Advanced book on Mathematics Olympiad

2.1 Identities and Inequalities 35

And now a list of problems, all of which are to be solved using the Cauchy–Schwarz
inequality.

103.Ifa, b, care positive numbers, prove that

9 a^2 b^2 c^2 ≤(a^2 b+b^2 c+c^2 a)(ab^2 +bc^2 +ca^2 ).

104.Ifa 1 +a 2 +···+an=nprove thata^41 +a 24 +···+a^4 n≥n.

105.Leta 1 ,a 2 ,...,anbe distinct real numbers. Find the maximum of

a 1 aσ(a)+a 2 aσ( 2 )+···+anaσ (n)

over all permutations of the set{ 1 , 2 ,...,n}.

106.Letf 1 ,f 2 ,...,fnbe positive real numbers. Prove that for any real numbers
x 1 ,x 2 ,...,xn, the quantity

f 1 x^21 +f 2 x^22 +···+fnxn^2 −

(f 1 x 1 +f 2 x 2 +···+fnxn)^2

f 1 +f 2 +···+fn

is nonnegative.

107.Find all positive integersn, k 1 ,...,knsuch thatk 1 +···+kn= 5 n−4 and

1

k 1

+···+

1

kn

= 1.

108.Prove that the finite sequencea 0 ,a 1 ,...,anof positive real numbers is a geometric
progression if and only if

(a 0 a 1 +a 1 a 2 +···+an− 1 an)^2 =(a 02 +a 12 +···+an^2 − 1 )(a 12 +a^22 +···+a^2 n).

109.LetP(x)be a polynomial with positive real coefficients. Prove that

√

P(a)P(b)≥P(

√

ab),

for all positive real numbersaandb.

110. Consider the real numbersx 0 >x 1 >x 2 >···>xn. Prove that

x 0 +

1

x 0 −x 1

+

1

x 1 −x 2

+···+

1

xn− 1 −xn

≥xn+ 2 n.

When does equality hold?