This problem can be proved using the Intermediate Value Theorem (IVT), which applies to continuous functions.

Steps:

1. Define the Situation:

   • Let $v_1(t)$ and $v_2(t)$ represent the velocities of the two bicyclists at time $t$, where $t$ is the time elapsed since the race started at 8:00 A.M.
   • Both bicyclists finish the race at the same time, 2 hours and 15 minutes later, which is at $t = 2.25$ hours.

2. Consider the Function $f(t) = v_1(t) - v_2(t)$:

   • $f(t)$ represents the difference in their velocities at time $t$.
   • The function $f(t)$ is continuous because the velocity of each bicyclist is assumed to be continuous over time (i.e., there are no sudden jumps or drops in their velocities).

3. Check the Values of $f(t)$:

   • At the start of the race ($t = 0$):
     • The velocity of both bicyclists could be different, so $f(0) = v_1(0) - v_2(0)$ might not be zero.
   • At the end of the race ($t = 2.25$ hours):
     • Both bicyclists finish the race at the same time, which means they must have traveled the same distance.
     • Since the time is the same, their average velocities over the entire race are the same.
     • However, their instantaneous velocities could still be different, so $f(2.25) = v_1(2.25) - v_2(2.25)$ might also not be zero.

4. Apply the Intermediate Value Theorem:

   • Since $f(t)$ is a continuous function, and the cyclists finish the race simultaneously, $f(t)$ must take every value between $f(0)$ and $f(2.25)$.
   • If $f(0)$ and $f(2.25)$ are of opposite signs or one of them is zero, by the Intermediate Value Theorem, there must be some time $t = c$ within the interval $0 < c < 2.25$ where $f(c) = 0$. This implies $v_1(c) = v_2(c)$.

Conclusion:

Therefore, at some time during the race, the two bicyclists must have been traveling at the same velocity.

---

Do you want any more details or have questions about this proof?

Here are five related questions:

1. How does the Intermediate Value Theorem apply to continuous functions in general?
2. Can this proof be applied to situations where the velocities are not continuous?
3. What assumptions about the bicyclists' motion are necessary for this proof to hold?
4. How would the proof change if the cyclists didn't finish at the same time?
5. What is the significance of the IVT in other areas of calculus?

Tip: When proving statements about continuous functions, always consider whether the Intermediate Value Theorem might be applicable—it’s a powerful tool in analysis.