Advanced book on Mathematics Olympiad

646 Geometry and Trigonometry

Figure 87

Remark.More is true, namely that the total curvature is equal to 2πif and only if the
curve is planar and convex. A result of Milnor and Fáry shows that the total curvature of
aknottedcurve in space exceeds 4π.
(W. Fenchel)

645.Consider a coordinate system with axes parallel to the sides ofR(and hence to the
sides of all rectangles of the tiling). It is not hard to see that ifD=[a, b]×[c, d]is a
rectangle whose sides are parallel to the axes, then the four integrals
∫∫

D

sin 2πxsin 2πydxdy,

∫∫

D

sin 2πxcos 2πydxdy,

∫∫

D

cos 2πxsin 2πydxdy,

∫∫

D

cos 2πxcos 2πydxdy

are simultaneously equal to zero if and only if eitherb−aord−cis an integer. Indeed,
this is equivalent to the fact that

(cos 2πb−cos 2πa)(cos 2πd−cos 2πc)= 0 ,

(cos 2πb−cos 2πa)(sin 2πd−sin 2πc)= 0 ,

(sin 2πb−sin 2πa)(cos 2πd−cos 2πc)= 0 ,

(sin 2πb−sin 2πa)(sin 2πd−sin 2πc)= 0 ,

and a case check shows that either cos 2πb= cos 2πaand sin 2πb=sin 2πa,or
cos 2πd=cos 2πcand sin 2πd=sin 2πc, which then implies that eitheraandborc
andddiffer by an integer. Because the four integrals are zero on each rectangle of the
tiling, by adding they are zero onR. Hence at least one of the sides ofRhasinteger
length.
(short list of the 30th International Mathematical Olympiad, 1989, proposed by
France)

646.We denote byA(XY Z)the area of triangleXY Z. Look first at the degenerate
situation described in Figure 88, whenPis on one side of the triangle. With the notation