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Tuymaada Olympiad
2005 Tuymaada Olympiad
6
6
Part of
2005 Tuymaada Olympiad
Problems
(1)
fractions, integers
Source: Tuymaada 2005, Day 2, Problem 6
7/30/2005
Given are a positive integer
n
n
n
and an infinite sequence of proper fractions
x
0
=
a
0
n
x_0 = \frac{a_0}{n}
x
0
=
n
a
0
,
…
\ldots
…
,
x
i
=
a
i
n
+
i
x_i=\frac{a_i}{n+i}
x
i
=
n
+
i
a
i
, with
a
i
<
n
+
i
a_i < n+i
a
i
<
n
+
i
. Prove that there exist a positive integer
k
k
k
and integers
c
1
c_1
c
1
,
…
\ldots
…
,
c
k
c_k
c
k
such that
c
1
x
1
+
…
+
c
k
x
k
=
1.
c_1 x_1 + \ldots + c_k x_k = 1.
c
1
x
1
+
…
+
c
k
x
k
=
1.
Proposed by M. Dubashinsky
algebra unsolved
algebra