0.1.1. 2004 USAMO Problem 5 (\(\bullet \circ \circ \))
Show that the following holds for \(a, b, c > 0\). \[ (a^5 - a^2 + 3)(b^5 - b^2 + 3)(c^5 - c^2 + 3) \geq (a + b + c)^3 \]

0.1.2. 2001 IMO Shortlist A3 (\(\bullet \circ \circ \))
Show that the following inequality holds for arbitrary reals \(x_1, x_2, \cdots , x_n\). \[ \frac {x_1}{1 + x_1^2} + \frac {x_2}{1 + x_1^2 + x_2^2} + \cdots + \frac {x_n}{1+ x_1^2 + \cdots + x_n^2} < \sqrt {n} \]

0.1.3. 2025 KMO Problem 6 (\(\bullet \bullet \circ \))
For arbitrary reals \(x_1, x_2, \dots , x_{99}\) that satisfy \(0 < x_1 < x_2 < \cdots < x_{99}\), determine the minimum value of \(c > 0\) such that the following inequality always holds. \begin{align*} &3\sqrt {x_1} + 4 \left ( \sqrt {x_2 - x_1} + \sqrt {x_3 - x_2} + \cdots + \sqrt {x_{99} - x_{98}} \right ) \\ &\qquad \qquad \qquad \qquad \qquad \leq c \left ( \sqrt {x_2} + \sqrt {x_3} + \cdots + \sqrt {x_{98}} \right ) + 5\sqrt {x_{99}} \end{align*}

0.1.4. 2023 IMO Problem 4 (\(\bullet \bullet \circ \))
Let \(x_1, x_2, \cdots , x_{2023} > 0\) be pairwise different numbers such that \(a_n\) defined as the following is an integer for all \(n = 1, 2, \cdots , 2023\). \[ a_n = \sqrt { \left ( x_1 + x_2 + \cdots + x_n \right ) \left ( \frac {1}{x_1} + \frac {1}{x_2} + \cdots + \frac {1}{x_n} \right ) } \] Show that \(a_{2023} \geq 3034\).