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Korea Junior Mathematics Olympiad
2002 Korea Junior Math Olympiad
1
2002 KJMO P1
2002 KJMO P1
Source: 2002 KJMO P1
June 29, 2024
algebra
Problem Statement
Find the value of
x
2
+
y
2
+
z
2
x^2+y^2+z^2
x
2
+
y
2
+
z
2
where
x
,
y
,
z
x, y, z
x
,
y
,
z
are non-zero and satisfy the following:(1)
x
+
y
+
z
=
3
x+y+z=3
x
+
y
+
z
=
3
(2)
x
2
(
1
y
+
1
z
)
+
y
2
(
1
z
+
1
x
)
+
z
2
(
1
x
+
1
y
)
=
−
3
x^2(\frac{1}{y}+\frac{1}{z})+y^2(\frac{1}{z}+\frac{1}{x})+z^2(\frac{1}{x}+\frac{1}{y})=-3
x
2
(
y
1
+
z
1
)
+
y
2
(
z
1
+
x
1
)
+
z
2
(
x
1
+
y
1
)
=
−
3
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