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RMO KV 2024 Q2

Source: RMO KV 2024 Q2

November 3, 2024
algebra

Problem Statement

Show that there do not exist non-zero real numbers a,b,ca,b,c such that the following statements hold simultaneously: ∙\bullet the equation ax2+bx+c=0ax^2 + bx + c = 0 has two distinct roots x1,x2x_1,x_2; ∙\bullet the equation bx2+cx+a=0bx^2 + cx + a = 0 has two distinct roots x2,x3x_2,x_3; ∙\bullet the equation cx2+ax+b=0cx^2 + ax + b = 0 has two distinct roots x3,x1x_3,x_1. (Note that x1,x2,x3x_1,x_2,x_3 may be real or complex numbers.)
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