2005 Indonesia MO Problems/Problem 3

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Let $k$ and $m$ be positive integers such that $\frac12\left(\sqrt{k+4\sqrt{m}}-\sqrt{k}\right)$ is an integer.

(a) Prove that $\sqrt{k}$ is rational.

(b) Prove that $\sqrt{k}$ is a positive integer.


Let $\frac12\left(\sqrt{k+4\sqrt{m}}-\sqrt{k}\right) = n$, where all variables are integers. Rearranging the expression results in $\sqrt{k+4\sqrt{m}} = 2n + \sqrt{k}$.

Squaring both sides results in $k+4\sqrt{m} = 4n^2 + 4n\sqrt{k} + k$, and rearranging terms results in $\sqrt{m} = n^2 + n\sqrt{k}$.

Squaring both sides results in $m = n^4 + 2n^3 \sqrt{k} + n^2 k$. Solving for $\sqrt{k}$ results in $\sqrt{k} = \frac{m - n^4 - n^2 k}{2n^3}$. Since the right side is rational, the left side must be rational. Therefore, $\sqrt{k}$ is rational, and since $k$ is a positive integer, $\sqrt{k}$ must be a positive integer.

See Also

2005 Indonesia MO (Problems)
Preceded by
Problem 2
1 2 3 4 5 6 7 8 Followed by
Problem 4
All Indonesia MO Problems and Solutions
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