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2021 The Chinese Mathematics Competition
Problem 8
2021 CMC (Non-Math Major) Problem 8
2021 CMC (Non-Math Major) Problem 8
Source: 2021 CMC (Non-Math Major) Problem 8
November 25, 2022
calculus
Problem Statement
Consider a homogeneous function with degree
4
4
4
.
f
(
x
,
y
,
z
)
=
a
1
x
4
+
a
2
y
4
+
a
3
z
4
+
3
a
4
x
2
y
2
+
3
a
5
y
2
z
2
+
3
a
6
x
2
z
2
f(x,y,z)=a_1x^4+a_2y^4+a_3z^4+3a_4x^2y^2+3a_5y^2z^2+3a_6x^2z^2
f
(
x
,
y
,
z
)
=
a
1
x
4
+
a
2
y
4
+
a
3
z
4
+
3
a
4
x
2
y
2
+
3
a
5
y
2
z
2
+
3
a
6
x
2
z
2
. Find
∯
∑
f
(
x
,
y
,
z
)
d
S
\oiint_{\sum} f(x,y,z)dS
∬
∑
f
(
x
,
y
,
z
)
d
S
, where
∑
:
x
2
+
y
2
+
z
2
=
1
\sum: x^2+y^2+z^2=1
∑
:
x
2
+
y
2
+
z
2
=
1
.
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