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National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2016 Korea Junior Math Olympiad
8
8
Part of
2016 Korea Junior Math Olympiad
Problems
(1)
number of paths
Source: 2016 KJMO #8
11/14/2016
One moving point in the coordinate plane can move right or up one position.
N
N
N
is a number of all paths : paths that moving point starts from
(
0
,
0
)
(0, 0)
(
0
,
0
)
, without passing
(
1
,
0
)
,
(
2
,
1
)
,
.
.
.
,
(
n
,
n
−
1
)
(1, 0), (2, 1), . . . , (n, n-1)
(
1
,
0
)
,
(
2
,
1
)
,
...
,
(
n
,
n
−
1
)
and moves
2
n
2n
2
n
times to
(
n
,
n
)
(n, n)
(
n
,
n
)
.
a
k
a_k
a
k
is a number of special paths : paths include in
N
N
N
, but
k
k
k
th moves to the right,
k
+
1
k+1
k
+
1
th moves to the up. find
1
N
(
a
1
+
a
2
+
.
.
.
+
a
2
n
−
1
)
\frac{1}{N} (a_1+a_2+ . . . + a_{2n-1})
N
1
(
a
1
+
a
2
+
...
+
a
2
n
−
1
)
combinatorics