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Problems
Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
1999 National Olympiad First Round
35
35
Part of
1999 National Olympiad First Round
Problems
(1)
Combinatorics
Source: 0
4/21/2009
Flights are arranged between 13 countries. For
k
≥
2
k\ge 2
k
≥
2
, the sequence
A
1
,
A
2
,
…
A
k
A_{1} ,A_{2} ,\ldots A_{k}
A
1
,
A
2
,
…
A
k
is said to a cycle if there exist a flight from
A
1
A_{1}
A
1
to
A
2
A_{2}
A
2
, from
A
2
A_{2}
A
2
to
A
3
A_{3}
A
3
,
…
\ldots
…
, from A_{k \minus{} 1} to
A
k
A_{k}
A
k
, and from
A
k
A_{k}
A
k
to
A
1
A_{1}
A
1
. What is the smallest possible number of flights such that how the flights are arranged, there exist a cycle?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
14
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
53
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
66
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
79
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
156
<span class='latex-bold'>(A)</span>\ 14 \qquad<span class='latex-bold'>(B)</span>\ 53 \qquad<span class='latex-bold'>(C)</span>\ 66 \qquad<span class='latex-bold'>(D)</span>\ 79 \qquad<span class='latex-bold'>(E)</span>\ 156
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
14
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
53
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
66
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
79
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
156
pigeonhole principle