A triangle $ABC$ with $AC=20$ is inscribed in a circle $\omega$. A tangent $t$ to $\omega$ is drawn through $B$. The distance $t$ from $A$ is $25$ and that from $C$ is $16$.If $S$ denotes the area of the triangle $ABC$, find the largest integer not exceeding $\frac{S}{20}$
1. **Identify the given information and set up the problem:**
   - Triangle \(ABC\) is inscribed in a circle \(\omega\).
   - \(AC = 20\).
   - A tangent \(t\) to \(\omega\) is drawn through \(B\).
   - The distance from \(A\) to \(t\) is \(25\).
   - The distance from \(C\) to \(t\) is \(16\).

2. **Define the feet of the perpendiculars:**
   - Let \(Q\) be the foot of the altitude from \(B\) to \(AC\).
   - Let \(E\) and \(F\) be the feet of the perpendiculars from \(A\) and \(C\) to the tangent \(t\), respectively.

3. **Use the properties of similar triangles:**
   - Since \(\angle ACB = \angle (\ell, AC)\) and \(\angle CAB = \angle (\ell, BC)\) due to the angle in alternate segments, we have:
     \[
     \triangle BFC \sim \triangle AQB \quad \text{and} \quad \triangle ABE \sim \triangle BCQ
     \]

4. **Set up the ratios from the similar triangles:**
   - From \(\triangle BFC \sim \triangle AQB\):
     \[
     \frac{BQ}{AC} = \frac{BF}{AQ} \implies \frac{BQ}{20} = \frac{16}{AQ}
     \]
   - From \(\triangle ABE \sim \triangle BCQ\):
     \[
     \frac{BQ}{AB} = \frac{BE}{BC} \implies \frac{BQ}{AB} = \frac{25}{BC}
     \]

5. **Equate the expressions for \(BQ\):**
   - From the first ratio:
     \[
     BQ = \frac{16 \cdot 20}{AQ}
     \]
   - From the second ratio:
     \[
     BQ = \frac{25 \cdot AB}{BC}
     \]

6. **Solve for \(BQ\):**
   - Since \(BQ\) is the same in both expressions, we equate them:
     \[
     \frac{16 \cdot 20}{AQ} = \frac{25 \cdot AB}{BC}
     \]
   - Let \(AB = a\) and \(BC = c\). Then:
     \[
     \frac{16 \cdot 20}{AQ} = \frac{25 \cdot a}{c}
     \]
   - From the given distances, we know:
     \[
     AQ = 25 \quad \text{and} \quad BQ = 20
     \]

7. **Calculate the area of \(\triangle ABC\):**
   - The area \(\Delta\) of \(\triangle ABC\) is given by:
     \[
     \Delta = \frac{1}{2} \cdot AC \cdot BQ = \frac{1}{2} \cdot 20 \cdot 20 = 200
     \]

8. **Find the largest integer not exceeding \(\frac{S}{20}\):**
   - \(\frac{S}{20} = \frac{200}{20} = 10\)

The final answer is \(\boxed{10}\)