14. Let $A, B, C$ be distinct points on the parabola $y=x^{2}$, and $R$ be the radius of the circumcircle of $\triangle A B C$. Find the range of values for $R$.

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Given $A\left(a, a^{2}\right), B\left(-a, a^{2}\right), C(0,0)$, then
the center $M(0, b)$ is on the $y$-axis, and $b^{2}=a^{2}+\left(a^{2}-b\right)^{2}$
$$
\Rightarrow b=\frac{a^{2}+1}{2}>\frac{1}{2} \Rightarrow R \in\left(\frac{1}{2},+\infty\right) \text {. }
$$

For the general case, let the equation of the circle be
$(x-a)^{2}+(y-b)^{2}=R^{2}$, substituting $y=x^{2}$ gives
$$
x^{4}+(1-2 b) x^{2}-2 a x+a^{2}+b^{2}-R^{2}=0 \text {. }
$$

Let the four roots be $x_{1}, x_{2}, x_{3}, x_{4}$, where $x_{1}, x_{2}, x_{3}$ are distinct, then $\sum_{i=1}^{4} x_{i}=0, \sum_{1 \leqslant i2 \Rightarrow R>\frac{1}{2}$. In summary, the range of $R$ is $\left(\frac{1}{2},+\infty\right)$.