MathNet 2014 Question 41

Let $ABC$ be an equilateral triangle with sides of length $1$. The point $X$ on the ray $AB$ and the point $Y$ on the ray $AC$ are chosen, so that $|AX|$ and $|AY|$ are positive integers.
Can the radius of the circumcircle of the triangle $AXY$ be $\sqrt{2014}$?