不要写的太深奥,已知四个整数a,b,c,d满足a^4+b^4+c^4+d^4=4abcd.求证a=b=c=d.
来源：学生作业帮助网 编辑：六六作业网 时间：2024/12/19 04:46:45
不要写的太深奥,已知四个整数a,b,c,d满足a^4+b^4+c^4+d^4=4abcd.求证a=b=c=d.
不要写的太深奥,
已知四个整数a,b,c,d满足a^4+b^4+c^4+d^4=4abcd.
求证a=b=c=d.
不要写的太深奥,已知四个整数a,b,c,d满足a^4+b^4+c^4+d^4=4abcd.求证a=b=c=d.
a^4+b^4+c^4+d^4=4abcd
a^4+b^4+c^4+d^4-2a²b²-2c²d²=4abcd-2a²b²-2c²d²
（a²-b²)²+(c²-d²)²+2(ab-cd)²=0
∴a²=b²,c²=d²,ab=cd
∴a=b=c=d
a^4+b^4+c^4+d^4=4abcd
2a^4 +2b^4+2c^4+2d^4=8abcd
a^4-2a^2b^2 + b^4 +b^4 - 2b^2c^2+c^4+c^4-2c^2d^2+d^4+d^4-2d^2a^2+a^4 =8abcd-2a^2b^2-2b^2c^2-2c^2d^2-2d^2a^2
(a^2-b^2)^2+(b^2-c^2)^2+(c^2-d^...
全部展开
a^4+b^4+c^4+d^4=4abcd
2a^4 +2b^4+2c^4+2d^4=8abcd
a^4-2a^2b^2 + b^4 +b^4 - 2b^2c^2+c^4+c^4-2c^2d^2+d^4+d^4-2d^2a^2+a^4 =8abcd-2a^2b^2-2b^2c^2-2c^2d^2-2d^2a^2
(a^2-b^2)^2+(b^2-c^2)^2+(c^2-d^2)^2+(d^2-a^2)^2=- 2*{(a^2b^2-2abcd+c^2d^2)+(b^2c^2-2abcd+a^2d^2)}
(a^2-b^2)^2+(b^2-c^2)^2+(c^2-d^2)^2+(d^2-a^2)^2=- 2*{(ab-cd)^2+(bc-ad)^2
(a^2-b^2)^2+(b^2-c^2)^2+(c^2-d^2)^2+(d^2-a^2)^2 + 2*{(ab-cd)^2+(bc-ad)^2 = 0
等式的左边均为平方项大于等于0， 等式最后的结果等于0，只能各项均等于0
故，a^2 = b^2 ....., 所以，a=b=c=d
收起
a^4+b^4+c^4+d^4=4abcd
2a^4 +2b^4+2c^4+2d^4=8abcd
a^4-2a^2b^2 + b^4 +b^4 - 2b^2c^2+c^4+c^4-2c^2d^2+d^4+d^4-2d^2a^2+a^4 =8abcd-2a^2b^2-2b^2c^2-2c^2d^2-2d^2a^2
(a^2-b^2)^2+(b^2-c^2)^2+(c^2-d^...
全部展开
a^4+b^4+c^4+d^4=4abcd
2a^4 +2b^4+2c^4+2d^4=8abcd
a^4-2a^2b^2 + b^4 +b^4 - 2b^2c^2+c^4+c^4-2c^2d^2+d^4+d^4-2d^2a^2+a^4 =8abcd-2a^2b^2-2b^2c^2-2c^2d^2-2d^2a^2
(a^2-b^2)^2+(b^2-c^2)^2+(c^2-d^2)^2+(d^2-a^2)^2=- 2*{(a^2b^2-2abcd+c^2d^2)+(b^2c^2-2abcd+a^2d^2)}
(a^2-b^2)^2+(b^2-c^2)^2+(c^2-d^2)^2+(d^2-a^2)^2=- 2*{(ab-cd)^2+(bc-ad)^2
(a^2-b^2)^2+(b^2-c^2)^2+(c^2-d^2)^2+(d^2-a^2)^2 + 2*{(ab-cd)^2+(bc-ad)0
收起