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Jozsef Wildt International Math Competition
2019 Jozsef Wildt International Math Competition
W. 68
Prove these two inequalities of tetrahedron
Prove these two inequalities of tetrahedron
Source: 2019 Jozsef Wildt International Math Competition
May 20, 2020
tetrahedron
3D geometry
inequalities
exradius
inradius
Problem Statement
In all tetrahedron
A
B
C
D
ABCD
A
BC
D
holds[*]
∑
c
y
c
h
a
−
r
h
a
+
r
≥
∑
c
y
c
h
a
t
−
r
t
(
h
a
+
r
)
t
\displaystyle{\sum \limits_{cyc}\frac{h_a-r}{h_a+r}\geq \sum \limits_{cyc}\frac{h_a^t-r^t}{(h_a+r)^t}}
cyc
∑
h
a
+
r
h
a
−
r
≥
cyc
∑
(
h
a
+
r
)
t
h
a
t
−
r
t
[*]
∑
c
y
c
2
r
a
−
r
2
r
a
+
r
≥
∑
c
y
c
2
r
a
t
−
r
t
(
2
r
a
+
r
)
t
\displaystyle{\sum \limits_{cyc}\frac{2r_a-r}{2r_a+r}\geq \sum \limits_{cyc}\frac{2r_a^t-r^t}{(2r_a+r)^t}}
cyc
∑
2
r
a
+
r
2
r
a
−
r
≥
cyc
∑
(
2
r
a
+
r
)
t
2
r
a
t
−
r
t
for all
t
∈
[
0
,
1
]
t\in [0,1]
t
∈
[
0
,
1
]
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