Let $\Omega_1$ be a circle with centre $O$ and let $AB$ be diameter of $\Omega_1$. Let $P$ be a point on the segment $OB$ different from $O$. Suppose another circle $\Omega_2$ with centre $P$ lies in the interior of $\Omega_1$. Tangents are drawn from $A$ and $B$ to the circle $\Omega_2$ intersecting $\Omega_1$ again at $A_1$ and B1 respectively such that $A_1$ and $B_1$ are on the opposite sides of $AB$. Given that $A_1 B = 5, AB_1 = 15$ and $OP = 10$, find the radius of $\Omega_1$.
1. **Define the problem and given values:**
   - Let $\Omega_1$ be a circle with center $O$ and diameter $AB$.
   - Let $P$ be a point on the segment $OB$ different from $O$.
   - Another circle $\Omega_2$ with center $P$ lies in the interior of $\Omega_1$.
   - Tangents are drawn from $A$ and $B$ to the circle $\Omega_2$ intersecting $\Omega_1$ again at $A_1$ and $B_1$ respectively.
   - Given: $A_1B = 5$, $AB_1 = 15$, and $OP = 10$.

2. **Identify the tangents and their properties:**
   - Let $AA_1$ and $BB_1$ be tangents to $\Omega_2$ at points $A_2$ and $B_2$ respectively.
   - Note that $\angle AA_2P = \angle AA_1B = \angle BB_2P = \angle BB_1A = \frac{\pi}{2}$ because tangents to a circle from a point are perpendicular to the radius at the point of tangency.

3. **Use the given lengths and properties of tangents:**
   - Since $A_1$ and $B_1$ are points where the tangents from $A$ and $B$ intersect $\Omega_1$ again, we can use the given lengths $A_1B = 5$ and $AB_1 = 15$ to set up relationships.
   - Let $R$ be the radius of $\Omega_1$. Since $AB$ is the diameter, $AB = 2R$.

4. **Set up the equations using the given lengths:**
   - From the given, $A_1B = 5$ and $AB_1 = 15$.
   - Since $A_1$ and $B_1$ are on opposite sides of $AB$, we can write:
     \[
     A_1B + AB_1 = AB = 2R
     \]
     Substituting the given values:
     \[
     5 + 15 = 2R \implies 20 = 2R \implies R = 10
     \]

5. **Verify the solution:**
   - Given $OP = 10$, and since $P$ lies on $OB$, $P$ is at a distance of $10$ from $O$.
   - The radius of $\Omega_1$ is $R = 10$, which is consistent with the given conditions and the derived relationships.

The final answer is $\boxed{10}$.