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2010 Turkey Team Selection Test
1
Turkey TST 2010 Q4
Turkey TST 2010 Q4
Source:
September 1, 2010
modular arithmetic
number theory proposed
number theory
Problem Statement
Let
0
≤
k
<
n
0 \leq k < n
0
≤
k
<
n
be integers and
A
=
{
a
:
a
≡
k
(
m
o
d
n
)
}
.
A=\{a \: : \: a \equiv k \pmod n \}.
A
=
{
a
:
a
≡
k
(
mod
n
)}
.
Find the smallest value of
n
n
n
for which the expression
a
m
+
3
m
a
2
−
3
a
+
1
\frac{a^m+3^m}{a^2-3a+1}
a
2
−
3
a
+
1
a
m
+
3
m
does not take any integer values for
(
a
,
m
)
∈
A
×
Z
+
.
(a,m) \in A \times \mathbb{Z^+}.
(
a
,
m
)
∈
A
×
Z
+
.
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