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Jozsef Wildt International Math Competition
2019 Jozsef Wildt International Math Competition
W. 50
Prove this 3 variable inequality
Prove this 3 variable inequality
Source: 2019 Jozsef Wildt International Math Competition
May 19, 2020
inequalities
Problem Statement
Let
x
x
x
,
y
y
y
,
z
>
0
z > 0
z
>
0
,
λ
∈
(
−
∞
,
0
)
∪
(
1
,
+
∞
)
\lambda \in (-\infty, 0) \cup (1,+\infty)
λ
∈
(
−
∞
,
0
)
∪
(
1
,
+
∞
)
such that
x
+
y
+
z
=
1
x + y + z = 1
x
+
y
+
z
=
1
. Then
∑
c
y
c
x
λ
y
λ
∑
c
y
c
1
(
x
+
y
)
2
λ
≥
9
(
1
4
−
1
9
∑
c
y
c
1
(
x
+
1
)
2
)
λ
\sum \limits_{cyc} x^{\lambda}y^{\lambda}\sum \limits_{cyc}\frac{1}{(x+y)^{2\lambda}}\geq 9\left(\frac{1}{4}-\frac{1}{9}\sum \limits_{cyc}\frac{1}{(x+1)^2} \right)^{\lambda}
cyc
∑
x
λ
y
λ
cyc
∑
(
x
+
y
)
2
λ
1
≥
9
(
4
1
−
9
1
cyc
∑
(
x
+
1
)
2
1
)
λ
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