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Austrian-Polish
2005 Austrian-Polish Competition
10
Inequality with powers
Inequality with powers
Source: Austrian-Polish 2005, Problem 10
July 5, 2015
inequalities
powers
Problem Statement
Determine all pairs
(
k
,
n
)
(k,n)
(
k
,
n
)
of non-negative integers such that the following inequality holds
∀
x
,
y
>
0
\forall x,y>0
∀
x
,
y
>
0
:
1
+
y
n
x
k
≥
(
1
+
y
)
n
(
1
+
x
)
k
.
1+ \frac{y^n}{x^k} \geq \frac{(1+y)^n}{(1+x)^k}.
1
+
x
k
y
n
≥
(
1
+
x
)
k
(
1
+
y
)
n
.
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