GR-HMS 2017

An acute triangle $ABC$ with $AB<AC<BC$ is inscribed in a circle $c(O,R)$ . The circle $c_1(A,AC)$ intersects the circle $c$ at point $D$ and intersects $CB$ at $E$ . If the line $AE$ intersects $c$ at $F$ and $G$ lies in $BC$ such that $EB=BG$ , prove that $F,E,D,G$ are concyclic.

Let $A$ be a point in the plane and $3$ lines which pass through this point divide the plane in $6$ regions. In each region there are $5$ points. We know that no three of the $30$ points existing in these regions are collinear. Prove that there exist at least $1000$ triangles whose vertices are points of those regions such that $A$ lies either in the interior or on the side of the triangle.

Find all integer triples $(a,b,c)$ with $a>0>b>c$ whose sum equal $0$ such that the number $$N=2017-a^3b-b^3c-c^3a$$ is a perfect square of an integer.

Let $u$ be the positive root of the equation $x^2+x-4=0$ . The polynomial $$P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_0$$ where $n$ is positive integer has non-negative integer coefficients and $P(u)=2017$ .

1. Prove that $a_0+a_1+...+a_n\equiv 1\mod 2$ .
2. Find the minimum possible value of $a_0+a_1+...+a_n$ .