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Bosnia and Herzegovina TST 1996 Day 1 Problem 1

Source: Bosnia and Herzegovina Team Selection Test 1996

September 20, 2018

inequalitiesalgebraparameter

Problem Statement

a)

Let

a

b

and

c

be positive real numbers. Prove that for all positive integers

m

holds:

(a+b)^m+(b+c)^m+(c+a)^m \leq 2^m(a^m+b^m+c^m)

b)

Does inequality

a)

holds for

1)

arbitrary real numbers

a

b

and

c

2)

any integer

m