Can Mulder Escape?

Mulder is inside a circular field of radius R. An alien moves only on the perimeter, but it is 4 times faster than Mulder. If Mulder reaches any unguarded point on the fence, he escapes instantly.

Geometry View

Blue = Mulder, Orange = Alien, dashed inner circle = radius R/4
r = R/4 alien must cover half the fence Mulder Alien center

Key Idea

  1. First, stay close enough to the center.
    If Mulder is at distance r from the center, his maximum angular speed is v/r. The alien’s angular speed along the fence is 4v/R.
v/r ≥ 4v/R  ⇒  r ≤ R/4
So while Mulder is inside the circle of radius R/4, he can maneuver so that the alien stays exactly opposite him.
  1. Then Mulder runs straight to the boundary.
    If the alien is on the opposite side, it must travel half the circumference to intercept him.
Alien time = (πR)/(4v)
Mulder time = (R - r)/v
(R - r)/v < (πR)/(4v)

⇒ 1 - r/R < π/4
⇒ r/R > 1 - π/4 ≈ 0.2146
So Mulder should start his sprint from a radius r satisfying:
0.2146R < r < 0.25R
Therefore, yes — Mulder can escape.

Numerical Check

Try a value of r between 0.2146R and 0.25R.
Mulder’s sprint time
0.760 R/v
Alien’s shortest intercept time
0.785 R/v
Conclusion: Mulder wins by circling near the center until the alien is diametrically opposite, then sprinting outward.