1/Ta có(ax2+by2+cz2)/2000= (ax2+by2+cz2)(a+b+c)=
=a2x2+abx2+acx2+aby2+b2y2+bcy2+acz2+bc...
=(abx2+aby2+bcy2+bcz2+acx2+acz2)+(a2x2... (1)
từ ax + by + cz = 0
=> a2x2+b2y2+c2z2+2(abxy+bcyz+acxz)=0
=> a2x2+b2y2+c2z2= - 2(abxy+bcyz+acxz) (2)
Thay (2) vào (1) có
(ax2+by2+cz2)/2000=
=(abx2+aby2+bcy2+bcz2+acx2+acz2)-2(abx...
=ab(x-y)2+bc(y-z)2+ca(z-x)2
=>dpcm
2/Xem bài 5 trước
3/ a2 + b2 + (a - b)2 = c2 + d2 + (c - d)2.
=> a4+b4+(a-b)4+2[a2b2+a2(a-b)2+b2(a-b)2]=
=c4+d4+(c-d)4+2[c2d2+c2(c-d)2+d2(c-d)2...
<=>a4+b4+(a-b)4+2[a2b2+(a2+b2)(a-b)2]
=c4+d4+(c-d)4+2[c2d2+(c2+d2)(c-d)2 (1)
a2 + b2 + (a - b)2 = c2 + d2 + (c - d)2.
=> 2(a2+b2-ab) =2(c2+d2-cd)
=>(a2+b2-ab) =(c2+d2-cd)
=>(a2+b2)2+a2b2-2ab(a2+b2)=(c2+d2)2+c2...
=>a2b2+(a2+b2)(a2+b2-2ab)=c2d2+(c2+d2)...
=>a2b2+(a2+b2)(a-b)2=c2d2+(c2+d2)(c-d)... (2)
từ (1) (2) => dpcm
4/B = a4 + b4 + c4=(a2+b2+c2)^2-2(a2b2+b2c2+c2a2)
B= 14^2 -2(a2b2+b2c2+c2a2) (1)
từ a+b+c=0 =>a= -(b+c)
=>a2=b2+c2+2bc
=> a2-b2-c2=2bc
=> a4+b4+c4-2a2b2-2a2c2+2b2c2=4b2c2
=>B=a4+b4+c4=2(a2b2+a2c2+b2c2) (2)
(1) (2) => 2B= 14^2 =196=>B=98
5/ a3+b3+c3-3abc= (a+b)^3-3ab(a+b)+c^3-3abc
=(a+b)^3+c^3-3ab(a+b+c)
=(a+b+c)[(a+b)^2+c^2-(a+b)c-3ab]
=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)=0
=>a+b+c=0 (1) hoặc a^2+b^2+c^2-ab-bc-ac=0 (2)
+nếu (1) xảy ra
=> a+b=-c;b+c=-a;c+a=-b
=>A =[(b+a)/b]x[(c+b)/c]x[(a+c)/a]=-abc/abc=...
+Nếu (2) xảy ra =>2a^2+2b^2+2c^2-2ab-2bc-2ca=0
=> (a-b)^2+(b-c)^2+(c-a)^2=0
=>a=b=c
=>A=2x2x2=8
