Problem

Source: MMC 2014 Problem 3

Tags: algebra proposed, algebra



Prove that for every integer S100 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 their product is P.'' The mathematician thinks and complains: This is not enough information to determine the three prices!'' (Proposed by Gerhard Woeginger, Austria)