PMO 2017

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 $C_2$.