MathDB
Problems
Contests
National and Regional Contests
Russia Contests
Russian Team Selection Tests
Russian TST 2014
P1
An asymmetric inequality
An asymmetric inequality
Source: Russian TST 2015, Day 7 P1 (Groups A & B)
April 21, 2023
algebra
inequalities
Problem Statement
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be positive real numbers. Prove that
x
y
+
y
z
+
z
x
⩾
z
(
x
+
y
)
y
(
y
+
z
)
+
x
(
y
+
z
)
z
(
z
+
x
)
+
y
(
z
+
x
)
x
(
x
+
y
)
.
\frac{x}{y}+\frac{y}{z}+\frac{z}{x}\geqslant\frac{z(x+y)}{y(y+z)}+\frac{x(y+z)}{z(z+x)}+\frac{y(z+x)}{x(x+y)}.
y
x
+
z
y
+
x
z
⩾
y
(
y
+
z
)
z
(
x
+
y
)
+
z
(
z
+
x
)
x
(
y
+
z
)
+
x
(
x
+
y
)
y
(
z
+
x
)
.
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