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Source: Romania District Olympiad 2013,grade IX(problem 1)

March 14, 2013
inequalitiesalgebra proposedalgebra

Problem Statement

a) Prove that, whatever the real number x would be, the following inequality takes place x4−x3−x+1≥0.{{x}^{4}}-{{x}^{3}}-x+1\ge 0. b) Solve the following system in the set of real numbers: x1+x2+x3=3,x13+x23+x33=x14+x24+x34{{x}_{1}}+{{x}_{2}}+{{x}_{3}}=3,x_{1}^{3}+x_{2}^{3}+x_{3}^{3}=x_{1}^{4}+x_{2}^{4}+x_{3}^{4}. The Mathematical Gazette
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