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2008 Princeton University Math Competition
A9
A9
Part of
2008 Princeton University Math Competition
Problems
(2)
2008 PUMaC Geometry A9
Source:
11/9/2012
In tetrahedron
A
B
C
D
ABCD
A
BC
D
with circumradius
2
2
2
,
A
B
=
2
AB = 2
A
B
=
2
,
C
D
=
7
CD = \sqrt{7}
C
D
=
7
, and
∠
A
B
C
=
∠
B
A
D
=
π
2
\angle ABC = \angle BAD = \frac{\pi}{2}
∠
A
BC
=
∠
B
A
D
=
2
π
. Find all possible angles between the planes containing
A
B
C
ABC
A
BC
and
A
B
D
ABD
A
B
D
.
geometry
3D geometry
tetrahedron
circumcircle
2008 PUMaC Number Theory A9
Source:
10/4/2019
Find the number of positive integer solutions of
(
x
2
+
2
)
(
y
2
+
3
)
(
z
2
+
4
)
=
60
x
y
z
(x^2 + 2)(y^2 + 3)(z^2 + 4) = 60xyz
(
x
2
+
2
)
(
y
2
+
3
)
(
z
2
+
4
)
=
60
x
yz
.
number theory