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Find the edge of a cube inscribed in a sphere of radius $R$.

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Since the center of the sphere circumscribed around the cube lies on the line passing through the center of the face and perpendicular to it, the centers of the sphere and the cube coincide. Therefore, the diagonal of the cube is the diameter of the circumscribed sphere. Let the edge of the cube be $x$. Then the diagonal of the cube is $x \sqrt{3}$. From the equation $x \sqrt{3} = 2R$

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