Question

[Vũ Anh Quân]

Cho các số nguyên a,b,c thỏa mãn: (a-b)³+(b-c)³+(c-a)³=210

Tính GT của BT: A=|a-b|+|b-c|+|c-a|

Answer 1

[Lê Thị Cẩm Tú]

(a−b)3=a3−3a2b+3ab2−b3(a−b)3=a3−3a2b+3ab2−b3
(b−c)3=b3−3b2c+3bc2−c3(b−c)3=b3−3b2c+3bc2−c3
(c−a)3=c3−3c2a+3ca2−a3(c−a)3=c3−3c2a+3ca2−a3
=>(a−b)3+(b−c)3+(c−a)3=−3a2b+3ab2−3b2c+3bc2−3c2a+3ca2=210=>(a−b)3+(b−c)3+(c−a)3=−3a2b+3ab2−3b2c+3bc2−3c2a+3ca2=210
<=>−3(a2b−ab2+b2c−bc2+c2a−ca2)=210<=>−3(a2b−ab2+b2c−bc2+c2a−ca2)=210
<=>−3[a2(b−c)+bc(b−c)−a(b2−c2)]=210<=>−3[a2(b−c)+bc(b−c)−a(b2−c2)]=210
<=>−3(b−c)[a2+bc−a(b+c)]=210<=>−3(b−c)[a2+bc−a(b+c)]=210
<=>−3(b−c)(a2+bc−ab−ac)=210<=>−3(b−c)(a2+bc−ab−ac)=210
<=>−3(b−c)[a(a−c)−b(a−c)]=−3(b−c)(a−c)(a−b)=210<=>−3(b−c)[a(a−c)−b(a−c)]=−3(b−c)(a−c)(a−b)=210
<=>3(b−c)(c−a)(a−b)<=>3(b−c)(c−a)(a−b)
<=>(b−c)(a−b)(c−a)=70<=>(b−c)(a−b)(c−a)=70
=>b−c=2,a−b=5,c−a=7=>b−c=2,a−b=5,c−a=7
=>|a−b|+|b−c|+|c−a|=14
 

Answer 2

[thanh ngọc]

a-b)^3=a^3-3a^2b+3ab^2-b^3
(b-c)^3=b^3-3b^2c+3bc^2-c^3
(c-a)^3=c^3-3c^2a+3ca^2-a^3
Cộng ba pt, ta được
-3a^2b+3ab^2-3b^2c+3bc^2-3c^2a+3ca^2
-3(a^2b-ab^2+b^2c-bc^2+c^2a-ca^2)
-3(a^2(b-c)+bc(b-c)-a(b^2-c^2))
-3(b-c)(a^2+bc-a(b+c))
-3(b-c)(a-b)(a-c)=210
(b-c)(a-b)(a-c)=-70
(b-c)(a-b)(a-c)=2*5*(-7)
=>b-c=2, a-b=5, a-c=-7
=>|a-b|+|b-c|+|c-a|=14