MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2005 National Olympiad First Round
2
2
Part of
2005 National Olympiad First Round
Problems
(1)
P02 [Number Theory] - Turkish NMO 1st Round - 2005
Source:
10/26/2013
Let
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \dots, a_n
a
1
,
a
2
,
…
,
a
n
be positive integers such that none of them is a multiple of
5
5
5
. What is the largest integer
n
<
2005
n<2005
n
<
2005
, such that
a
1
4
+
a
2
4
+
⋯
+
a
n
4
a_1^4 + a_2^4 + \cdots + a_n^4
a
1
4
+
a
2
4
+
⋯
+
a
n
4
is divisible by
5
5
5
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
2000
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
2001
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
2002
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
2003
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
2004
<span class='latex-bold'>(A)</span>\ 2000 \qquad<span class='latex-bold'>(B)</span>\ 2001 \qquad<span class='latex-bold'>(C)</span>\ 2002 \qquad<span class='latex-bold'>(D)</span>\ 2003 \qquad<span class='latex-bold'>(E)</span>\ 2004
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
2000
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
2001
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
2002
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
2003
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
2004
modular arithmetic