### Questions (6)

 3 Complex numbers problem with absolute value property 2 Show that $\sum_{\text{cyc}} \frac{1}{b^2+c^2+5bc-a^2} \leq \frac{\sqrt3}{8S}$ for a triangle with sides $a$, $b$, $c$ and area $S$ 2 Prove that for some real numbers $a$ and $b$: $(a^2+a+1)^{-1/2}+(b^2+b+1)^{-1/2}+4((a+b)^2-2(a+b)+4)^{-1/2} \leq 4$ 1 Prove that $\sqrt{1-x^2} + \sqrt{1-y^2} + \sqrt{1-z^2}\geq 4\sqrt{\frac{3-(x^2+y^2+z^2)}{5+x^2+y^2+z^2}}$. 0 Complex function property

### Reputation (60)

 +15 Complex numbers problem with absolute value property -2 Inequality for contests +10 Show that $\sum_{\text{cyc}} \frac{1}{b^2+c^2+5bc-a^2} \leq \frac{\sqrt3}{8S}$ for a triangle with sides $a$, $b$, $c$ and area $S$ +5 Prove that for some real numbers $a$ and $b$: $(a^2+a+1)^{-1/2}+(b^2+b+1)^{-1/2}+4((a+b)^2-2(a+b)+4)^{-1/2} \leq 4$

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### Tags (13)

 0 inequality × 4 0 radicals 0 contest-math × 3 0 a.m.-g.m.-inequality 0 substitution × 2 0 summation 0 jensen-inequality 0 complex-analysis 0 geometric-inequalities 0 absolute-value

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