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(m+n)a_{m+n }<= a_m +a_n , prove 1 / a_{200} > 4 x 10^7

Source: Norwegian Mathematical Olympiad 2011 - Abel Competition p3a

September 4, 2019

inequalitiesalgebraSequence

Problem Statement

The positive numbers

a_1, a_2,...

satisfy

a_1 = 1

and

(m+n)a_{m+n }\le a_m +a_n

for all positive integers

m

and

n

. Show that

\frac{1}{a_{200}} > 4 \cdot 10^7

. .