Found problems: 30
2015 Iran Team Selection Test, 6
If $a,b,c$ are positive real numbers such that $a+b+c=abc$ prove that
$$\frac{abc}{3\sqrt{2}}\left ( \sum_{cyc}\frac{\sqrt{a^3+b^3}}{ab+1} \right )\geq \sum_{cyc}\frac{a}{a^2+1}$$
2021 Cyprus JBMO TST, 1
Let $x,y,z$ be positive real numbers such that $x^2+y^2+z^2=3$. Prove that
\[ xyz(x+y+z)+2021\geqslant 2024xyz\]
2007 Nicolae Păun, 4
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]
2022 Bundeswettbewerb Mathematik, 1
Find all quadrupels $(a, b, c, d)$ of positive real numbers that satisfy the following two equations:
\begin{align*}
ab + cd &= 8,\\
abcd &= 8 + a + b + c + d.
\end{align*}
2002 Canada National Olympiad, 3
Prove that for all positive real numbers $a$, $b$, and $c$,
\[ \frac{a^3}{bc} + \frac{b^3}{ca} + \frac{c^3}{ab} \geq a+b+c \]
and determine when equality occurs.