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National and Regional Contests
Turkey Contests
National Olympiad First Round
2012 National Olympiad First Round
20
20
Part of
2012 National Olympiad First Round
Problems
(1)
Turkish NMO First Round - 2012 Problem - 20 {Combinatorics}
Source:
7/1/2012
For each permutation
(
a
1
,
a
2
,
…
,
a
11
)
(a_1,a_2,\dots,a_{11})
(
a
1
,
a
2
,
…
,
a
11
)
of the numbers
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
10
,
11
1,2,3,4,5,6,7,8,9,10,11
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
10
,
11
, we can determine at least
k
k
k
of
a
i
a_i
a
i
s when we get
(
a
1
+
a
3
,
a
2
+
a
4
,
a
3
+
a
5
,
…
,
a
8
+
a
10
,
a
9
+
a
11
)
(a_1+a_3, a_2+a_4,a_3+a_5,\dots,a_8+a_{10},a_9+a_{11})
(
a
1
+
a
3
,
a
2
+
a
4
,
a
3
+
a
5
,
…
,
a
8
+
a
10
,
a
9
+
a
11
)
.
k
k
k
can be at most ?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
11
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
6
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
5
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
None
<span class='latex-bold'>(A)</span>\ 11 \qquad <span class='latex-bold'>(B)</span>\ 6 \qquad <span class='latex-bold'>(C)</span>\ 5 \qquad <span class='latex-bold'>(D)</span>\ 2 \qquad <span class='latex-bold'>(E)</span>\ \text{None}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
11
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
6
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
5
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None