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National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
2009 Czech and Slovak Olympiad III A
3
czech republic,third round,2009,problem 3
czech republic,third round,2009,problem 3
Source:
February 25, 2012
inequalities
inequalities proposed
Problem Statement
Find the least value of
x
>
0
x>0
x
>
0
such that for all positive real numbers
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
satisfying
a
b
c
d
=
1
abcd=1
ab
c
d
=
1
, the inequality
a
x
+
b
x
+
c
x
+
d
x
≥
1
a
+
1
b
+
1
c
+
1
d
a^x+b^x+c^x+d^x\ge\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}
a
x
+
b
x
+
c
x
+
d
x
≥
a
1
+
b
1
+
c
1
+
d
1
is true.
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