Between 0 to 9
Let be sides of a (non-degenerate) triangle satisfying . Let be the set of all integers that can be the value of . What is the arithmetic mean of the elements in ?
The answer is 5.
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1 solution
½{(a-b)²+(b-c)²+(c-a)²}≥0
a²+b²+c²≥ab+ bc+ ca
ab+ bc + ca≤6
a²>b²+c²-2bc cosA > b²+c²-2bc(since cosA <1 )
b²>a²+c²-2ac and c²>a²+b²-2ab
a²+b²+c²>2(a²+b²+c²-ab-bc-ca)
ab+bc+ca> ½(a²+b²+c²) =3
hence
ab+bc+ca belongs to (3,6]