若p+q+r=1,x的平方-yz分之p=y的平方-zx分之q=z的平方-xy分之r,求x+y+z分之px+qy+rz的值
问题描述：
若p+q+r=1,x的平方-yz分之p=y的平方-zx分之q=z的平方-xy分之r,求x+y+z分之px+qy+rz的值
答
设p/(x^2-yz)=q/(y^2-xz)=r/(z^2-xy)=k
得p=k(x^2-yz)
q=k(y^2-xz)
r=k(z^2-xy)
p+q+r=k(x^2+y^2+z^2-xy-yz-xz)=1
(px+qy+rz)/(x+y+z) 将p q r 代入
=k(x^3+y^3+z^3-3xyz)/(x+y+z)
=k(x^2+y^2+z^2-xy-yz-xz)(x+y+z)/(x+y+z)=k(x^2+y^2+z^2-xy-yz-xz)=1
答
P/(X^2-YZ)=Q/(Y^2-ZX)=R/(Z^2-XY)=KP=K(X^2-YZ)Q=K(Y^2-ZX)R=K(Z^2-XY)P+Q+R=1K(X^2-YZ)+K(Y^2-ZX)+K(Z^2-XY)=1K(X^2-YZ+Y^2-ZX+Z^2-XY)=1K=1/(X^2-YZ+Y^2-ZX+Z^2-XY)XP=K(X^3-XYZ)YQ=K(Y^3-XYZ)ZR=K(Z^3-XYZ)XP+YQ...