An ancient Inca legend tells that a monster lives among the mountains that when wakes up, eats everyone who read this issue. After such a task, the monster returns to the mountains and sleeps for a number of years equal to the sum of its digits of the year in which you last woke up. The monster woke up for the first time in the year $234$. a) Would the monster have woken up between the years $2005$ and $2015$? b) Will we be safe in the next $10$ years?
2017 Ecuador Juniors
Day 1
Find all pairs of real numbers $x, y$ that satisfy the following system of equations $$\begin{cases} x^2 + 3y = 10 \\ 3 + y = \frac{10}{ x} \end{cases}$$
Given an isosceles triangle $ABC$ with $AB = AC$. Let $O$ be the circumcenter of $ABC$, $D$ the midpoint of $AB$ and $E$ the centroid of $ACD$. Prove that $CD \perp EO$.
Day 2
Indicate whether it is possible to write the integers $1, 2, 3, 4, 5, 6, 7, 8$ at the vertices of an regular octagon such that the sum of the numbers of any $3$ consecutive vertices is greater than: a) $11$. b) $13$.
Two positive integers are coprime if their greatest common divisor is $1$. Let $C$ be the set of all divisors of the number $8775$ that are greater than $ 1$. A set of $k$ consecutive positive integers satisfies that each of them is coprime with some element of $C$. Determine the largest possible value of $K$.
Find all primes $p$ such that $p^2- p + 1$ is a perfect cube.