Define the sequence $\{x_n\}$ by $x_1=a\ge1$ and
$$x_{n+1}=1+\ln\frac{x_n(x_n^2+3)}{1+3x_n^2}.$$
Prove that the sequence converges and determine its limit.
Let $O$ be the center of the circumsphere of tetrahedron $ABCD$. Let $A',B',C',D'$ lie on that sphere so that $AA',BB',CC',DD'$ are diameters. Let $A''$ be the centroid of triangle $BCD$, and define $B'',C'',D''$ analogously. Prove that the lines $A'A'',B'B'',C'C'',D'D''$ are concurrent. Suppose they meet at $X$. Prove that the line through $X$ and the midpoint of $AB$ is perpendicular to $CD$.
The sequence $a_0,a_1,a_2,\ldots$ is defined by $a_0=20$, $a_1=100$, and
$$a_{n+2}=4a_{n+1}+5a_n+20.$$
Find the smallest positive integer $m$ such that $a_m-a_0,a_{m+1}-a_1,a_{m+2}-a_2,\ldots$ are all divisible by $1998$.
Does there exist an infinite sequence of real numbers $\{x_n\}_{n=1}^{+\infty}$ such that $|x_n|\le0.666$ and
$$|x_m-x_n|\ge\frac1{n^2+n+m^2+m}$$
for all distinct positive integers $m,n$?
Find all positive integers $n$ for which there exists a polynomial $p(x)$ with real coefficients satisfying
$$p\left(x^{1998}+\frac1{x^{1998}}\right)=x^n+\frac1{x^n}.$$