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Maximum of sum given two conditions

Source: Chinese TST 1 2012 Day 2 Q1

March 15, 2013

inequalities proposedinequalitiesChina TST

Problem Statement

Let

n

and

k

be two integers which are greater than

1

. Let

a_1,a_2,\ldots,a_n,c_1,c_2,\ldots,c_m

be non-negative real numbers such that i)

a_1\ge a_2\ge\ldots\ge a_n

and

a_1+a_2+\ldots+a_n=1

; ii) For any integer

m\in\{1,2,\ldots,n\}

, we have that

c_1+c_2+\ldots+c_m\le m^k

. Find the maximum of

c_1a_1^k+c_2a_2^k+\ldots+c_na_n^k

.

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