Let $ABCD$ be a tetrahedron such that the sums of the three face angles at $A$ and at $B$ are both $180^\circ$. The two face angles $\angle ACD$ and $\angle BCD$ at $C$ have sum $180^\circ$. Determine the sum of the areas of its four faces in terms of $k=AC+CB$ and $\angle ACB=x$.
For every positive integer $n$, let $f(n)$ be the number of positive divisors of $n$ congruent to $\pm1$ modulo $10$, and let $g(n)$ be the number of positive divisors of $n$ congruent to $\pm3$ modulo $10$. Prove that $f(n)\ge g(n)$.
Let $a,b,c$ be positive numbers. Define sequences $\{a_n\},\{b_n\},\{c_n\}$ by $a_0=a$, $b_0=b$, $c_0=c$, and
$$a_{n+1}=a_n+\frac2{b_n+c_n},\quad b_{n+1}=\frac2{c_n+a_n},\quad c_{n+1}=c_n+\frac2{a_n+b_n}$$
for every $n\in\mathbb Z_+$. Prove that $a_n$ tends to infinity as $n$ tends to infinity.
A $1991\times1992$ rectangle is divided into unit squares, indexed by $(m,n)$ with $1\le m\le1991$, $1\le n\le1992$ (row $m$, column $n$). Initially the squares $(m,n)$, $(m+1,n+1)$ and $(m+2,n+2)$ are colored, where $m\lt 1989$, $n\lt 1991$. Each subsequent move colors three uncolored squares lying in one row or one column. Can all the squares of the rectangle be colored in this way?
Let $ABCD$ be a rectangle with center $O$ and $\angle AOB\le45^\circ$. Rotate it about $O$ through an angle $x$ with $0^\circ\lt x\lt 360^\circ$. Determine the values of $x$ for which the intersection of the rotated rectangle and the original rectangle has the smallest possible area.