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2017 Turkey EGMO TST
3
Turkey EGMO TST 2017 P3
Turkey EGMO TST 2017 P3
Source: Turkey EGMO TST 2017 P3
June 1, 2017
Turkey
EGMO
TST
algebra
inequalities
contest problem
Problem Statement
For all positive real numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
satisfying the inequality
x
y
z
+
y
z
x
+
z
x
y
≤
3
,
\frac{xy}{z}+\frac{yz}{x}+\frac{zx}{y}\leq 3,
z
x
y
+
x
yz
+
y
z
x
≤
3
,
prove that
x
2
y
3
+
y
2
z
3
+
z
2
x
3
≥
x
y
+
y
z
+
z
x
.
\frac{x^2}{y^3}+\frac{y^2}{z^3}+\frac{z^2}{x^3}\geq \frac{x}{y}+\frac{y}{z}+\frac{z}{x}.
y
3
x
2
+
z
3
y
2
+
x
3
z
2
≥
y
x
+
z
y
+
x
z
.
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