MathDB

Prove that for every integer

S\ge100

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)

Problem Statement