OMM 2014

Each of the integers from 1 to 4027 has been colored either green or red. Changing the color of a number is making it red if it was green and making it green if it was red. Two positive integers $m$ and $n$ are said to be cuates if either $\frac{m}{n}$ or $\frac{n}{m}$ is a prime number. A step consists in choosing two numbers that are cuates and changing the color of each of them. Show it is possible to apply a sequence of steps such that every integer from 1 to 2014 is green.

A positive integer $a$ is said to reduce to a positive integer $b$ if when dividing $a$ by its units digits the result is $b$ . For example, 2015 reduces to $\frac{2015}{5} = 403$ . Find all the positive integers that become 1 after some amount of reductions. For example, 12 is one such number because 12 reduces to 6 and 6 reduces to 1.

Let $\Gamma_1$ be a circle and $P$ a point outside of $\Gamma_1$ . The tangents from $P$ to $\Gamma_1$ touch the circle at $A$ and $B$ . Let $M$ be the midpoint of $PA$ and $\Gamma_2$ the circle through $P$ , $A$ and $B$ . Line $BM$ cuts $\Gamma_2$ at $C$ , line $CA$ cuts $\Gamma_1$ at $D$ , segment $DB$ cuts $\Gamma_2$ at $E$ and line $PE$ cuts $\Gamma_1$ at $F$ , with $E$ in segment $PF$ . Prove lines $AF$ , $BP$ , and $CE$ are concurrent.

Problem 4

Let $ABCD$ be a rectangle with diagonals $AC$ and $BD$ . Let $E$ be the intersection of the bisector of $\angle CAD$ with segment $CD$ , $F$ on $CD$ such that $E$ is midpoint of $DF$ , and $G$ on $BC$ such that $BG = AC$ (with $C$ between $B$ and $G$ ). Prove that the circumference through $D$ , $F$ and $G$ is tangent to $BG$ .

Let $a, b, c$ be positive reals such that $a + b + c = 3$ . Prove:

$$\frac{a^2}{a + \sqrt[3]{bc}} + \frac{b^2}{b + \sqrt[3]{ca}} + \frac{c^2}{c + \sqrt[3]{ab}} \geq \frac{3}{2}$$

And determine when equality holds.

Let $d(n)$ be the number of positive divisors of a positive integer $n$ (including $1$ and $n$ ). Find all values of $n$ such that $n + d(n) = d(n)^2$ .