%⎝⎜⎜⎠⎟⎟⎟SUY%RA%P=(X−Y)( Y−1)(X−1)(XY+X+Y)≤(X−1)34 (XY+X+Y).%CHÚ%Ý%S...

4 .%⎝⎜⎜⎠⎟⎟⎟Suy%ra%P=(x−y)( y−1)(x−1)(xy+x+y)≤(x−1)

³

4 (xy+x+y).%Chú%ý%sử%dụng%bất%đẳng%thức%Cauchy%–Schwarz%ta%có:%(x−y)

²

+(y−1)

²

≥12(x−1)

²

.%⇒32(x−1)

²

≤(x−1)

²

+(x−y)

²

+(y−1)

²

=10−2(x+y+xy)⇒(x−1)

²

≤43(5−xy−x−y)Đặt%t=x+y+xy≤x

²

+y

²

+1=5⇒t∈⎡3;5⎦⎥.%⎣⎢ ⎤(5−t)

³

Khi%đó%P

²

≤(x−1)

⁶

16 (xy+x+y)

²

≤.4

³

16 t

²

=4t

²

(5−t)

³