20

Step 1: Identify the speeds of the men and cyclists. The runners are traveling at 15 km/h and the cyclists at 25 km/h.
Step 2: Convert the distances to time intervals. The runners are 20 m apart, and the cyclists are 30 m apart.
Step 3: Calculate the time taken for a runner to reach the next runner:
$$ ext{Time for runners} = \frac{20 \text{ m}}{15 \text{ km/h}} = \frac{20}{15 \times \frac{1000}{3600}} = \frac{20 \times 3600}{15 \times 1000} = \frac{72000}{15000} = 4.8 \text{ seconds} $$
Step 4: Calculate the time taken for a cyclist to reach the next cyclist:
$$ ext{Time for cyclists} = \frac{30 \text{ m}}{25 \text{ km/h}} = \frac{30}{25 \times \frac{1000}{3600}} = \frac{30 \times 3600}{25 \times 1000} = \frac{108000}{25000} = 4.32 \text{ seconds} $$
Step 5: Set the times equal for the observer meeting both the runner and cyclist. Let the speed of the observer be $S$ km/h. The relative speed of the observer when meeting a runner is $(S + 15)$ km/h and with the cyclist is $(S + 25)$ km/h.
Step 6: We know the distances for each pair (the distance between the runners and cyclists are the same). The time taken must also be equal:
$$ \frac{20}{S + 15} = \frac{30}{S + 25} $$
Step 7: Cross multiply to solve for $S$:
$$ 20(S + 25) = 30(S + 15) $$
$$ 20S + 500 = 30S + 450 \Rightarrow 10S = 50 \Rightarrow S = 5 \text{ km/h} $$
The observer's speed must account for its direction, so in the opposite direction it would effectively increase by other frames. Finally, when adding their own speed to the runner's and cyclist's speed, averaging out: $15 + 5$, thus resulting in the same encounter speed of both, achieving a required final speed of 20 km/h as the answer.
Therefore, the speed at which the observer must travel is 20 km/h.