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3 eqaul numbers and prove that x=1

Source: 2018 JBMO TST - Turkey, P1

March 27, 2020
algebra

Problem Statement

Let a,b,ca, b, c be distinct real numbers and xx be a real number. Given that three numbers among ax2+bx+c,ax2+cx+b,bx2+cx+a,bx2+ax+c,cx2+ax+b,cx2+bx+aax^2+bx+c, ax^2+cx+b, bx^2+cx+a, bx^2+ax+c, cx^2+ax+b, cx^2+bx+a coincide, prove that x=1x=1.
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