MathDB
Problems
Contests
National and Regional Contests
China Contests
China Team Selection Test
2013 China Team Selection Test
1
2013 China IMO Team Selection Test 3 Day 1 Q1
2013 China IMO Team Selection Test 3 Day 1 Q1
Source:
April 1, 2013
number theory proposed
number theory
Problem Statement
Let
n
≥
2
n\ge 2
n
≥
2
be an integer.
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\dotsc,a_n
a
1
,
a
2
,
…
,
a
n
are arbitrarily chosen positive integers with
(
a
1
,
a
2
,
…
,
a
n
)
=
1
(a_1,a_2,\dotsc,a_n)=1
(
a
1
,
a
2
,
…
,
a
n
)
=
1
. Let
A
=
a
1
+
a
2
+
⋯
+
a
n
A=a_1+a_2+\dotsb+a_n
A
=
a
1
+
a
2
+
⋯
+
a
n
and
(
A
,
a
i
)
=
d
i
(A,a_i)=d_i
(
A
,
a
i
)
=
d
i
. Let
(
a
2
,
a
3
,
…
,
a
n
)
=
D
1
,
(
a
1
,
a
3
,
…
,
a
n
)
=
D
2
,
…
,
(
a
1
,
a
2
,
…
,
a
n
−
1
)
=
D
n
(a_2,a_3,\dotsc,a_n)=D_1, (a_1,a_3,\dotsc,a_n)=D_2,\dotsc, (a_1,a_2,\dotsc,a_{n-1})=D_n
(
a
2
,
a
3
,
…
,
a
n
)
=
D
1
,
(
a
1
,
a
3
,
…
,
a
n
)
=
D
2
,
…
,
(
a
1
,
a
2
,
…
,
a
n
−
1
)
=
D
n
. Find the minimum of
∏
i
=
1
n
A
−
a
i
d
i
D
i
\prod\limits_{i=1}^n\dfrac{A-a_i}{d_iD_i}
i
=
1
∏
n
d
i
D
i
A
−
a
i
Back to Problems
View on AoPS