Advanced book on Mathematics Olympiad

172 3 Real Analysis

off a horizontal mirror represented schematically in Figure 22. Denote byCandD
the projections ofAandBonto the mirror, and byPthe point where the ray hits the
mirror. The angles of incidence and reflection are, respectively, the angles formed by
APandBPwith the normal to the mirror. To prove that they are equal it suffices
to show that∠AP C=∠BPD. Letx =CPandy =DP. We have to minimize
f (x, y)=AP+BPwith the constraintg(x, y)=x+y=CD.

P

A

Cx y D

B

Figure 22

Using the Pythagorean theorem we find that

f (x, y)=

√

x^2 +AC^2 +

√

y^2 +BD^2.

The method of Lagrange multipliers yields the system of equations

x

√

x^2 +CP^2

=λ,

y

√

y^2 +DP^2

=λ,

x+y=CD.

From the first two equations, we obtain

x

√

x^2 +CP^2

=

y

√

y^2 +DP^2

,

i.e., CPAP = DPBP. This shows that the right trianglesCAPandDBPare similar, so
∠AP C=∠BPDas desired. 

The following example was proposed by C. Niculescu forMathematics Magazine.

Example.Find the smallest constantk>0 such that

ab

a+b+ 2 c

+

bc

b+c+ 2 a

+

ca

c+a+ 2 b

≤k(a+b+c)

for everya, b, c > 0.