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Contests
National and Regional Contests
Iran Contests
Iran Team Selection Test
2014 Iran Team Selection Test
3
hard question 3
hard question 3
Source: iran tst 2014 third exam
May 22, 2014
algebra
polynomial
function
algebra unsolved
Problem Statement
let
m
,
n
∈
N
m,n\in \mathbb{N}
m
,
n
∈
N
and
p
(
x
)
,
q
(
x
)
,
h
(
x
)
p(x),q(x),h(x)
p
(
x
)
,
q
(
x
)
,
h
(
x
)
are polynomials with real Coefficients such that
p
(
x
)
p(x)
p
(
x
)
is Descending. and for all
x
∈
R
x\in \mathbb{R}
x
∈
R
p
(
q
(
n
x
+
m
)
+
h
(
x
)
)
=
n
(
q
(
p
(
x
)
)
+
h
(
x
)
)
+
m
p(q(nx+m)+h(x))=n(q(p(x))+h(x))+m
p
(
q
(
n
x
+
m
)
+
h
(
x
))
=
n
(
q
(
p
(
x
))
+
h
(
x
))
+
m
. prove that dont exist function
f
:
R
→
R
f:\mathbb{R}\rightarrow \mathbb{R}
f
:
R
→
R
such that for all
x
∈
R
x\in \mathbb{R}
x
∈
R
f
(
q
(
p
(
x
)
)
+
h
(
x
)
)
=
f
(
x
)
2
+
1
f(q(p(x))+h(x))=f(x)^{2}+1
f
(
q
(
p
(
x
))
+
h
(
x
))
=
f
(
x
)
2
+
1
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