An integer is square-free if it is not divisible by $a^2$ for any integer $a\gt 1$. Let $S$ be the set of positive square-free integers. Determine, with justification, the value of
$$\sum_{k\in S}\left\lfloor\sqrt{\frac{10^{10}}{k}}\right\rfloor,$$
where $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$. For example, $\lfloor2.5\rfloor=2$.
Let $G$ be the centroid of $\triangle ABC$. Through $G$ draw a line parallel to $BC$ and intersecting the sides $AB$ and $AC$ at $P$ and $Q$, respectively. Let $BQ$ intersect $GC$ at $E$ and $CP$ intersect $GB$ at $F$. Prove that triangles $ABC$ and $DEF$ are similar.
Editorial source erratum: the printed statement does not define $D$. The supplied solution and diagram use $D$ as the midpoint of $BC$. The original statement above is preserved; the solution below is conditional on this intended definition. This question requires review.
Place $2005$ points on the circumference of a circle. Two points $P,Q$ are said to form a pair of neighbours if the chord $PQ$ subtends an angle of at most $10^\circ$ at the centre. Find the smallest number of pairs of neighbours.