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Cartesian Product

3.If a, b, c>...

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3.If a, b, c>0 , then prove that a/b + b/c + c/a > ab+bc+ca

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\frac{a}{b + c} = \frac{b}{c + a} = \frac{c}{a + b}

a (b - c) + b (c - a) + c (a - b) = 0

∣ ∣ ∣ ∣ ∣ \begin{matrix} b c - a^{2} & c a - b^{2} & a b - c^{2} - b c + c a + a b & b c - c a + a b & b c + c a - a b (a + b) (a + c) & (b + c) (b + a) & (c + a) (c + b) \end{matrix} ∣ ∣ ∣ ∣ ∣ = 3. (b - c) (c - a) (a - b) (a + b + c) (a b + b c + c a)

a > b > c > 0

then prove that

{cot}^{- 1} (\frac{a b + 1}{a - b}) + {cot}^{- 1} (\frac{b c + 1}{b - c}) + {cot}^{- 1} (\frac{c a + 1}{c - a}) = π

\frac{{log}_{e} a}{b - c} = \frac{{log}_{e} b}{c - a} = \frac{{log}_{e} c}{a - b}

, then prove that

a^{b + c} + b^{c + a} + c^{a + b} \geq 3

.

a + b + c = 0

, prove that identities

(b^{2} c + c^{2} a + a^{2} b - 3 a b c) (b c^{2} + c a^{2} + a b^{2} - 3 a b c) = (b c + c a + a b)^{3} + 27 a^{2} b^{2} c^{2}

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