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National and Regional Contests
Ireland Contests
Ireland National Math Olympiad
2014 Irish Math Olympiad
8
8
Part of
2014 Irish Math Olympiad
Problems
(1)
if Q(i),Q(i+1),Q(i+2),Q(i+3) are integers for 1 integer i, then Q(n) is integer
Source: Irmo 2014 p2 q8
9/15/2018
(a) Let
a
0
,
a
1
,
a
2
a_0, a_1,a_2
a
0
,
a
1
,
a
2
be real numbers and consider the polynomial
P
(
x
)
=
a
0
+
a
1
x
+
a
2
x
2
P(x) = a_0 + a_1x + a_2x^2
P
(
x
)
=
a
0
+
a
1
x
+
a
2
x
2
. Assume that
P
(
−
1
)
,
P
(
0
)
P(-1), P(0)
P
(
−
1
)
,
P
(
0
)
and
P
(
1
)
P(1)
P
(
1
)
are integers. Prove that
P
(
n
)
P(n)
P
(
n
)
is an integer for all integers
n
n
n
. (b) Let
a
0
,
a
1
,
a
2
,
a
3
a_0,a_1, a_2, a_3
a
0
,
a
1
,
a
2
,
a
3
be real numbers and consider the polynomial
Q
(
x
)
=
a
0
+
a
1
x
+
a
2
x
2
+
a
3
x
3
Q(x) = a0 + a_1x + a_2x^2 + a_3x^3
Q
(
x
)
=
a
0
+
a
1
x
+
a
2
x
2
+
a
3
x
3
. Assume that there exists an integer
i
i
i
such that
Q
(
i
)
,
Q
(
i
+
1
)
,
Q
(
i
+
2
)
Q(i),Q(i+1),Q(i+2)
Q
(
i
)
,
Q
(
i
+
1
)
,
Q
(
i
+
2
)
and
Q
(
i
+
3
)
Q(i+3)
Q
(
i
+
3
)
are integers. Prove that
Q
(
n
)
Q(n)
Q
(
n
)
is an integer for all integers
n
n
n
.
algebra
polynomial
Integer