已知ax3=by3=cz3，且1x+1y+1z=1．求证：3ax2+by2+cz2=3a+3b+3c．

Question

已知ax³=by³=cz³，且(1)/(x)+(1)/(y)+(1)/(z)=1．
求证：

3	ax2+by2+cz2

=

3

a

+

3

b

+

3

c

．

Answer 1

证明：设ax³=by³=cz³=t³，则a=(t3)/(x3)，b=(t3)/(y3)，c=(t3)/(z3)，
因为

3

a

+

3

b

+

3

c

=t（(1)/(x)+(1)/(y)+(1)/(z)）=t，
又因为

3	ax2+by2+cz2

=

3	ax3•(1)/(x)+by3•(1)/(y)+cz3•(1)/(z)

=

3	t3((1)/(x)+(1)/(y)+(1)/(z))

=t．
所以原式成立．