Advanced book on Mathematics Olympiad

3.2 Continuity, Derivatives, and Integrals 145

432.Prove that if a sequence of positive real numbers(bn)nhas the property that(anbn)n
is a convex sequence for all real numbersa, then the sequence(lnbn)nis also
convex.

433.Find the largest constantCsuch that for everyn≥3 and every positive concave
sequence(ak)nk= 1 ,
(n
∑

k= 1

ak

) 2

≥C(n− 1 )

∑n

k= 1

ak^2.

A convex function on a closed interval attains its maximum at an endpoint of the inter-
val. We illustrate how this fact can be useful with a problem fromTimi ̧soara Mathematics
Gazette, proposed by V. Cârtoaje and M. Lascu.

Example.Leta, b, c, d∈[ 1 , 3 ]. Prove that

(a+b+c+d)^2 ≥ 3 (a^2 +b^2 +c^2 +d^2 ).

Solution.Divide by 2 and move everything to one side to obtain the equivalent inequality

a^2 +b^2 +c^2 +d^2 − 2 ab− 2 ac− 2 ad− 2 bc− 2 bd− 2 cd≤ 0.

Now we recognize the expression on the left to be a convex function in each variable.
So the maximum is attained for some choice ofa, b, c, d=1or3. Ifkof these numbers
are equal to 3, and 4−kare equal to 1, wherekcould be 1, 2, 3, or 4, then the original
inequality becomes

( 3 k+ 4 −k)^2 = 3 ( 9 k+ 4 −k).

Dividing by 3, we obtaink^2 + 4 k+ 4 ≥ 6 k+3, or(k− 1 )^2 ≥0, which is clearly true.
The inequality is proved. Equality occurs when one of the numbersa, b, c, dis equal to
3 and the other three are equal to 1. 

Here are additional problems of this kind.

434.Letα,β, andγ be three fixed positive numbers and[a, b]a given interval. Find
x, y, zin[a, b]for which the expression
E(x, y, z)=α(x−y)^2 +β(y−z)^2 +γ(z−x)^2
has maximal value.

435.Let 0<a<bandti ≥0,i= 1 , 2 ,...,n. Prove that for anyx 1 ,x 2 ,...,xn∈
[a, b],
(n
∑

i= 1

tixi

)(n

∑

i= 1

ti

xi

)

≤

(a+b)^2

4 ab

(n

∑

i= 1

ti

) 2

.