# 2005 Indonesia MO Problems/Problem 3

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## Problem

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.

## Solution

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 byProblem 2 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 Followed byProblem 4 All Indonesia MO Problems and Solutions
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