Advanced book on Mathematics Olympiad

3.2 Continuity, Derivatives, and Integrals 147

The generalized mean inequality.Given the positive numbersx 1 ,x 2 ,...,xnand the
positive weightsλ 1 ,λ 2 ,...,λnwithλ 1 +λ 2 + ··· +λn = 1 , the following inequal-
ity holds:

λ 1 x 1 +λ 2 x 2 +···+λnxn≥x 1 λ^1 xλ 22 ···xnλn.

Solution.Simply write Jensen’s inequality for the concave functionf(x)=lnx, then
exponentiate. 

Forλ 1 =λ 2 = ··· =λn=^1 none obtains the AM–GM inequality.

439.Show that ifA, B, Care the angles of a triangle, then

sinA+sinB+sinC≥

3

√

3

2

.

440.Letai,i = 1 , 2 ,...,n, be nonnegative numbers with

∑n

i= 1 ai = 1, and let

0 <xi≤1,i= 1 , 2 ,...,n. Prove that

∑n

i= 1

ai

1 +xi

≤

1

1 +x 1 a^1 x 2 a^2 ···xnan

441.Prove that for any three positive real numbersa 1 ,a 2 ,a 3 ,

a 12 +a^22 +a^23

a 13 +a^32 +a^33

≥

a 13 +a 23 +a^33

a 14 +a 24 +a^43

442.Let 0<xi<π,i= 1 , 2 ,...,n, and setx=x^1 +x^2 +···+n xn. Prove that

∏n

i= 1

(

sinxi

xi

)

≤

(

sinx

x

)n

443.Letn>1 andx 1 ,x 2 ,...,xn>0 be such thatx 1 +x 2 +···+xn=1. Prove that

x 1

√

1 −x 1

+

x 2

√

1 −x 2

+···+

xn

√

1 −xn

≥

√

x 1 +

√

x 2 +···+

√

xn

√

n− 1

3.2.7 Indefinite Integrals

“Anyone who stops learning is old, whether at twenty or eighty. Anyone who keeps
learning stays young. The greatest thing in life is to keep your mind young.’’ Following
this advice of Henry Ford, let us teach you some clever tricks for computing indefinite
integrals.