This is a riff on a classic problem, given in Challenging Problems in Algebra.
“N. Bank and S. Bank are, respectively, the north and south banks of a river with a uniform width of one mile. Town A is 3 miles north of N. Bank, town B is 5 miles south of S. Bank and 15 miles east of A. If crossing at the river banks is only at right angles to the banks, find the length of the shortest path from A to B.
Challenge. If the rate of land travel is uniformly 8 mph, and the rowing rate on the river is 1 2/3 mph (in still water) with a west to east current of 1 1/3 mph, find the shortest time it takes to go from A to B. [The path across the river must still be perpendicular to the banks.]”
See the River Crossing for a solution.
Here is another imaginative geometry problem from 
The issue 7 of the Chalkdust mathematics magazine had an interesting geometric problem presented by Matthew Scroggs.
This problem posted by Presh Talwalkar offers a variety of solutions, but I didn’t quite see my favorite approach for such problems. So I thought I would add it to the mix.
This was a nice geometric problem from Poo-Sung Park 
This is another UKMT Senior Challenge problem, this time from 2006.
Normally I don’t care for combinatorial problems, but this problem from 
The following is a famous problem of Bachet as recounted by Heinrich Dörrie in his book 100 Great Problems of Elementary Mathematics:
This is a fun little problem from the United Kingdom Mathematics Trust (UKMT) Senior Math Challenge of 2008.
This is an interesting problem from the United Kingdom Mathematics Trust (UKMT) Senior Math Challenge of 2008.
