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Problems
Contests
National and Regional Contests
Brazil Contests
Brazil National Olympiad
1993 Brazil National Olympiad
1993 Brazil National Olympiad
Part of
Brazil National Olympiad
Subcontests
(5)
2
1
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Real numbers on a chessboard
A real number with absolute value less than
1
1
1
is written in each cell of an
n
×
n
n\times n
n
×
n
array, so that the sum of the numbers in each
2
×
2
2\times 2
2
×
2
square is zero. Show that for odd
n
n
n
the sum of all the numbers is less than
n
n
n
.
3
1
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Construct the right-angled triangle!
Given a circle and its center
O
O
O
, a point
A
A
A
inside the circle and a distance
h
h
h
, construct a triangle
B
A
C
BAC
B
A
C
with
∠
B
A
C
=
9
0
∘
\angle BAC = 90^\circ
∠
B
A
C
=
9
0
∘
,
B
B
B
and
C
C
C
on the circle and the altitude from
A
A
A
length
h
h
h
.
4
1
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Angles in a quadrilateral (beautiful and almost famous)
A
B
C
D
ABCD
A
BC
D
is a convex quadrilateral with
∠
B
A
C
=
3
0
∘
\angle BAC = 30^\circ
∠
B
A
C
=
3
0
∘
∠
C
A
D
=
2
0
∘
\angle CAD = 20^\circ
∠
C
A
D
=
2
0
∘
∠
A
B
D
=
5
0
∘
\angle ABD = 50^\circ
∠
A
B
D
=
5
0
∘
∠
D
B
C
=
3
0
∘
\angle DBC = 30^\circ
∠
D
BC
=
3
0
∘
If the diagonals intersect at
P
P
P
, show that
P
C
=
P
D
PC = PD
PC
=
P
D
.
1
1
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Find the perfect squares
The sequence
(
a
n
)
n
∈
N
(a_n)_{n \in\mathbb{N}}
(
a
n
)
n
∈
N
is defined by
a
1
=
8
,
a
2
=
18
,
a
n
+
2
=
a
n
+
1
a
n
a_1 = 8, a_2 = 18, a_{n+2} = a_{n+1}a_{n}
a
1
=
8
,
a
2
=
18
,
a
n
+
2
=
a
n
+
1
a
n
. Find all terms which are perfect squares.
5
1
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F(2x+1) = 3f(x) + 5
Find at least one function
f
:
R
→
R
f: \mathbb R \rightarrow \mathbb R
f
:
R
→
R
such that
f
(
0
)
=
0
f(0)=0
f
(
0
)
=
0
and
f
(
2
x
+
1
)
=
3
f
(
x
)
+
5
f(2x+1) = 3f(x) + 5
f
(
2
x
+
1
)
=
3
f
(
x
)
+
5
for any real
x
x
x
.