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fancy formulation, but easy concept

Source: Romanian District Olympiad 2015, Grade VIII, Problem 2

September 25, 2018
algebra

Problem Statement

For every real number a, a, define the set Aa={n∈{0}∪N∣n2+an∈{0}∪N}. A_a=\left\{ n\in\{ 0\}\cup\mathbb{N}\bigg|\sqrt{n^2+an}\in\{ 0\}\cup\mathbb{N}\right\} .
a) Show the equivalence: #Aa∈N  ⟺  a≠0, \# A_a\in\mathbb{N}\iff a\neq 0, where #B \# B is the cardinal of B. B. b) Determine max⁡A40. \max A_{40} .
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