I nearly ran over a mountain lion (we call them cougars here in the NW) once while riding down from Hurricane Ridge in Olympic Nat Pk, on my road bike, as it is a paved road. Fortunately he had his eye on other prey. On another ride, a dirt-road tour in the wilds of Idaho in 1983, a black bear ran parallel to us but about 50 feet away down one road for about a mile, near the town of Yellow Pine. I think he was just racing us.
Bob Freeman
Seattle