3.
x^3+y^3+z^3-3xyz=(x+y+z)(x^2+y^2+z^2-xy-yz-zx)=0
-3xyz-18=0
xyz=-6
(x,y,z)=(1,2,-3), (1,-3,2),(2,1,-3),(2,-3,1),(-3,1,2),(-3,2,1)
+ / -檢視主題
由 GFIF 於 星期日 七月 30, 2006 5:22 pm
由 aa2191943 於 星期二 七月 25, 2006 12:57 pm
#ed_op#DIV#ed_cl#第二題我直接說好了:#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#明顯地, 答案是(1,2,3)等六組變換#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#用1,2兩式求出 ab + bc + ca ;#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#再求出 abc#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#最後用根與係數去說明#ed_op#/DIV#ed_cl#
由 aa2191943 於 星期二 七月 25, 2006 8:55 am
#ed_op#DIV#ed_cl#第三題的作法有類似的:#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#利用a#ed_op#SUP#ed_cl#3#ed_op#/SUP#ed_cl#+b#ed_op#SUP#ed_cl#3#ed_op#/SUP#ed_cl#+c#ed_op#SUP#ed_cl#3#ed_op#/SUP#ed_cl#-3abc#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#第一題可以更快......#ed_op#/DIV#ed_cl#
由 lcflcflcf 於 星期二 七月 25, 2006 1:01 am
3.
x+y+z=0
x^3=-(y^3+3y^2z+3yz^2+z^3)
x^3+y^3+z^3=-18
y^3+z^3-(y^3+3y^2z+3yz^2+z^3)=-18
yz(y+z)=6
解方程且符合兩原方程，得
(x,y,z)=(2,1,-3),(1,2,-3),(2,-3,1),(1,-3,2),(-3,1,2),(-3,2,1)
x+y+z=0
x^3=-(y^3+3y^2z+3yz^2+z^3)
x^3+y^3+z^3=-18
y^3+z^3-(y^3+3y^2z+3yz^2+z^3)=-18
yz(y+z)=6
解方程且符合兩原方程，得
(x,y,z)=(2,1,-3),(1,2,-3),(2,-3,1),(1,-3,2),(-3,1,2),(-3,2,1)
由 lcflcflcf 於 星期二 七月 25, 2006 12:51 am
1.
by Cauchy-Schwarz Inequality
(x^2+y^2+z^2)(1+1+1)≧(x+y+z)^2
9≧9
equality hold if and only if when x^2/1=y^2/1=z^2/1
=>|x|=|y|=|z|
from x^2+y^2+z^2=3
=>|x|=|y|=|z|=1
from x+y+z=3
=>x=y=z=1
by Cauchy-Schwarz Inequality
(x^2+y^2+z^2)(1+1+1)≧(x+y+z)^2
9≧9
equality hold if and only if when x^2/1=y^2/1=z^2/1
=>|x|=|y|=|z|
from x^2+y^2+z^2=3
=>|x|=|y|=|z|=1
from x+y+z=3
=>x=y=z=1
[數學]特殊方程祖
由 aa2191943 於 星期一 七月 24, 2006 11:35 pm
#ed_op#DIV#ed_cl#1.試解該方程組#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl# x + y + z = 3,#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl# x#ed_op#SUP#ed_cl#2 #ed_op#/SUP#ed_cl#+ y#ed_op#SUP#ed_cl#2 #ed_op#/SUP#ed_cl#+ z#ed_op#SUP#ed_cl#2 #ed_op#/SUP#ed_cl#= 3,#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl##ed_op#DIV#ed_cl# x#ed_op#SUP#ed_cl#3 #ed_op#/SUP#ed_cl#+ y#ed_op#SUP#ed_cl#3 #ed_op#/SUP#ed_cl#+ z#ed_op#SUP#ed_cl#3 #ed_op#/SUP#ed_cl#= 3#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl# #ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#2.試解該方程組#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl# x + y + z = 6, #ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl##ed_op#DIV#ed_cl# x#ed_op#SUP#ed_cl#2 #ed_op#/SUP#ed_cl#+ y#ed_op#SUP#ed_cl#2 #ed_op#/SUP#ed_cl#+ z#ed_op#SUP#ed_cl#2 #ed_op#/SUP#ed_cl#= 14,#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl##ed_op#DIV#ed_cl# x#ed_op#SUP#ed_cl#3 #ed_op#/SUP#ed_cl#+ y#ed_op#SUP#ed_cl#3 #ed_op#/SUP#ed_cl#+ z#ed_op#SUP#ed_cl#3 #ed_op#/SUP#ed_cl#= 36#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl# #ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#3.試解該方程組 (x, y, z為整數)#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl# x + y + z = 0, #ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl##ed_op#DIV#ed_cl# x#ed_op#SUP#ed_cl#3 #ed_op#/SUP#ed_cl#+ y#ed_op#SUP#ed_cl#3 #ed_op#/SUP#ed_cl#+ z#ed_op#SUP#ed_cl#3 #ed_op#/SUP#ed_cl#= -18#ed_op#/DIV#ed_cl##ed_op#/DIV#ed_cl##ed_op#/DIV#ed_cl##ed_op#/DIV#ed_cl##ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl##ed_op#/DIV#ed_cl#