In a triangle $ABC$, the external bisector of $\angle BAC$ intersects the ray $BC$ at $D$. The feet of the perpendiculars from $B$ and $C$ to line $AD$ are $E$ and $F$, respectively and the foot of the perpendicular from $D$ to $AC$ is $G$. Show that $\angle DGE + \angle DGF = 180^{\circ}$.

$3m$ balls numbered $1, 1, 1, 2, 2, 2, 3, 3, 3, \ldots, m, m, m$ are distributed into $8$ boxes so that any two boxes contain identical balls. Find the minimal possible value of $m$.

Find all pairs $(m, n)$ of positive integers satsifying $m^6+5n^2=m+n^3$.

Determine the smallest value of $(a+5)^2+(b-2)^2+(c-9)^2$ for all real numbers $a, b, c$ satisfying $a^2+b^2+c^2-ab-bc-ca=3$.

Find all real values of $a$ for which the equation $x(x+1)^3=(2x+a)(x+a+1)$ has four distinct real roots.

Find all triples of positive integers $(a, b, c)$ satisfying $(a^3+b)(b^3+a)=2^c$.

Let a line $\ell$ intersect the line $AB$ at $F$, the sides $AC$ and $BC$ of a triangle $ABC$ at $D$ and $E$, respectively and the internal bisector of the angle $BAC$ at $P$. Suppose that $F$ is at the opposite side of $A$ with respect to the line $BC$, $CD = CE$ and $P$ is in the interior the triangle $ABC$. Prove that \[FB \cdot FA+CP^2 = CF^2 \iff AD \cdot BE = PD^2.\]

Alice and Bob play a game on a complete graph $G$ with $2014$ vertices. They take moves in turn with Alice beginning. At each move Alice directs one undirected edge of $G$. At each move Bob chooses a positive integer number $m,$ $1 \le m \le 1000$ and after that directs $m$ undirected edges of $G$. The game ends when all edges are directed. If there is some directed cycle in $G$ Alice wins. Determine whether Alice has a winning strategy.