228 4 Geometry and Trigonometry
639.Two convex polygons are placed one inside the other. Prove that the perimeter of
the polygon that lies inside is smaller.
640.There arenline segments in the plane with the sum of the lengths equal to 1. Prove
that there exists a straight line such that the sum of the lengths of the projections of
the segments onto the line is equal toπ^2.
641.In a triangleABCfor a variable pointPonBCwithPB=xlett(x)be the
measure of∠PAB. Compute
∫a
0cost(x)dxin terms of the sides and angles of triangleABC.642.Letf:[ 0 ,a]→Rbe a continuous and increasing function such thatf( 0 )=0.
Define byRthe region bounded byf(x)and the linesx=aandy=0. Now
consider the solid of revolution obtained whenRis rotated around they-axis as a
sort of dish. Determinefsuch that the volume of water the dish can hold is equal
to the volume of the dish itself, this happening for alla.
643.Consider a unit vector starting at the origin and pointing in the direction of the
tangent vector to a continuously differentiable curve in three-dimensional space.
The endpoint of the vector describes the spherical image of the curve (on the unit
sphere). Show that if the curve is closed, then its spherical image intersects every
great circle of the unit sphere.
644.With the hypothesis of the previous problem, if the curve is twice differentiable,
then the length of the spherical image of the curve is called the total curvature.
Prove that the total curvature of a closed curve is at least 2π.
645.A rectangleRis tiled by finitely many rectangles each of which has at least one side
of integral length. Prove thatRhas at least one side of integral length.
4.1.6 Other Geometry Problems.................................
We conclude with problems from elementary geometry. They are less in the spirit of
Euclid, being based on algebraic or combinatorial considerations. Here “imagination is
more important than knowledge’’ (A. Einstein).
Example.Find the maximal number of triangles of area 1 with disjoint interiors that can
be included in a disk of radius 1. Describe all such configurations.
Solution.Let us first solve the following easier problem:
Find all triangles of area 1 that can be placed inside a half-disk of radius 1.