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Polish MO Final 2010, 6th problem (sequence with properties)

Source:

November 7, 2010

logarithmslimitalgebra proposedalgebraInequalityinequalities

Problem Statement

Real number

C > 1

is given. Sequence of positive real numbers

a_1, a_2, a_3, \ldots

, in which

a_1=1

and

a_2=2

, satisfy the conditions

a_{mn}=a_ma_n,

a_{m+n} \leq C(a_m + a_n),

for

m, n = 1, 2, 3, \ldots

. Prove that

a_n = n

for

n=1, 2, 3, \ldots