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2022 South Africa National Olympiad
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Part of
2022 South Africa National Olympiad
Problems
(1)
Nice problem on polynomials, x^2+y^2+z^2+2xyz=1
Source: SAMO, 2022, R3, P6
7/28/2022
Show that there are infinitely many polynomials P with real coefficients such that if x, y, and z are real numbers such that
x
2
+
y
2
+
z
2
+
2
x
y
z
=
1
x^2+y^2+z^2+2xyz=1
x
2
+
y
2
+
z
2
+
2
x
yz
=
1
, then
P
(
x
)
2
+
P
(
y
)
2
+
P
(
z
)
2
+
2
P
(
x
)
P
(
y
)
P
(
z
)
=
1
P\left(x\right)^2+P\left(y\right)^2+P\left(z\right)^2+2P\left(x\right)P\left(y\right)P\left(z\right) = 1
P
(
x
)
2
+
P
(
y
)
2
+
P
(
z
)
2
+
2
P
(
x
)
P
(
y
)
P
(
z
)
=
1
algebra
polynomial