Which Idaho city lies within 50 miles of the borders of both Montana and Washington state?

Coeur d'Alene is the correct answer because it is strategically located in the northern part of Idaho, making it approximately 30 miles away from the Canada border and notably close to both the Washington border to the west and the Montana border to the east. It lies within the 50-mile radius of both states due to its position near the junction of these borders, with direct access to key highways that connect Idaho with Washington and Montana.

Other cities mentioned do not share this geographical advantage. Boise, located in the west-central region of Idaho, is significantly further from the borders of both states. Pocatello is situated further south and has a greater distance to travel to either border. Similarly, Idaho Falls is located in the southeast section of the state and does not fall within this 50-mile range from both borders. Thus, Coeur d'Alene stands out as the only city that meets the criteria set out in the question.