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Official Ukraine Selection Cycle
Ukraine Team Selection Test
2010 Ukraine Team Selection Test
12
12
Part of
2010 Ukraine Team Selection Test
Problems
(1)
r=b+\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}
Source: Ukraine TST 2010 p12
5/5/2020
Is there a positive integer
n
n
n
for which the following holds: for an arbitrary rational
r
r
r
there exists an integer
b
b
b
and non-zero integers
a
1
,
a
2
,
.
.
.
,
a
n
a _1, a_2, ..., a_n
a
1
,
a
2
,
...
,
a
n
such that
r
=
b
+
1
a
1
+
1
a
2
+
.
.
.
+
1
a
n
r=b+\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}
r
=
b
+
a
1
1
+
a
2
1
+
...
+
a
n
1
?
algebra
rational
Sum
reciprocal sum
number theory