Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP
have equal areas. Prove X and Y divide the sides of PQRS in the
If xyz = 1 and x+y+z =1/x + 1/y + 1/z show that at least one of
these numbers must be 1. Now for the complexity! When are the other
numbers real and when are they complex?
Two cubes, each with integral side lengths, have a combined volume equal to the total of the lengths of their edges. How big are the cubes? [If you find a result by 'trial and error' you'll need to prove you have found all possible solutions.]
The diagram shows a regular pentagon with sides of unit length.
Find all the angles in the diagram. Prove that the quadrilateral
shown in red is a rhombus. Prove that triangles EFB and DFC are similar and hence show that the length EB is equal to the golden ratio and find the ratio BF : FD.