MathDB
Problems
Contests
National and Regional Contests
Taiwan Contests
TST Round 3
2016 Taiwan TST Round 3
1
Inequality
Inequality
Source: 2016 Taiwan TST Round 3
July 23, 2016
inequalities
Problem Statement
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be positive real numbers satisfying
x
+
y
+
z
=
1
x+y+z=1
x
+
y
+
z
=
1
. Find the smallest
k
k
k
such that
x
2
y
2
1
−
z
+
y
2
z
2
1
−
x
+
z
2
x
2
1
−
y
≤
k
−
3
x
y
z
\frac{x^2y^2}{1-z}+\frac{y^2z^2}{1-x}+\frac{z^2x^2}{1-y}\leq k-3xyz
1
−
z
x
2
y
2
+
1
−
x
y
2
z
2
+
1
−
y
z
2
x
2
≤
k
−
3
x
yz
.
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