Consider two parabolas $y = x^2$ and $y = x^2 - 1$. Let $U$ be the set of points between the parabolas (including the points on the parabolas themselves). Does $U$ contain a line segment of length greater than $10^6$ ?

Alexey Tolpygo
1. **Identify the points on the parabolas:**
   - Consider the points \( M_1(t-1, (t-1)^2 - 1) \) and \( M_2(t+1, (t+1)^2 - 1) \) on the parabola \( y = x^2 - 1 \).

2. **Calculate the coordinates of \( M_1 \) and \( M_2 \):**
   - For \( M_1 \):
     \[
     M_1 = (t-1, (t-1)^2 - 1) = (t-1, t^2 - 2t + 1 - 1) = (t-1, t^2 - 2t)
     \]
   - For \( M_2 \):
     \[
     M_2 = (t+1, (t+1)^2 - 1) = (t+1, t^2 + 2t + 1 - 1) = (t+1, t^2 + 2t)
     \]

3. **Find the midpoint \( N \) of \( M_1M_2 \):**
   - The midpoint \( N \) is:
     \[
     N = \left( \frac{(t-1) + (t+1)}{2}, \frac{(t^2 - 2t) + (t^2 + 2t)}{2} \right) = (t, t^2)
     \]
   - Note that \( N \) lies on the parabola \( y = x^2 \).

4. **Verify that \( M_1M_2 \) is tangent to \( y = x^2 \) at \( N \):**
   - The slope of the tangent line to \( y = x^2 \) at \( N(t, t^2) \) is given by the derivative \( \frac{dy}{dx} = 2t \).
   - The slope of the line segment \( M_1M_2 \) is:
     \[
     \text{slope of } M_1M_2 = \frac{(t^2 + 2t) - (t^2 - 2t)}{(t+1) - (t-1)} = \frac{4t}{2} = 2t
     \]
   - Since the slopes match, \( M_1M_2 \) is indeed tangent to \( y = x^2 \) at \( N \).

5. **Calculate the length of the line segment \( M_1M_2 \):**
   - The distance between \( M_1 \) and \( M_2 \) is:
     \[
     \text{Length of } M_1M_2 = \sqrt{((t+1) - (t-1))^2 + ((t^2 + 2t) - (t^2 - 2t))^2} = \sqrt{(2)^2 + (4t)^2} = \sqrt{4 + 16t^2} = 2\sqrt{4t^2 + 1}
     \]

6. **Determine if the length can be greater than \( 10^6 \):**
   - We need \( 2\sqrt{4t^2 + 1} > 10^6 \).
   - Dividing both sides by 2:
     \[
     \sqrt{4t^2 + 1} > 5 \times 10^5
     \]
   - Squaring both sides:
     \[
     4t^2 + 1 > (5 \times 10^5)^2
     \]
   - Simplifying:
     \[
     4t^2 + 1 > 25 \times 10^{10}
     \]
     \[
     4t^2 > 25 \times 10^{10} - 1
     \]
     \[
     t^2 > \frac{25 \times 10^{10} - 1}{4}
     \]
     \[
     t^2 > 6.25 \times 10^{10} - 0.25
     \]
     \[
     t^2 > 6.25 \times 10^{10}
     \]
     \[
     t > \sqrt{6.25 \times 10^{10}}
     \]
     \[
     t > 2.5 \times 10^5
     \]

Since \( t \) can be any real number, we can always choose \( t \) such that \( t > 2.5 \times 10^5 \), ensuring that the length of \( M_1M_2 \) is greater than \( 10^6 \).

The final answer is \( \boxed{ U } \) contains a line segment of length greater than \( 10^6 \).