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National and Regional Contests
Brazil Contests
Brazil National Olympiad
1991 Brazil National Olympiad
5
Find the common point!
Find the common point!
Source: Problem 5, Brazilian MO, 1991
March 19, 2006
geometry proposed
geometry
Problem Statement
P
0
=
(
1
,
0
)
,
P
1
=
(
1
,
1
)
,
P
2
=
(
0
,
1
)
,
P
3
=
(
0
,
0
)
P_0 = (1,0), P_1 = (1,1), P_2 = (0,1), P_3 = (0,0)
P
0
=
(
1
,
0
)
,
P
1
=
(
1
,
1
)
,
P
2
=
(
0
,
1
)
,
P
3
=
(
0
,
0
)
.
P
n
+
4
P_{n+4}
P
n
+
4
is the midpoint of
P
n
P
n
+
1
P_nP_{n+1}
P
n
P
n
+
1
.
Q
n
Q_n
Q
n
is the quadrilateral
P
n
P
n
+
1
P
n
+
2
P
n
+
3
P_{n}P_{n+1}P_{n+2}P_{n+3}
P
n
P
n
+
1
P
n
+
2
P
n
+
3
.
A
n
A_n
A
n
is the interior of
Q
n
Q_n
Q
n
. Find
∩
n
≥
0
A
n
\cap_{n \geq 0}A_n
∩
n
≥
0
A
n
.
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