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Turkey Team Selection Test
1990 Turkey Team Selection Test
6
n_i | (2^n_(i-1) - 1)
n_i | (2^n_(i-1) - 1)
Source: Turkey TST 1990 - P6
September 11, 2013
abstract algebra
induction
number theory proposed
number theory
Problem Statement
Let
k
≥
2
k\geq 2
k
≥
2
and
n
1
,
…
,
n
k
∈
Z
+
n_1, \dots, n_k \in \mathbf{Z}^+
n
1
,
…
,
n
k
∈
Z
+
. If
n
2
∣
(
2
n
1
−
1
)
n_2 | (2^{n_1} -1)
n
2
∣
(
2
n
1
−
1
)
,
n
3
∣
(
2
n
2
−
1
)
n_3 | (2^{n_2} -1)
n
3
∣
(
2
n
2
−
1
)
,
…
\dots
…
,
n
k
∣
(
2
n
k
−
1
−
1
)
n_k | (2^{n_{k-1}} -1)
n
k
∣
(
2
n
k
−
1
−
1
)
,
n
1
∣
(
2
n
k
−
1
)
n_1 | (2^{n_k} -1)
n
1
∣
(
2
n
k
−
1
)
, show that
n
1
=
⋯
=
n
k
=
1
n_1 = \dots = n_k =1
n
1
=
⋯
=
n
k
=
1
.
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