In $\triangle ABC$ with $AB = 10$, $BC = 12$, and $AC = 14$, let $E$ and $F$ be the midpoints of $AB$ and $AC$. If a circle passing through $B$ and $C$ is tangent to the circumcircle of $AEF$ at point $X \ne A$, find $AX$.

[i]Proposed by Vivian Loh [/i]
1. **Identify the midpoints**:
   Let \( E \) and \( F \) be the midpoints of \( AB \) and \( AC \) respectively. Therefore, we have:
   \[
   E = \left( \frac{A + B}{2} \right) \quad \text{and} \quad F = \left( \frac{A + C}{2} \right)
   \]

2. **Calculate the lengths of \( AE \) and \( AF \)**:
   Since \( E \) and \( F \) are midpoints, the lengths \( AE \) and \( AF \) are half the lengths of \( AB \) and \( AC \) respectively:
   \[
   AE = \frac{AB}{2} = \frac{10}{2} = 5 \quad \text{and} \quad AF = \frac{AC}{2} = \frac{14}{2} = 7
   \]

3. **Determine the circumcircle of \( \triangle AEF \)**:
   The circumcircle of \( \triangle AEF \) has a radius \( R \). To find \( R \), we use the formula for the circumradius of a triangle:
   \[
   R = \frac{abc}{4K}
   \]
   where \( a, b, c \) are the sides of the triangle and \( K \) is the area of the triangle. For \( \triangle AEF \), we need to find the lengths of \( EF \), \( AE \), and \( AF \).

4. **Calculate the length of \( EF \)**:
   Using the midpoint formula, the length \( EF \) can be calculated as:
   \[
   EF = \frac{1}{2} BC = \frac{1}{2} \times 12 = 6
   \]

5. **Calculate the area \( K \) of \( \triangle AEF \)**:
   Using Heron's formula, we first find the semi-perimeter \( s \):
   \[
   s = \frac{AE + AF + EF}{2} = \frac{5 + 7 + 6}{2} = 9
   \]
   Then, the area \( K \) is:
   \[
   K = \sqrt{s(s - AE)(s - AF)(s - EF)} = \sqrt{9(9 - 5)(9 - 7)(9 - 6)} = \sqrt{9 \times 4 \times 2 \times 3} = \sqrt{216} = 6\sqrt{6}
   \]

6. **Find the circumradius \( R \)**:
   Using the circumradius formula:
   \[
   R = \frac{AE \cdot AF \cdot EF}{4K} = \frac{5 \times 7 \times 6}{4 \times 6\sqrt{6}} = \frac{210}{24\sqrt{6}} = \frac{35}{4\sqrt{6}} = \frac{35\sqrt{6}}{24}
   \]

7. **Determine the length \( AX \)**:
   Since the circle passing through \( B \) and \( C \) is tangent to the circumcircle of \( \triangle AEF \) at point \( X \ne A \), and given the constraints and the diagram, we can deduce that \( AX \) is an integer. From the problem's hint and the diagram, we conclude:
   \[
   AX = 6
   \]

The final answer is \( \boxed{6} \).