Given \(n \in \mathbb{N}\), let \(\sigma (n)\) denote the sum of the divisors of \(n\) and \(\phi (n)\) denote the number of integers \(n \geq m\) for which \(\gcd(m,n) = 1\). Show that for all \(n \in \mathbb{N}\), \[\large \frac{1}{\sigma (n)} + \frac{1}{\phi (n)} \geq \frac{2}{n}\] and determine when equality holds.

Find all positive real numbers \((a,b,c) \leq 1\) which satisfy \[ \huge \min \Bigg\{ \sqrt{\frac{ab+1}{abc}}\, \sqrt{\frac{bc+1}{abc}}, \sqrt{\frac{ac+1}{abc}} \Bigg \} = \sqrt{\frac{1-a}{a}} + \sqrt{\frac{1-b}{b}} + \sqrt{\frac{1-c}{c}}\]

Each of the numbers in the set \(A = \{1,2, \cdots, 2017\}\) is colored either red or white. Prove that for \(n \geq 18\), there exists a coloring of the numbers in \(A\) such that any of its n-term arithmetic sequences contains both colors.

Circles \(\mathcal{C}_1\) and \(\mathcal{C}_2\) with centers at \(C_1\) and \(C_2\) respectively, intersect at two points \(A\) and \(B\). Points \(P\) and \(Q\) are varying points on \(\mathcal{C}_1\) and \(\mathcal{C}_2\), respectively, such that \(P\), \(Q\) and \(B\) are collinear and \(B\) is always between \(P\) and \(Q\). Let lines \(PC_1\) and \(QC_2\) intersect at \(R\), let \(I\) be the incenter of \(\Delta PQR\), and let \(S\) be the circumcenter of \(\Delta PIQ\). Show that as \(P\) and \(Q\) vary, \(S\) traces the arc of a circle whose center is concyclic with \(A\), \(C_1\) and \(C_2\).