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Rioplatense Mathematical Olympiad, Level 3
2018 Rioplatense Mathematical Olympiad, Level 3
5
5
Part of
2018 Rioplatense Mathematical Olympiad, Level 3
Problems
(1)
(a_1+d) (a_2+d)... (a_n + d) / (a_1a_2...a_n) is integer for every d>=0
Source: Rioplatense Olympiad 2018 level 3 p5
12/11/2018
Let
n
n
n
be a positive integer. Find all
n
n
n
- rows
(
a
1
,
a
2
,
.
.
.
,
a
n
)
( a_1 , a_2 ,..., a_n )
(
a
1
,
a
2
,
...
,
a
n
)
of different positive integers such that
(
a
1
+
d
)
(
a
2
+
d
)
⋅
⋅
⋅
(
a
n
+
d
)
a
1
a
2
⋅
⋅
⋅
a
n
\frac{(a_1 + d ) (a_2 + d ) \cdot\cdot\cdot ( a_n + d )}{a_1a_2\cdot \cdot \cdot a_n }
a
1
a
2
⋅
⋅
⋅
a
n
(
a
1
+
d
)
(
a
2
+
d
)
⋅
⋅
⋅
(
a
n
+
d
)
is integer for every integer
d
≥
0
d\ge 0
d
≥
0
number theory
Integer
Product
Products
positive integers