In a communications network between two regions $A$ and $B$, there are $n$ stations in $A$ and $k$ stations in $B$. Every station in $A$ can communicate with at least $k-p$ stations in $B$. Prove that if $np\lt k$, at least one station in $B$ can communicate with every station in $A$.
Given numbers $x\gt 0$, $y_1\gt 0$, $x_2\lt 0$, $y_2\gt 0$, $x_3\lt 0$, $y_3\lt 0$, $x_4\gt 0\gt y_4$. For each $i=1,\ldots,4$, suppose $(x_i-a)^2+(y_i-b)^2\le c^2$. Prove that $a^2+b^2\le c^2$. Express this assertion as a theorem in plane geometry.
Source note: the first symbol is printed $x\gt 0$, without subscript. The intended condition $x_1\gt 0$ is required and is explicitly adopted for the solution; the literal statement is defective.
Two circles with centers $O,O'$ and radii $R,R'$ respectively meet at two points $P,Q$. A variable line $\Delta$ is always cut by the two circles in a harmonic range of four points.
(a) From $O,O'$ drop perpendiculars $OH,O'H'$ to $\Delta$. Use logical reasoning to conjecture the loci of $H,H'$ as precisely as possible.
(b) Prove those conjectures and determine the limits of the loci. In what case does the locus pass through the two points $O,O'$?
(c) Assuming $OO'\lt \sqrt{R^2+R'^2}$, find a point $M$ on $\Delta$ for which the sum of the lengths $MO$ and $MO'$ is the shortest among all paths joining $O$ to $O'$ through a point on $\Delta$. Prove this shortest length is constant as $\Delta$ varies. How should the question be changed when $OO'\gt \sqrt{R^2+R'^2}$?
Convention: the two intersection points belonging to each circle form one of the two harmonic pairs; all four points must be distinct real intersections. Tangencies are excluded. The source does not specify whether “the locus” means its full supporting circle or its allowed arcs; the solution distinguishes them.