các chuyên đề toán chọn lọc - toán thcs

[...]... b, c = -2 ; b = 5; E [0 ; 1] ChUng minh rang a 2 + b2 + c2::; 1 + a 2b + b2c + c2a Loi gild Vi a, b e [0 ; 1] nen a(1 - b) ~ a 2(1 - b) TLJOng t'! b(1 - c) ~ b2(1 - c); c(1 - a);~ c2(1 - a) C¢ng thea ve ba BDT tren ta suy ra a + b + c - (ab + bc + cal ~ (a 2 + b2 + c 2) - (a 2b + b2c + c 2a) 1 - (1 - a)(1 - b)(1 - c) - abc ~ (a2 + b2 + c 2) ­ (a 2b + b2c + c2a) 1 + (a 2b + b2c + c 2a) - abc ~... y2 + z2 s 5 Loi giai VI x, y, Z E [0 ; 2] nen (x - 2)(y - 2)(z - 2) ::; O Sa dunq gia thiet x + y + Z = 3 ta dL/ . b(1 - c) ~ - c); c(1 - a);~ c2(1 - a). C¢ng thea ve ba BDT tren ta suy ra a + b + c - (ab + bc + cal 2) 2a) ~ (a 2 + b 2 + c - (a 2b + b 2c + c <=> 1 - (1 - a)(1 - b)(1 - c) - abc. (a 2b + b 2c + c 2a) ~ (a 2 + b 2 + c - 2a) <=> 1 + (a 2b + b 2c + c - abc ~ (a 2 + b 2 + c 2) + (1 - a)(1 - b)(1 - c). Ma(1 - a)(1 - b)(1 - c) ~ 0 va abc ~ 0 nen suy ra dpcm E [0 ; 2] nen (x - 2)(y - 2)(z - 2) ::; O. Sa dunq gia thiet x + y + Z = 3 ta dL/<;lc xyz - 2(xy + yz + zx) + 4(x + Y + z) - 8 ::; 0 <=> xyz + (x 2 + y2 + z2) - (x + Y + z)2 +