Let $A$, $B$, $C$, and $D$ be distinct points on a line, in that order. The circles with diameters $AC$ and $BD$ intersect at $X$ and $Y$. $O$ is an arbitrary point on the line $XY$ but not on $AD$. $CO$ intersects the circle with diameter $AC$ again at $M$, and $BO$ intersects the other circle again at $N$. Prove that the lines $AM$, $DN$, and $XY$ are concurrent.
Determine all integers $n\gt 3$ such that there are $n$ points $A_1,A_2,\ldots,A_n$ in the plane that satisfy the following two conditions simultaneously:
(a) No three lie on the same line.
(b) There exist real numbers $p_1,p_2,\ldots,p_n$ such that the area of $\triangle A_iA_jA_k$ is equal to $p_i+p_j+p_k$, for $1\le i\lt j\lt k\le n$.
The positive real numbers $x_0,x_1,\ldots,x_{1995}$ satisfy $x_0=x_{1995}$ and
$$x_{i-1}+\frac2{x_{i-1}}=2x_i+\frac1{x_i}$$
for $i=1,2,\ldots,1995$. Find the maximum value that $x_0$ can have.
Let $ABCDEF$ be a convex hexagon with $AB=BC=CD$, $DE=EF=FA$, and $\angle BCD=\angle EFA=\pi/3$ (that is, $60^\circ$). Let $G$ and $H$ be two points interior to the hexagon, such that angles $AGB$ and $DHE$ are both $2\pi/3$ (that is, $120^\circ$). Prove that $AG+GB+GH+DH+HE\ge CF$.
Let $p$ be an odd prime. Find the number of $p$-element subsets $A$ of $\{1,2,\ldots,2p\}$ such that the sum of all elements of $A$ is divisible by $p$.