Prove that for every integer S≥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 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
Source: MMC 2014 Problem 3
Tags: algebra proposed, algebra
BSJL
13.06.2014 16:20
Notice that if the statement is true for S then it's true for S+3, too. And we can proof it's true for S=23,24,25!
test20
17.06.2014 13:18
Even simpler: Notice that if the statement is true for S≥20 then it's true for S+1, too. And we can proof it's true for S=20.
chaotic_iak
17.06.2014 19:39
test20 wrote: Notice that if the statement is true for S≥20 then it's true for S+1, too. I'm not convinced. Can you show P always exists? Not many pairs of triples of positive integers sum to the same result as well as multiply to the same result.
Konigsberg
07.07.2015 14:29
test20 wrote: Even simpler: Notice that if the statement is true for S≥20 then it's true for S+1, too. And we can proof it's true for S=20. I got 20: 1, 9, 10 and 2, 3, 15. But I don't get the induction!