Given
$$S=1+\frac1{1+\frac13}+\frac1{1+\frac13+\frac16}+\cdots+\frac1{1+\frac13+\frac16+\cdots+\frac1{1993006}},$$
where the denominators are partial sums of the reciprocals of triangular numbers $k=n(n+1)/2$ for $n=1,2,\ldots,1996$, prove that $S\gt 1001$.
Let $ABC$ be a triangle inscribed in a circle. Put $l_a=m_a/M_a$, $l_b=m_b/M_b$, $l_c=m_c/M_c$, where $m_a,m_b,m_c$ are the internal angle-bisector lengths and $M_a,M_b,M_c$ are the lengths of these bisectors extended to the circle. Prove that
$$\frac{l_a}{\sin^2A}+\frac{l_b}{\sin^2B}+\frac{l_c}{\sin^2C}\ge3,$$
with equality if and only if $ABC$ is equilateral.
In the constructed diagram $D_A,D_B,D_C$ are the intersections with the opposite sides and $T_A,T_B,T_C$ the second intersections with the circle.
Triangle $A_1A_2A_3$ is right-angled at $A_3$. For $n\ge3$, draw the perpendicular from $A_n$ to the line $A_{n-2}A_{n-1}$, meeting it at $A_{n+1}$.
(a) Prove that if this process continues indefinitely, exactly one point $P$ is interior to every triangle $A_{n-2}A_{n-1}A_n$.
(b) With $A_1,A_3$ fixed, let $A_2$ vary over all permissible plane positions. Find the locus of $P$.
Suppose $n\ge3$ people $A_1,\ldots,A_n$ sit in a circle, and $A_i$ has $a_i$ objects, with $a_1+\cdots+a_n=nN$ for a positive integer $N$. To give everyone the same number of objects, each $A_i$ may give objects to or receive objects from the neighbors $A_{i-1},A_{i+1}$, with cyclic indices. How should redistribution be performed to minimize the total number of objects transferred?
