Question 100
Prove that for every integer $S \ge 100$ there exists an integer $P$ for which the following story could hold true: The mathematician asks the shop owner: "How much are the table, the cabinet and the bookshelf?" The shop owner replies: "Each item costs a (positive) integer amount of Euros. The table is more expensive than the cabinet, and the cabinet is more expensive than the bookshelf. The sum of the three prices is $S$ and the product is $P$." The mathematician thinks and complains: "This is not enough information to determine the three prices!"