How many lattice points are exactly twice as close to $(0,0)$ as they are to $(15,0)$? (A lattice point is a point $(a,b)$ such that both $a$ and $b$ are integers.)
1. **Set up the distance equations:**
   - The distance from a point \((x, y)\) to \((0, 0)\) is given by:
     \[
     d_1 = \sqrt{x^2 + y^2}
     \]
   - The distance from the point \((x, y)\) to \((15, 0)\) is given by:
     \[
     d_2 = \sqrt{(x - 15)^2 + y^2}
     \]
   - According to the problem, the point \((x, y)\) is twice as close to \((0, 0)\) as it is to \((15, 0)\). Therefore, we have:
     \[
     d_1 = \frac{1}{2} d_2
     \]

2. **Square both sides to eliminate the square roots:**
   \[
   x^2 + y^2 = \left(\frac{1}{2} \sqrt{(x - 15)^2 + y^2}\right)^2
   \]
   \[
   x^2 + y^2 = \frac{1}{4} ((x - 15)^2 + y^2)
   \]

3. **Multiply both sides by 4 to clear the fraction:**
   \[
   4(x^2 + y^2) = (x - 15)^2 + y^2
   \]

4. **Expand and simplify the equation:**
   \[
   4x^2 + 4y^2 = x^2 - 30x + 225 + y^2
   \]
   \[
   4x^2 + 4y^2 = x^2 + y^2 - 30x + 225
   \]

5. **Combine like terms:**
   \[
   4x^2 + 4y^2 - x^2 - y^2 = -30x + 225
   \]
   \[
   3x^2 + 3y^2 = -30x + 225
   \]

6. **Divide the entire equation by 3:**
   \[
   x^2 + y^2 = -10x + 75
   \]

7. **Rearrange the equation to form a circle equation:**
   \[
   x^2 + 10x + y^2 = 75
   \]
   \[
   (x + 5)^2 - 25 + y^2 = 75
   \]
   \[
   (x + 5)^2 + y^2 = 100
   \]

8. **Identify the circle's center and radius:**
   - The equation \((x + 5)^2 + y^2 = 100\) represents a circle centered at \((-5, 0)\) with a radius of 10.

9. **Find the lattice points on the circle:**
   - We need to find integer solutions \((x, y)\) such that \((x + 5)^2 + y^2 = 100\).

10. **List the possible integer pairs \((x + 5, y)\) that satisfy the equation:**
    - \((10, 0)\), \((-10, 0)\), \((0, 10)\), \((0, -10)\), \((6, 8)\), \((6, -8)\), \((-6, 8)\), \((-6, -8)\), \((8, 6)\), \((8, -6)\), \((-8, 6)\), \((-8, -6)\).

11. **Convert back to the original coordinates \((x, y)\):**
    - \((x, y) = (5, 0)\), \((-15, 0)\), \((-5, 10)\), \((-5, -10)\), \((1, 8)\), \((1, -8)\), \((-11, 8)\), \((-11, -8)\), \((3, 6)\), \((3, -6)\), \((-13, 6)\), \((-13, -6)\).

12. **Count the number of lattice points:**
    - There are 12 lattice points that satisfy the condition.

The final answer is \(\boxed{12}\)