In a triangle $ABC$ with $AB<AC$, $D$ is a point on segment $AC$ such that $BD = CD$. A line parallel to $BD$ meets segment $BC$ at $E$ and line $AB$ at $F$. Point $G$ is the intersection of $AE$ and $BD$. Show that $\angle BCG = \angle BCF$. (Côte d’Ivoire)

Find all positive integers $m$ and $n$ with no common divisor greater than 1 such that $m^3 + n^3$ divides $m^2 + 20mn + n^2$. (Professor Yongjin Song)

Consider a sequence of real numbers defined by: \begin{align*} x_{1} & = c \\ x_{n+1} & = cx_{n} + \sqrt{c^{2} - 1}\sqrt{x_{n}^{2} - 1} \quad \text{for all } n \geq 1. \end{align*}Show that if $c$ is a positive integer, then $x_{n}$ is an integer for all $n \geq 1$. (South Africa)

Manzi has $n$ stamps and an album with $10$ pages. He distributes the $n$ stamps in the album such that each page has a distinct number of stamps. He finds that, no matter how he does this, there is always a set of $4$ pages such that the total number of stamps in these $4$ pages is at least $\frac{n}{2}$. Determine the maximum possible value of $n$.

Let $a, b$ be reals with $a \neq 0$ and let $$P(x)=ax^4-4ax^3+(5a+b)x^2-4bx+b.$$Show that all roots of $P(x)$ are real and positive if and only if $a=b$.

Let $ABC$ be an acute triangle with $AB<AC$. Let $D, E,$ and $F$ be the feet of the perpendiculars from $A, B,$ and $C$ to the opposite sides, respectively. Let $P$ be the foot of the perpendicular from $F$ to line $DE$. Line $FP$ and the circumcircle of triangle $BDF$ meet again at $Q$. Show that $\angle PBQ = \angle PAD$.