In an acute triangle $ABC$, $AC>AB$, $D$ is the point on $BC$ such that $AD=AB$. Let $\omega_1$ be the circle through $C$ tangent to $AD$ at $D$, and $\omega_2$ the circle through $C$ tangent to $AB$ at $B$. Let $F\ (\ne C)$ be the second intersection of $\omega_1$ and $\omega_2$. Prove that $F$ lies on $AC$.
Find the smallest positive integer $n$ for which there exist integers $x_1\lt x_2\lt \cdots\lt x_n$ such that every integer from $1000$ to $2000$ can be written as a sum of some of the integers from $x_1,x_2,\ldots,x_n$ without repetition.
Suppose $p$ is a prime number and $x,y,z$ are integers satisfying $0\lt x\lt y\lt z\lt p$. If $x^3,y^3,z^3$ have equal remainders when divided by $p$, prove that $x^2+y^2+z^2$ is divisible by $x+y+z$.
Let $a_1,a_2,\ldots$ be a sequence of positive numbers satisfying, for any positive integers $k,l,m,n$ such that $k+n=m+l$,
$$\frac{a_k+a_n}{1+a_ka_n}=\frac{a_m+a_l}{1+a_ma_l}.$$
Show that there exist positive numbers $b,c$ so that $b\le a_n\le c$ for any positive integer $n$.