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Turkey Contests
Turkey Team Selection Test
2010 Turkey Team Selection Test
2
Turkey TST 2010 Q2
Turkey TST 2010 Q2
Source:
September 1, 2010
inequalities
inequalities proposed
Problem Statement
Show that
∑
c
y
c
(
a
2
+
b
2
)
(
a
2
−
a
b
+
b
2
)
2
4
≤
2
3
(
a
2
+
b
2
+
c
2
)
(
1
a
+
b
+
1
b
+
c
+
1
c
+
a
)
\sum_{cyc} \sqrt[4]{\frac{(a^2+b^2)(a^2-ab+b^2)}{2}} \leq \frac{2}{3}(a^2+b^2+c^2)\left(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right)
cyc
∑
4
2
(
a
2
+
b
2
)
(
a
2
−
ab
+
b
2
)
≤
3
2
(
a
2
+
b
2
+
c
2
)
(
a
+
b
1
+
b
+
c
1
+
c
+
a
1
)
for all positive real numbers
a
,
b
,
c
.
a, \: b, \: c.
a
,
b
,
c
.
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