Consider the sequence of integers $ \left( a_n\right)_{n\ge 0} $ defined as $$ a_n=\left\{\begin{matrix}n^6-2017, & 7|n\\ \frac{1}{7}\left( n^6-2017\right) , & 7\not | n\end{matrix}\right. . $$Determine the largest length a string of consecutive terms from this sequence sharing a common divisor greater than $ 1 $ may have.

Let $ ABC $ be a triangle, let $ O $ and $ \gamma $ be its circumcentre and circumcircle, respectively, and let $ P $ and $ Q $ be distinct points in the interior of $ \gamma $ such that $ O, P $ and $ Q $ are not collinear. Reflect $ O $ in the midpoint of the segment $ PQ $ to obtain $R,$ then reflect $R$ in the centre of the nine-point circle of the triangle $ABC$ to obtain $S.$ The circle through $P$ and $Q$ and orthogonal to $ \gamma , $ crosses the rays $OP$ and $OQ,$ emanating from $O,$ again at $P'$ and $Q'$ respectively. Let the lines $PQ'$ and $QP'$ cross at $T.$ Prove that, if $P$ and $Q$ are isogonally conjugate with respect to the triangle $ABC,$ then so are $S$ and $T.$ E.D. Camier

Let $$ 2^{-n_1}+2^{-n_2}+2^{-n_3}+\cdots,\quad1\le n_1\le n_2\le n_3\le\cdots $$be the binary representation of the golden ratio minus one. Prove that $ n_k\le 2^{k-1}-2, $ for all integers $ k\ge 4. $ American Mathematical Monthly

Let be distinct points on a plane, four of which form a quadrangle, and three of which are in the interior or boundary of this quadrangle. Show that the diagonals of this quadrangle are longer than the double of the minimum of the distances between any two of these seven points. Paul Erdős Side noteIf the quadrangle is convex, the constant from the inequality can be improved from $ 2 $ to $ \sqrt{\frac{3\pi}{2}}. $

How many natural numbers smaller than $ 2017 $ can be uniquely (order of summands are not relevant) written as a sum of three powers of $ 2? $ Andrei Eckstein

Let $ x,y,z $ be three positive real numbers such that $ x^2+y^2+z^2+3=2(xy+yz+zx) . $ Show that $$ \sqrt{xy}+\sqrt{yz}+\sqrt{zx}\ge 3, $$and determine in which circumstances equality happens. Vlad Robu

A certain frog that was placed on a vertex of a convex polygon chose to jump to another vertex, either clockwise skipping one vertex, either counterclockwise skipping two vertexes, and repeated the procedure. If the number of jumps that the frog made is equal to the number of sides of the polygon, the frog has passed through all its vertexes and ended up on the initial vertex, what´s the set formed by all the possible values that this number can take? Andrei Eckstein

Let $ ABC $ be an acute triangle having $ AB<AC, $ let $ M $ be the midpoint of the segment $ BC, D$ be a point on the segment $ AM, E $ be a point on the segment $ BD $ and $ F $ on the line $ AB $ such that $ EF $ is parallel to $ BC, $ and such that $ AE $ and $ DF $ pass through the orthocenter of $ ABC. $ Prove that the interior bisectors of $ \angle BAC $ and $ \angle BDC, $ together with $ BC $ are concurrent. Vlad Robu