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Problems
Contests
National and Regional Contests
Brazil Contests
Brazil National Olympiad
1984 Brazil National Olympiad
1984 Brazil National Olympiad
Part of
Brazil National Olympiad
Subcontests
(6)
6
1
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moving in the solitaire board
There is a piece on each square of the solitaire board shown except for the central square. A move can be made when there are three adjacent squares in a horizontal or vertical line with two adjacent squares occupied and the third square vacant. The move is to remove the two pieces from the occupied squares and to place a piece on the third square. (One can regard one of the pieces as hopping over the other and taking it.) Is it possible to end up with a single piece on the board, on the square marked
X
X
X
?
5
1
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4 squares outside a convex quadrilateral, centers's distances equal & _|_
A
B
C
D
ABCD
A
BC
D
is any convex quadrilateral. Squares center
E
,
F
,
G
,
H
E, F, G, H
E
,
F
,
G
,
H
are constructed on the outside of the edges
A
B
,
B
C
,
C
D
AB, BC, CD
A
B
,
BC
,
C
D
and
D
A
DA
D
A
respectively. Show that
E
G
EG
EG
and
F
H
FH
F
H
are equal and perpendicular.
4
1
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minimum segment of projections of a point in hypotenuse on other sides
A
B
C
ABC
A
BC
is a triangle with
∠
A
=
9
0
o
\angle A = 90^o
∠
A
=
9
0
o
. For a point
D
D
D
on the side
B
C
BC
BC
, the feet of the perpendiculars to
A
B
AB
A
B
and
A
C
AC
A
C
are
E
E
E
and
F
F
F
. For which point
D
D
D
is
E
F
EF
EF
a minimum?
3
1
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a rectangle constructed starting with a regular dodecahedron
Given a regular dodecahedron of side
a
a
a
. Take two pairs of opposite faces:
E
,
E
′
E, E'
E
,
E
′
and
F
,
F
′
F, F'
F
,
F
′
. For the pair
E
,
E
′
E, E'
E
,
E
′
take the line joining the centers of the faces and take points
A
A
A
and
C
C
C
on the line each a distance
m
m
m
outside one of the faces. Similarly, take
B
B
B
and
D
D
D
on the line joining the centers of
F
,
F
′
F, F'
F
,
F
′
each a distance
m
m
m
outside one of the faces. Show that
A
B
C
D
ABCD
A
BC
D
is a rectangle and find the ratio of its side lengths.
2
1
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each day 289 students are divided into 17 groups of 17 ...
Each day
289
289
289
students are divided into
17
17
17
groups of
17
17
17
. No two students are ever in the same group more than once. What is the largest number of days that this can be done?
1
1
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solve in positive integers (n+1)^k -1 = n!
Find all solutions in positive integers to
(
n
+
1
)
k
−
1
=
n
!
(n+1)^k -1 = n!
(
n
+
1
)
k
−
1
=
n
!