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Ques: 16 Let
ABC be a triangle. Prove that
Question 2, we have
The arithmeticâ€"geometric means inequality yields
Combining the last two equalities gives part (a).
Part (b) then follows from (a) and Question 15. Part (c ) then follows from part (b) by noting that Finally, by (c ) and by the arithmeticâ€" geometric means inequality, we have
Again by Question 2, we have
and analogous formulas for and . Then part (e) follows routinely from the arithmeticâ€"geometric means inequality.
Note: We present another approach to part (a). Note that are all positive. Let It suffices to show that By the arithmeticâ€"geometric means inequality, we have
By Question 15, we have Thus,
Consequently, establishing (a).
Ques: 17 In
triangle ABC, show that
Conversely, if x, y, z are positive real numbers such that show that there is an acute triangle ABC such that
Parts (c.) and (d) follow immediately from (b)
because Thus we show only (a) and
(a) Applying the sum-to-product formulas and the fact that we find that
(b) By the sum-to-product formulas, we have
Note that It suffices to show that
or which is evident by the sum-to-product formula
From the given equality, we have and thus we may set where Because is an increasing function of z, there is at most one non-negative value c such that the given equality holds. We know that one solution to this equality is where Because we know that Because we have implying that Thus, and Therefore, we must have as desired.
Nevertheless, we present a cool proof of part (d). Consider the system of equations
Using the addition and subtraction formulas, one can easily see that is a nontrivial solution. Hence the determinant of the system is 0; that is,