MathDB
Problems
Contests
National and Regional Contests
Japan Contests
Japan MO Finals
2024 Japan MO Finals
3
3
Part of
2024 Japan MO Finals
Problems
(1)
Z-Shaped Polyline
Source: 2024 Japan MO P3
2/11/2024
In the
x
y
xy
x
y
-plane, points with integer coordinates ranging from
1
1
1
to
2000
2000
2000
for both the
x
x
x
and
y
y
y
coordinates are called good points. Moreover, for any four points
A
(
x
1
,
y
1
)
,
B
(
x
2
,
y
2
)
,
C
(
x
3
,
y
3
)
,
D
(
x
4
,
y
4
)
A(x_1, y_1), B(x_2, y_2), C(x_3, y_3), D(x_4, y_4)
A
(
x
1
,
y
1
)
,
B
(
x
2
,
y
2
)
,
C
(
x
3
,
y
3
)
,
D
(
x
4
,
y
4
)
satisfying the following conditions, the polyline
A
B
C
D
ABCD
A
BC
D
is called a
Z
\mathbb{Z}
Z
-shaped polyline.[*] All
A
,
B
,
C
,
D
A, B, C, D
A
,
B
,
C
,
D
are good points. [*]
x
1
<
x
2
x_1<x_2
x
1
<
x
2
,
y
1
=
y
2
y_1=y_2
y
1
=
y
2
[*]
x
2
>
x
3
x_2>x_3
x
2
>
x
3
,
y
2
−
x
2
=
y
3
−
x
3
y_2-x_2=y_3-x_3
y
2
−
x
2
=
y
3
−
x
3
[*]
x
3
<
x
4
x_3<x_4
x
3
<
x
4
,
y
3
=
y
4
y_3=y_4
y
3
=
y
4
Find the smallest possible positive integer
n
n
n
such that
Z
\mathbb{Z}
Z
-shaped polylines
Z
1
,
Z
2
,
…
,
Z
n
Z_1, Z_2,\dots, Z_n
Z
1
,
Z
2
,
…
,
Z
n
satisfy the following condition: for any good point
P
P
P
, there exists an integer
i
i
i
between
1
1
1
and
n
n
n
inclusive such that
P
P
P
lies on
Z
i
Z_i
Z
i
.
combinatorics