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2014 Silk Road
4
Nice and Hard FE
Nice and Hard FE
Source: SRMC 2014
March 31, 2014
induction
inequalities
number theory proposed
number theory
Problem Statement
Find all
f
:
N
→
N
f:N\rightarrow N
f
:
N
→
N
, such that
∀
m
,
n
∈
N
\forall m,n\in N
∀
m
,
n
∈
N
2
f
(
m
n
)
≥
f
(
m
2
+
n
2
)
−
f
(
m
)
2
−
f
(
n
)
2
≥
2
f
(
m
)
f
(
n
)
2f(mn) \geq f(m^2+n^2)-f(m)^2-f(n)^2 \geq 2f(m)f(n)
2
f
(
mn
)
≥
f
(
m
2
+
n
2
)
−
f
(
m
)
2
−
f
(
n
)
2
≥
2
f
(
m
)
f
(
n
)
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