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Friday, March 17, 2017

More about sides of a triangle

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 n3 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 n13, show that the triangles are all acute.

Problem 3) If n12, the triangles are not necessarily acute.

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