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Sum of x_i and y_i is 0 [ILL 1977]
Sum of x_i and y_i is 0 [ILL 1977]
Source:
January 11, 2011
algebra proposed
algebra
Problem Statement
Given that
x
1
+
x
2
+
x
3
=
y
1
+
y
2
+
y
3
=
x
1
y
1
+
x
2
y
2
+
x
3
y
3
=
0
,
x_1+x_2+x_3=y_1+y_2+y_3=x_1y_1+x_2y_2+x_3y_3=0,
x
1
+
x
2
+
x
3
=
y
1
+
y
2
+
y
3
=
x
1
y
1
+
x
2
y
2
+
x
3
y
3
=
0
,
prove that:
x
1
2
x
1
2
+
x
2
2
+
x
3
2
+
y
1
2
y
1
2
+
y
2
2
+
y
3
2
=
2
3
\frac{x_1^2}{x_1^2+x_2^2+x_3^2}+\frac{y_1^2}{y_1^2+y_2^2+y_3^2}=\frac{2}{3}
x
1
2
+
x
2
2
+
x
3
2
x
1
2
+
y
1
2
+
y
2
2
+
y
3
2
y
1
2
=
3
2
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