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Mexico National Olympiad
1997 Mexico National Olympiad
6
6
Part of
1997 Mexico National Olympiad
Problems
(1)
infinite representations of 1 =1/5+Σ 1/a_i , from i=1,..., n & 5<a_1<...<a_n
Source: Mexican Mathematical Olympiad 1997 OMM P6
7/28/2018
Prove that number
1
1
1
has infinitely many representations of the form
1
=
1
5
+
1
a
1
+
1
a
2
+
.
.
.
+
1
a
n
1 =\frac{1}{5}+\frac{1}{a_1}+\frac{1}{a_2}+ ...+\frac{1}{a_n}
1
=
5
1
+
a
1
1
+
a
2
1
+
...
+
a
n
1
, where
n
n
n
and
a
i
a_i
a
i
are positive integers with
5
<
a
1
<
a
2
<
.
.
.
<
a
n
5 < a_1 < a_2 < ... < a_n
5
<
a
1
<
a
2
<
...
<
a
n
.
number theory
representation