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Romania District Olympiad 2012 - Grade XII

Source:

March 10, 2012
quadraticsalgebrapolynomialsuperior algebrasuperior algebra unsolved

Problem Statement

Let (A,+,⋅)(A,+,\cdot) a 9 elements ring. Prove that the following assertions are equivalent:
(a) For any x∈A\{0}x\in A\backslash\{0\} there are two numbers a∈{−1,0,1}a\in \{-1,0,1\} and b∈{−1,1}b\in \{-1,1\} such that x2+ax+b=0x^2+ax+b=0.
(b) (A,+,⋅)(A,+,\cdot) is a field.
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