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| z | < =a if | z + a | <= a and |z^2 + a | <= a

Source: 1998 Romania NMO X p2

August 14, 2024
complex numbersinequalitiesalgebra

Problem Statement

Let a≥1a \ge1 be a real number and zz be a complex number such that ∣z+a∣≤a| z + a | \le a and ∣z2+a∣≤a|z^2+ a | \le a. Show that ∣z∣≤a| z | \le a.
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