Advanced book on Mathematics Olympiad

174 3 Real Analysis

508.Using the method of Lagrange multipliers prove Snell’s law of optics: If a light ray
passes between two media separated by a planar surface, then
sinθ 1
sinθ 2

=

v 1

v 2

,

whereθ 1 andθ 2 are, respectively, the angle of incidence and the angle of refraction,

andv 1 andv 2 are the speeds of light in the first and second media, respectively.

509.LetABCbe a triangle such that
(
cot

A

2

) 2

+

(

2 cot

B

2

) 2

+

(

3 cot

C

2

) 2

=

(

6 s

7 r

) 2

,

wheresandrdenote its semiperimeter and its inradius, respectively. Prove that

triangleABCis similar to a triangleTwhose side lengths are all positive integers

with no common divisors and determine these integers.

510.Prove that of all quadrilaterals that can be formed from four given sides, the one
that is cyclic has the greatest area.

511. Of all triangles circumscribed about a given circle, find the one with the small-
     est area.

512.Leta, b, c, dbe four nonnegative numbers satisfyinga+b+c+d=1. Prove
the inequality

abc+bcd+cda+dab≤

1

27

+

176

27

abcd.

513.Given two triangles with anglesα,β,γ, respectively,α 1 ,β 1 ,γ 1 , prove that

cosα 1

sinα

+

cosβ 1

sinβ

+

cosγ 1

sinγ

≤cotα+cotβ+cotγ,

with equality if and only ifα=α 1 ,β=β 1 ,γ=γ 1.

3.3.2 Multivariable Integrals....................................

For multivariable integrals, the true story starts with a change of coordinates.

Theorem.Letf : D ⊂ Rn → Rbe an integrable function. Let alsox(u) =
(xi(uj))ni,j= 1 be a change of coordinates, viewed as a map from some domainD∗to

D, with Jacobian∂x∂u=det(∂x∂uij). Then

∫

D

f(x)dx=

∫

D∗

f (x(u))

∣∣

∣

∣

∂x

∂u

∣∣

∣

∣du.