 \sqrt{1 + \frac{1}{n^2} + \frac{1}{(n+1)^2}}
is rational for every positive integer nn.

Solution

As
1+1n2+1(n+1)2=n2(n+1)2+(n+1)2+n2n(n+1) \sqrt{1 + \frac{1}{n^2} + \frac{1}{(n+1)^2}} = \frac{\sqrt{n^2(n+1)^2 + (n+1)^2 + n^2}}{n(n+1)}
it suffices to show that n2(n+1)2+(n+1)2+n2n^2(n + 1)^2 + (n + 1)^2 + n^2 is a perfect square. This follows from
n2(n+1)2+(n+1)2+n2=n2((n+1)2+1)+(n+1)2=n4+2n2(n+1)+(n+1)2=(n2+n+1)2. \begin{aligned} n^2(n+1)^2 + (n+1)^2 + n^2 &= n^2((n+1)^2 + 1) + (n+1)^2 \\ &= n^4 + 2n^2(n+1) + (n+1)^2 \\ &= (n^2 + n + 1)^2. \end{aligned}

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