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Vojtěch Jarník IMC
2012 VJIMC
Problem 3
inequality, sum 1/(x+1)=1
inequality, sum 1/(x+1)=1
Source: VJIMC 2012 1.3
May 31, 2021
inequalities
Problem Statement
Determine the smallest real number
C
C
C
such that the inequality
x
y
z
⋅
1
x
+
1
+
y
z
x
⋅
1
y
+
1
+
z
x
y
⋅
1
x
+
1
≤
C
\frac x{\sqrt{yz}}\cdot\frac1{x+1}+\frac y{\sqrt{zx}}\cdot\frac1{y+1}+\frac z{\sqrt{xy}}\cdot\frac1{x+1}\le C
yz
x
⋅
x
+
1
1
+
z
x
y
⋅
y
+
1
1
+
x
y
z
⋅
x
+
1
1
≤
C
holds for all positive real numbers
x
,
y
x,y
x
,
y
and
z
z
z
with
1
x
+
1
+
1
y
+
1
+
1
z
+
1
=
1
\frac1{x+1}+\frac1{y+1}+\frac1{z+1}=1
x
+
1
1
+
y
+
1
1
+
z
+
1
1
=
1
.
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