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Poland - Second Round
1958 Poland - Second Round
4
(a + b + c)^2 = 3 (ab + bc + ac - x^2 - y^2 - z^2),
(a + b + c)^2 = 3 (ab + bc + ac - x^2 - y^2 - z^2),
Source: Polish MO second round 1958 p4
August 29, 2024
algebra
Problem Statement
Prove that if
(
a
+
b
+
c
)
2
=
3
(
a
b
+
b
c
+
a
c
−
x
2
−
y
2
−
z
2
)
,
(a + b + c)^2 = 3 (ab + bc + ac - x^2 - y^2 - z^2),
(
a
+
b
+
c
)
2
=
3
(
ab
+
b
c
+
a
c
−
x
2
−
y
2
−
z
2
)
,
where
a
a
a
,
b
b
b
,
c
c
c
,
x
x
x
,
y
y
y
,
z
z
z
denote real numbers, then
a
=
b
=
c
a = b = c
a
=
b
=
c
and
x
=
y
=
z
=
0
x = y = z = 0
x
=
y
=
z
=
0
.
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