We propose and study a new perpetual maintenance problem: The Snow Plow Problem. Inspired by snow removal in urban environments, we consider the perpetual maintenance by n mobile agents of a fixed-length interval over which some resource accumulates continuously over time (i.e. snow). In order to maintain the interval, agents must traverse it, collecting accumulated resources by plowing over continuous segments, and then return to some predefined destination point on the interval to dump collected resources. In a sense, our problem can be described as a combination of the well-studied patrolling and delivery problems. The maintenance cost for an agent is some combination of the distance traveled and the volume of resources collected between visits to the destination. We first study the case where the maintenance cost is simply the maximum amount of snow each agent must carry at any point in time and provide an optimal O(n) algorithm for computing optimal trajectories for the mobile agents for scenarios where the destination is at an endpoint and snow falls uniformly across the interval. Then, we generalize the problem for any maintenance cost which can be expressed as the product of a positive, non-decreasing travel cost function f(x) (the cost to travel to a distance x) and a positive resource cost function g(x) such that ∫y^z g(x) dx represents the volume of snowfall in one unit of time in the sub-interval (y, z] ⊆ (0,1]. We provide a (1+ε)-approximation algorithm that runs in O(n log 1/ε) time and an exact O(n) algorithm for the case where f(x)=ax and g(x)=b for some positive constants a and b. Finally, we generalize further to consider scenarios where the destination is at any point along the interval, providing another (1+ε)-approximation algorithm which runs in O(n log 1/ε) time.