An ant starts at $(0,0)$ in the southwest corner of a $5\times5$ grid and moves east or north along its lines to $(5,5)$. It cannot pass through the four broken intersections $(1,1),(1,4),(4,1),(4,4)$. How many different paths are possible?
On a circle are $n$ numbers with sum ninety-four. Each equals the absolute difference of the next two in clockwise order. Find the sum of all possible values of $n$.
Decreasing the thousands digit of a four-digit square by three and increasing its units digit by three gives another four-digit square. What is the original number?
The broken line $A-B-C-D$ consists of segments parallel to rectangle edges and bisects its area. Point $E$ on the perimeter makes $AE$ also bisect the area. Given $AB=30$, $BC=24$, $CD=10$ cm, find $DE$ in cm.
Two facing vertical mirrors form a thirty-degree angle at $V$. A horizontal beam from $S$, parallel to one mirror, strikes the other at $A$ and eventually returns to $S$. If $SA=AV=1$ cm, find its total travel distance in cm.
$ABC$ is equilateral, and $ABDE$ is a rectangle whose side $DE$ passes through $C$. The incircle of $ABC$ has radius one cm. Find the diameter of the circle through $A,B,D,E$ in cm.
Find $\left\lfloor\sqrt{2008+\sqrt{2008+\sqrt{2008+\cdots}}}\right\rfloor$, with exactly $2008$ occurrences of $2008$. Here the brackets denote the greatest integer not exceeding the number.
Triangle $ABC$ has $\angle ABC=60^\circ$, circumcentre $O$, orthocentre $H$. Points $D\in BC$, $E\in AB$ satisfy $BD=BH$, $BE=BO$. If $BO=1$ cm, find area $BED$ in square cm.
