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9(x- 1)(y- 1) + (y^2 + y+ 1)(x + 1) + (x^2-x+ 1)(y- 1) >=9

Source: 2021 IMOC qualification problem, A1

December 30, 2021
algebrainequalities

Problem Statement

Prove that if positive reals x,yx,y satisfy x+y=3x+y= 3, x,y≥1x,y \ge 1 then 9(x−1)(y−1)+(y2+y+1)(x+1)+(x2−x+1)(y−1)≥99(x- 1)(y- 1) + (y^2 + y+ 1)(x + 1) + (x^2-x+ 1)(y- 1) \ge 9
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