Advanced book on Mathematics Olympiad

640 Geometry and Trigonometry

633.The equation of the locus can be expressed in a simple form using determinants as

∣∣

∣∣

∣∣

∣

∣∣

x 1 x 2 ··· xn

xn x 1 ···xn− 1

··· ···

...

···

x 2 x 3 ··· x 1

∣∣

∣∣

∣∣

∣∣

∣

= 0.

Adding all rows to the first, we see that the determinant has a factor ofx 1 +x 2 +···+xn.
Hence the planex 1 +x 2 +···+xn=0 belongs to the locus.
634.Without loss of generality, we may assume that the edges of the cube have length
equal to 2, in which case the cube consists of the points(x 1 ,x 2 ,...,xn)with max|xi|≤1.
The intersection of the cube with the plane determined by−→a and

−→

b is

P=

{

s−→a +t

−→

b

∣

∣

∣∣max

k

∣∣

∣∣scos^2 kπ

n

+tsin

2 kπ

n

∣∣

∣∣≤ 1

}

.

This set is a convex polygon with at most 2nsides, being the intersection ofnstrips

determined by parallel lines, namely the strips

Pk=

{

s−→a +t

−→

b

∣∣

∣∣

∣∣

∣∣scos^2 kπ

n

+tsin

2 kπ

n

∣∣

∣∣≤ 1

}

.

Adding^2 nπto all arguments in the coordinates of−→a and

−→

b permutes thePk’s, leaving

Pinvariant. This corresponds to the transformation

−→a −→cos^2 π

n

−→a −sin^2 π

n

−→

b,

−→

b −→sin

2 π

n

−→a +cos^2 π

n

−→

b,

which is a rotation by^2 nπin the plane of the two vectors. HencePis invariant under a

rotation by^2 nπ, and being a polygon with at most 2nsides, it must be a regular 2n-gon.

(V.V. Prasolov, V.M. Tikhomirov,Geometry, AMS, 2001)

635.Consider the unit sphere inRn,

Sn−^1 =

{

(x 1 ,x 2 ,...,xn)∈Rn|

∑n

k= 1

x^2 k= 1

}

.

The distance between two pointsX =(x 1 ,x 2 ,...,xn)andY =(y 1 ,y 2 ,...,yn)is

given by

d(X,Y)=

(n

∑

k= 1

(xk−yk)^2

) 1 / 2

.