The pilot reported that he was in cruise flight at 5,500 ft in his amphibious airplane with the landing gear down over a lake on the last leg of an 8-day cross-country trip to his home airport, when the engine “sputtered” several times and then stopped operating. The pilot reported that at the same time the engine lost power, a “burnt” smell also entered the cockpit. The pilot conducted a forced landing to the lake. During the landing flare the landing gear caught in the water and the airplane nosed forward into the lake. The airplane came to a stop on the surface upright but began to fill with water. The pilot was rescued before the airplane sank. The pilot had recently purchased the airplane and was flying it back to his home airport. The pilot reported several issues with the airplane that resulted in several mishaps during the 8-day trip. During the forced landing, the pilot was unable to put down flaps, or retract the landing gear, because the airplane had no hydraulic pressure. In a conversation with the FAA, and because of a previous issue with the landing gear, the pilot agreed to fly with the landing gear down to ensure a good landing at his destination. The airplane was not recovered; therefore, the engine could not be examined, and the reason for the loss of engine power could not be determined. Additionally, based on the pilot’s flight, it’s likely the airplane was not airworthy before the pilot’s initial departure.