A coloring of the set of integers greater than or equal to $1$, must be done according to the following rule: Each number is colored blue or red, so that the sum of any two numbers (not necessarily different) of the same color is blue. Determine all the possible colorings of the set of integers greater than or equal to $1$ that follow this rule.

Octavio writes an integer $n \geq 1$ on a blackboard and then he starts a process in which, at each step he erases the integer $k$ written on the blackboard and replaces it with one of the following numbers: $$3k-1, \quad 2k+1, \quad \frac{k}{2}.$$provided that the result is an integer. Show that for any integer $n \geq 1$, Octavio can write on the blackboard the number $3^{2023}$ after a finite number of steps.

Let $a,\ b$ and $c$ be positive real numbers such that $a b+b c+c a=1$. Show that $$ \frac{a^3}{a^2+3 b^2+3 a b+2 b c}+\frac{b^3}{b^2+3 c^2+3 b c+2 c a}+\frac{c^3}{c^2+3 a^2+3 c a+2 a b}>\frac{1}{6\left(a^2+b^2+c^2\right)^2} . $$

A four-digit number $n=\overline{a b c d}$, where $a, b, c$ and $d$ are digits, with $a \neq 0$, is said to be guanaco if the product $\overline{a b} \times \overline{c d}$ is a positive divisor of $n$. Find all guanaco numbers.

Let $ABC$ be an acute-angled triangle with $AB < AC$ and $\Gamma$ the circumference that passes through $A,\ B$ and $C$. Let $D$ be the point diametrically opposite $A$ on $\Gamma$ and $\ell$ the tangent through $D$ to $\Gamma$. Let $P, Q$ and $R$ be the intersection points of $B C$ with $\ell$, of $A P$ with $\Gamma$ such that $Q \neq A$ and of $Q D$ with the $A$-altitude of the triangle $ABC$, respectively. Define $S$ to be the intersection of $AB$ with $\ell$ and $T$ to be the intersection of $A C$ with $\ell$. Show that $S$ and $T$ lie on the circumference that passes through $A, Q$ and $R$.

In a pond there are $n \geq 3$ stones arranged in a circle. A princess wants to label the stones with the numbers $1, 2, \dots, n$ in some order and then place some toads on the stones. Once all the toads are located, they start jumping clockwise, according to the following rule: when a toad reaches the stone labeled with the number $k$, it waits for $k$ minutes and then jumps to the adjacent stone. What is the greatest number of toads for which the princess can label the stones and place the toads in such a way that at no time are two toads occupying a stone at the same time? Note: A stone is considered occupied by two toads at the same time only if there are two toads that are on the stone for at least one minute.