Advanced book on Mathematics Olympiad

Real Analysis 531

(a^21 +a 22 +a^23 )(a 14 +a^42 +a 34 )≥(a^31 +a^32 +a 33 )^2.

This is just the Cauchy–Schwarz inequality applied toa 1 ,a 2 ,a 3 , anda 12 ,a^22 ,a^23.

(Gazeta Matematica (Mathematics Gazette), Bucharest ̆ )

442.Take the natural logarithm of both sides, which are positive becausexi ∈( 0 ,π),

i= 1 , 2 ,...,n, to obtain the equivalent inequality

∑n

i= 1

ln

sinxi

xi

≤nln

sinx

x

.

All we are left to check is that the functionf(t)=lnsinttis concave on( 0 ,π).
Becausef(t)=ln sint−lnt, its second derivative is

f′′(t)=−

1

sin^2 t

+

1

t^2

.

The fact that this is negative follows from sint<tfort>0, and the inequality is proved.

(39th W.L. Putnam Mathematical Competition, 1978)

443.The functionf:( 0 , 1 )→R,f(x)=√ 1 x−xis convex. By Jensen’s inequality,

1

n

∑n

i= 1

xi

√

1 −xi

≥

1

n

∑n

i= 1

xi

√√

√

√ 1 −^1

n

∑n

i= 1

xi

=

1

√

n(n− 1 )

.

We have thus found that

x 1

√

1 −x 1

+

x 2

√

1 −x 2

+···+

xn

√

1 −xn

≥

√

n

n− 1

.

On the other hand, by the Cauchy–Schwarz inequality

n=n

∑n

i= 1

xi≥

( n

∑

i= 1

√

xi

) 2

,

whence

∑n

i= 1

√

xi≤

√

n. It follows that

√

x 1 +

√

x 2 +···+

√

xn

√

n− 1

≤

√

n

n− 1

.

Combining the two inequalities, we obtain the one from the statement.