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
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 $S\ge20$ 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 $S\ge20$ 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 $S\ge20$ 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!