Advanced book on Mathematics Olympiad

4.1 Geometry 225

634.Prove that the intersection of ann-dimensional cube centered at the origin and with
edges parallel to the coordinate axes with the plane determined by the vectors

−→a=

(

cos

2 π

n

,cos

4 π

n

,...,cos

2 nπ

n

)

and

−→

b=

(

sin

2 π

n

,sin

4 π

n

,...,sin

2 nπ

n

)

is a regular 2n-gon.

635.Find the maximum number of points on a sphere of radius 1 inRnsuch that the
distance between any two points is strictly greater than

√

2.

4.1.5 Integrals in Geometry.....................................

We now present various applications of integral calculus to geometry problems. Here is
a classic.

Example.A disk of radiusRis covered bymrectangular strips of width 2. Prove that
m≥R.

Solution.Since the strips have different areas, depending on the distance to the center of
the disk, a proof using areas will not work. However, if we move to three dimensions the
problem becomes easy. The argument is based on the following property of the sphere.

Lemma.The area of the surface cut from a sphere of radiusRby two parallel planes at
distancedfrom each other is equal to 2 πRd.

To prove this result, let us assume that the sphere is centered at the origin and the
planes are perpendicular to thex-axis. The surface is obtained by rotating the graph of
the functionf :[a, b]→R,f(x)=

√

R^2 −x^2 about thex-axis, where[a, b]is an
interval of lengthd. The area of the surface is given by

2 π

∫b

a

f(x)

√

(f′(x))^2 + 1 dx= 2 π

∫b

a

√

R^2 −x^2

R

√

R^2 −x^2

dx

= 2 π

∫b

a

Rdx= 2 πRd.

Returning to the problem, the sphere has area 4πR^2 and is covered bymsurfaces, each
having area 4πR. The inequality 4πmR≥ 4 πR^2 implies thatm≥R, as desired. 

The second example, suggested to us by Zh. Wang, is even more famous. We present
the proof from H. Solomon,Geometric Probability(SIAM 1978).