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Problems
Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2008 National Olympiad First Round
20
20
Part of
2008 National Olympiad First Round
Problems
(1)
Turkey NMO 2008 1st Round - P20 (Combinatorics)
Source:
8/25/2012
Each of the integers
a
1
,
a
2
,
a
3
,
…
,
a
2008
a_1,a_2,a_3,\dots,a_{2008}
a
1
,
a
2
,
a
3
,
…
,
a
2008
is at least
1
1
1
and at most
5
5
5
. If
a
n
<
a
n
+
1
a_n < a_{n+1}
a
n
<
a
n
+
1
, the pair
(
a
n
,
a
n
+
1
)
(a_n, a_{n+1})
(
a
n
,
a
n
+
1
)
will be called as an increasing pair. If
a
n
>
a
n
+
1
a_n > a_{n+1}
a
n
>
a
n
+
1
, the pair
(
a
n
,
a
n
+
1
)
(a_n, a_{n+1})
(
a
n
,
a
n
+
1
)
will be called as an decreasing pair. If the sequence contains
103
103
103
increasing pairs, at least how many decreasing pairs are there?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
21
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
24
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
36
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
102
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
None of the above
<span class='latex-bold'>(A)</span>\ 21 \qquad<span class='latex-bold'>(B)</span>\ 24 \qquad<span class='latex-bold'>(C)</span>\ 36 \qquad<span class='latex-bold'>(D)</span>\ 102 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the above}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
21
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
24
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
36
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
102
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of the above
ceiling function