Competition problem — Number theory

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Number theory Difficulty 5.3 AIME, harder Prove it Spain

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!"

Solution

Write $S$ in the form $S = 6k + r$ for integers $k$ and $r$ with $1 \le r \le 6$ , and note that $k > 2r$ . We claim that the number
$P = 6k(k - r)(k + r)$
is an appropriate choice.
Denote the prices of table, cabinet and shelf by $x$ , $y$ and $z$ , respectively. Then $x = 3(k+r)$ , $y = 2(k-r)$ , $z = k$ is one possibility, and $x' = 3k$ , $y' = 2(k+r)$ , $z' = k-r$ is another possibility. One easily verifies $x > y > z$ and $x' > y' > z'$ , and $x' < x$ implies that these two possibilities are indeed distinct. □

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.