Problem:
A set of positive numbers has the triangle property if it has three distinct elements that are the lengths of the sides of a triangle whose area is positive. Consider sets {4,5,6,…nn} of consecutive positive integers, all of whose ten-element subsets have the triangle property. What is the largest possible value of n?
Solution:
The set {4,5,9,14,23,37,60,97,157,254} is a ten-element subset of {4,5,6,…,254} that does not have the triangle property. Let N be the smallest integer for which {4,5,6,…,N} has a ten-element subset that lacks the triangle property. Let {a1,a2,a3,…,a10} be such a subset, with a1<a2<a3<⋯<a10. Because none of its three-element subsets define triangles, the following must be true:
N≥a10≥a9+a8≥(a8+a7)+a8=2a8+a7≥2(a7+a6)+a7=3a7+2a6≥3(a6+a5)+2a6=5a6+3a5≥8a5+5a4≥13a4+8a3≥21a3+13a2≥34a2+21a1≥34⋅5+21⋅4=254
Thus the largest possible value of n is N−1=253.
The problems on this page are the property of the MAA's American Mathematics Competitions