Question 5
An equilateral triangle is inscribed in the ellipse whose equation is $x^2 + 4y^2 = 4.$ One vertex of the triangle is $(0, 1),$ one altitude is contained in the $y$-axis, and the length of each side is $\sqrt{\frac{m}{n}},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$