Advanced book on Mathematics Olympiad

222 4 Geometry and Trigonometry

O

y

x

z

Figure 29

Figure 30

of the 1968 Olympic stadium in Mexico City). There is one more nondegenerate ruled
quadric, which makes the object of the first problem in our list. And if you find some
of the problems below too difficult, remember Winston Churchill’s words: “Success
consists of going from failure to failure without loss of enthusiasm.’’

626.A cube is rotated about the main diagonal. What kind of surfaces do the edges
describe?

627.Prove that the plane

x

a

+

y

b

−

z

c

= 1

is tangent to the hyperboloid of one sheet

x^2

a^2

+

y^2

b^2

−

z^2

c^2

= 1.