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any natural number can be given as a_i+a_j, a _n>n^2 /16, increasing

Source: Ukraine TST 2011 p8

May 7, 2020

SequencealgebraNatural Numbers

Problem Statement

Is there an increasing sequence of integers

0 = {{a} _{0}} <{{a} _{1}} <{{a} _{2}} <\ldots

for which the following two conditions are satisfied simultaneously: 1) any natural number can be given as

{{a} _{i}} + {{a} _{j}}

for some (possibly equal)

i \ge 0

j \ge 0

, 2)

{{a} _ {n}}> \tfrac {{{n} ^ {2}}} {16}

for all natural

n