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1980 All Soviet Union Mathematical Olympiad
299
ASU 299 All Soviet Union MO 1980 x<1/3(p/4-sqrt(d^2-s/2)) in parallelepiped
ASU 299 All Soviet Union MO 1980 x<1/3(p/4-sqrt(d^2-s/2)) in parallelepiped
Source:
July 19, 2019
geometric inequality
inequalities
geometry
3D geometry
parallelepiped
Problem Statement
Let the edges of rectangular parallelepiped be
x
,
y
x,y
x
,
y
and
z
z
z
(
x
<
y
<
z
x<y<z
x
<
y
<
z
). Let
p
=
4
(
x
+
y
+
z
)
,
s
=
2
(
x
y
+
y
z
+
z
x
)
a
n
d
d
=
x
2
+
y
2
+
z
2
p=4(x+y+z), s=2(xy+yz+zx) \,\,\, and \,\,\, d=\sqrt{x^2+y^2+z^2}
p
=
4
(
x
+
y
+
z
)
,
s
=
2
(
x
y
+
yz
+
z
x
)
an
d
d
=
x
2
+
y
2
+
z
2
be its perimeter, surface area and diagonal length, respectively. Prove that
x
<
1
3
(
p
4
−
d
2
−
s
2
)
a
n
d
z
>
1
3
(
p
4
−
d
2
−
s
2
)
x < \frac{1}{3}\left( \frac{p}{4}- \sqrt{d^2 - \frac{s}{2}}\right )\,\,\, and \,\,\, z > \frac{1}{3}\left( \frac{p}{4}- \sqrt{d^2 - \frac{s}{2}}\right )
x
<
3
1
(
4
p
−
d
2
−
2
s
)
an
d
z
>
3
1
(
4
p
−
d
2
−
2
s
)
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