Found problems: 33
If $a,b,c$ are positive real numbers such that $a+b+c=abc$ prove that
$$\frac{abc}{3\sqrt{2}}\left ( \sum_{cyc}\frac{\sqrt{a^3+b^3}}{ab+1} \right )\geq \sum_{cyc}\frac{a}{a^2+1}$$
Let $a$, $b$ and $c$ be positive real numbers with $abc = 1$. Determine the minimum possible value of
$$ \left(\frac{a}{b} + \frac{b}{c} + \frac{c}{a}\right) \cdot \left(\frac{ab}{a+b} + \frac{bc}{b+c} + \frac{ca}{c+a}\right) $$
as well as all triples $(a,b,c)$ which attain the minimum.
With positive $ a,b,c, $ prove:
$$ \frac{a}{8a^2+5b^2+3c^2} +\frac{b}{8b^2+5c^2+3a^2} +\frac{c}{8c^2+5a^2+3b^2}\le\frac{1}{16}\left( \frac{1}{a} +\frac{1}{b} +\frac{1}{c} \right) $$
[i]Titu Zvonaru[/i]
Let $a,b$ and $c$ be positive real numbers satisfying $abc=1.$ Prove that$$\frac{a^2-b^2}{a+bc}+\frac{b^2-c^2}{b+ca}+\frac{c^2-a^2}{c+ab}\leq a+b+c-3.$$
$ \sum_{cyc}\frac{1}{\left(\text{tg} y+\text{tg} z\right) \text{cos}^2 x} \ge 3, $ for any $ x,y,z\in (0,\pi/2) $
[i]Carmen[/i] and [i]Viorel Botea[/i]
Let $a, b, c$ be positive real numbers such that $abc = 1$. Prove that
$$(a + b + c)(ab + bc + ca) + 3\ge 4(a + b + c).$$
Prove that for any natural number $ n, $ there exists a number having $ n+1 $ decimal digits, namely, $ k_0,k_1,k_2,\ldots ,k_n $, and a $ \text{(n+1)-tuple}, $ say $\left( \epsilon_0 ,\epsilon_1 ,\epsilon_2\ldots ,\epsilon_n \right)\in\{-1,1\}^{n+1} , $ that satisfies:
$$ 1\le \prod_{j=0}^n (2+j)^{k_j\cdot \epsilon_j}\le \sqrt[10^n-1]{2} $$
[i]Sorin Rădulescu[/i] and [i]Ion Savu[/i]
Let $0 \le a_1 \le a_2 \le \dots \le a_n \le 1$ be $n$ real numbers with $n \ge 2$. Assume $a_1 + a_2 + \dots + a_n \ge n-1$. Prove that
\[ a_2a_3\dots a_n \ge \left( 1 - \frac 1n \right)^{n-1} \]