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Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
1970 Canada National Olympiad
10
10
Part of
1970 Canada National Olympiad
Problems
(1)
Never 8
Source: Canada 1970, Problem 10
5/14/2006
Given the polynomial
f
(
x
)
=
x
n
+
a
1
x
n
−
1
+
a
2
x
n
−
2
+
⋯
+
a
n
−
1
x
+
a
n
f(x)=x^n+a_{1}x^{n-1}+a_{2}x^{n-2}+\cdots+a_{n-1}x+a_n
f
(
x
)
=
x
n
+
a
1
x
n
−
1
+
a
2
x
n
−
2
+
⋯
+
a
n
−
1
x
+
a
n
with integer coefficients
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\ldots,a_n
a
1
,
a
2
,
…
,
a
n
, and given also that there exist four distinct integers
a
a
a
,
b
b
b
,
c
c
c
and
d
d
d
such that
f
(
a
)
=
f
(
b
)
=
f
(
c
)
=
f
(
d
)
=
5
,
f(a)=f(b)=f(c)=f(d)=5,
f
(
a
)
=
f
(
b
)
=
f
(
c
)
=
f
(
d
)
=
5
,
show that there is no integer
k
k
k
such that
f
(
k
)
=
8
f(k)=8
f
(
k
)
=
8
.
algebra
polynomial