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Vojtěch Jarník IMC
2012 VJIMC
Problem 4
inequality in 7 variables, [1,4]
inequality in 7 variables, [1,4]
Source: VJIMC 2012 2.4
June 1, 2021
inequalities
Problem Statement
Let
a
,
b
,
c
,
x
,
y
,
z
,
t
a,b,c,x,y,z,t
a
,
b
,
c
,
x
,
y
,
z
,
t
be positive real numbers with
1
≤
x
,
y
,
z
≤
4
1\le x,y,z\le4
1
≤
x
,
y
,
z
≤
4
. Prove that
x
(
2
a
)
t
+
y
(
2
b
)
t
+
z
(
2
c
)
t
≥
y
+
z
−
x
(
b
+
c
)
t
+
z
+
x
−
y
(
c
+
a
)
t
+
x
+
y
−
z
(
a
+
b
)
t
.
\frac x{(2a)^t}+\frac y{(2b)^t}+\frac z{(2c)^t}\ge\frac{y+z-x}{(b+c)^t}+\frac{z+x-y}{(c+a)^t}+\frac{x+y-z}{(a+b)^t}.
(
2
a
)
t
x
+
(
2
b
)
t
y
+
(
2
c
)
t
z
≥
(
b
+
c
)
t
y
+
z
−
x
+
(
c
+
a
)
t
z
+
x
−
y
+
(
a
+
b
)
t
x
+
y
−
z
.
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