12

Part of 2010 AIME Problems

Problems(2)

Partition with Product

Source: AIME 2010 Problem 12

M \ge 3

be an integer and let S \equal{} \{3,4,5,\ldots,m\}. Find the smallest value of

m

such that for every partition of

S

into two subsets, at least one of the subsets contains integers

a

b

c

(not necessarily distinct) such that ab \equal{} c.
Note: a partition of

S

is a pair of sets

A

B

such that A \cap B \equal{} \emptyset, A \cup B \equal{} S.

Related Isosceles Triangles

Source: AIME 2010II Problem 12

Two noncongruent integer-sided isosceles triangles have the same perimeter and the same area. The ratio of the lengths of the bases of the two triangles is

8: 7

. Find the minimum possible value of their common perimeter.

geometryperimeterratioAMCAIME