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Contests
National and Regional Contests
Turkey Contests
Akdeniz University MO
1999 Akdeniz University MO
3
3
Part of
1999 Akdeniz University MO
Problems
(2)
inequality
Source:
1/31/2016
Let
a
a
a
,
b
b
b
,
c
c
c
and
d
d
d
positive reals. Prove that
1
a
+
b
+
c
+
d
≤
1
64
(
1
a
+
1
b
+
4
c
+
16
d
)
\frac{1}{a+b+c+d} \leq \frac{1}{64}(\frac{1}{a}+\frac{1}{b}+\frac{4}{c}+\frac{16}{d})
a
+
b
+
c
+
d
1
≤
64
1
(
a
1
+
b
1
+
c
4
+
d
16
)
inequalities
algebra
inequalities proposed
inequality
Source:
1/31/2016
For all
x
>
2
x> \sqrt 2
x
>
2
,
y
>
2
y> \sqrt 2
y
>
2
numbers, prove that
x
4
−
x
3
y
+
x
2
y
2
−
x
y
3
+
y
4
>
x
2
+
y
2
x^4-x^3y+x^2y^2-xy^3+y^4>x^2+y^2
x
4
−
x
3
y
+
x
2
y
2
−
x
y
3
+
y
4
>
x
2
+
y
2
inequalities