A = 2x2 + y2 - 2x - 4y + 2018 - 2xy
A = x2 - 2xy + y2 + 4x - 4y + 4 + x2 - 6x + 9 + 2005
A = ( x - y)2 + 4( x - y) + 4 + ( x - 3)2 + 2005
A = ( x - y + 2)2 + ( x - 3)2 + 2005
Do : ( x - y + 2)2 ≥ 0 ∀x,y
( x - 3)2 ≥ 0 ∀x,y
⇒ ( x - y + 2)2 + ( x - 3)2 ≥ 0
⇒ ( x - y + 2)2 + ( x - 3)2 + 2005 ≥ 2005
⇒ AMIN = 2005 ⇔ x = 3 ; y = 5