This problem is inspired by an IMO (don't know which year) and was the inspiration for the previous problem.
Anyway. Here goes:
Suppose t1,t2,…,tn are n≥3 positive real numbers such that
(t1+t2+⋯+tn)(1t1+1t2+⋯+1tn)<n2+1
Show that:
Problem 1) For any distinct i,j,k the numbers ti,tj,tk form the sides of some triangle. [This is the original IMO problem]
Problem 2) If n≥13, show that the triangles are all acute.
Problem 3) If n≤12, the triangles are not necessarily acute.
Anyway. Here goes:
Suppose t1,t2,…,tn are n≥3 positive real numbers such that
(t1+t2+⋯+tn)(1t1+1t2+⋯+1tn)<n2+1
Show that:
Problem 1) For any distinct i,j,k the numbers ti,tj,tk form the sides of some triangle. [This is the original IMO problem]
Problem 2) If n≥13, show that the triangles are all acute.
Problem 3) If n≤12, the triangles are not necessarily acute.
No comments:
Post a Comment