RIOPLATENSE 2018

Determine if there are $2018$ different positive integers such that the sum of their squares is a perfect cube and the sum of their cubes is a perfect square.

Let $P$ be a point outside a circumference $\Gamma$ , and let $PA$ be one of the tangents from $P$ to $\Gamma$ . Line $l$ passes through $P$ and intersects $\Gamma$ at $B$ and $C$ , with $B$ between $P$ and $C$ . Let $D$ be the symmetric of $B$ with respect to $P$ . Let $\omega_1$ and $\omega_2$ be the circles circumscribed to the triangles $DAC$ and $PAB$ respectively. $\omega_1$ and $\omega _2$ intersect at $E \neq A$ . Line $EB$ cuts back to $\omega _1$ in $F$ . Prove that $CF = AB$ .

Determine all the triples $\{a, b, c \}$ of positive integers coprime (not necessarily pairwise prime) such that $a + b + c$ simultaneously divides the three numbers $a^{12} + b^{12}+ c^{12}$ , $a^{23} + b^{23} + c^{23}$ and $a^{11004} + b^{11004} + c^{11004}$

Let $ABC$ be an acute triangle with $AC> AB$ . be $\Gamma$ the circumcircle circumscribed to the triangle $ABC$ and $D$ the midpoint of the smallest arc $BC$ of this circle. Let $E$ and $F$ points of the segments $AB$ and $AC$ respectively such that $AE = AF$ . Let $P \neq A$ be the second intersection point of the circumcircle circumscribed to $AEF$ with $\Gamma$ . Let $G$ and $H$ be the intersections of lines $PE$ and $PF$ with $\Gamma$ other than $P$ , respectively. Let $J$ and $K$ be the intersection points of lines $DG$ and $DH$ with lines $AB$ and $AC$ respectively. Show that the $JK$ line passes through the midpoint of $BC$

Let $n$ be a positive integer. Find all $n$ - rows $( a_1 , a_2 ,..., a_n )$ of different positive integers such that $$\frac{(a_1 + d ) (a_2 + d ) \cdot\cdot\cdot ( a_n + d )}{a_1a_2\cdot \cdot \cdot a_n }$$ is integer for every integer $d\ge 0$

A company has $n$ employees. It is known that each of the employees works at least one of the $7$ days of the week, with the exception of an employee who does not work any of the $7$ days. Furthermore, for any two of these $n$ employees, there are at least $3$ days of the week in which one of the two works that day and the other does not (it is not necessarily the same employee who works those days). Determine the highest possible value of $n$ .