Advanced book on Mathematics Olympiad

644 Geometry and Trigonometry

cost=

c−xcosB

√

x^2 +c^2 − 2 xccosB

.

The integral from the statement is
∫a

0

cost(x)dx=

∫a

0

c−xcosB

√

x^2 +c^2 − 2 xccosB

dx.

Using the standard integration formulas
∫
dx
√
x^2 +αx+β

=ln

(

2 x+α+ 2

√

x^2 +αx+β

)

,

∫

xdx

√

x^2 +αx+β

=

√

x^2 +αx+β−

α

2

ln

(

2 x+α+ 2

√

x^2 +αx+β

)

,

we obtain
∫a

0

cost(x)dx=csin^2 Bln

(

2 x+ 2 ccosB+ 2

√

x^2 − 2 cxcosB+c^2

)∣∣

∣

a

0

−cosB

√

x^2 − 2 cxcosB+c^2

∣

∣

∣

a

0

=csin^2 Bln

a−ccosB+b

c( 1 −cosB)

+cosB(c−b).

642.It is equivalent to ask that the volume of the dish be half of that of the solid of
revolution obtained by rotating the rectangle 0≤x≤aand 0≤y≤f(a). Specifically,
this condition is
∫a

0

2 πxf(x)dx=

1

2

πa^2 f(a).

Because the left-hand side is differentiable with respect toafor alla>0, the right-hand
side is differentiable, too. Differentiating, we obtain

2 πaf(a)=πaf(a)+

1

2

πa^2 f′(a).

This is a differential equation inf, which can be written asf′(a)/f (a)=^2 a. Integrating,
we obtain lnf(a)=2lna,orf(a)=ca^2 for some constantc>0. This solves the
problem.
(Math Horizons)

643.Parametrize the curve by its length as(x(s), y(s), z(s)),0≤ s≤ L. Then the
coordinates(ξ,η,ζ)of its spherical image are given by