MathDB
Problems
Contests
National and Regional Contests
Brazil Contests
Brazil National Olympiad
2004 Brazil National Olympiad
2004 Brazil National Olympiad
Part of
Brazil National Olympiad
Subcontests
(6)
2
1
Hide problems
Combinatorial Geometry
Determine all values of
n
n
n
such that it is possible to divide a triangle in
n
n
n
smaller triangles such that there are not three collinear vertices and such that each vertex belongs to the same number of segments.
6
1
Hide problems
Periodic points
Let
a
a
a
and
b
b
b
be real numbers. Define
f
a
,
b
:
R
2
→
R
2
f_{a,b}\colon R^2\to R^2
f
a
,
b
:
R
2
→
R
2
by
f
a
,
b
(
x
;
y
)
=
(
a
−
b
y
−
x
2
;
x
)
f_{a,b}(x;y)=(a-by-x^2;x)
f
a
,
b
(
x
;
y
)
=
(
a
−
b
y
−
x
2
;
x
)
. If
P
=
(
x
;
y
)
∈
R
2
P=(x;y)\in R^2
P
=
(
x
;
y
)
∈
R
2
, define
f
a
,
b
0
(
P
)
=
P
f^0_{a,b}(P) = P
f
a
,
b
0
(
P
)
=
P
and
f
a
,
b
k
+
1
(
P
)
=
f
a
,
b
(
f
a
,
b
k
(
P
)
)
f^{k+1}_{a,b}(P)=f_{a,b}(f_{a,b}^k(P))
f
a
,
b
k
+
1
(
P
)
=
f
a
,
b
(
f
a
,
b
k
(
P
))
for all nonnegative integers
k
k
k
. The set
p
e
r
(
a
;
b
)
per(a;b)
p
er
(
a
;
b
)
of the periodic points of
f
a
,
b
f_{a,b}
f
a
,
b
is the set of points
P
∈
R
2
P\in R^2
P
∈
R
2
such that
f
a
,
b
n
(
P
)
=
P
f_{a,b}^n(P) = P
f
a
,
b
n
(
P
)
=
P
for some positive integer
n
n
n
. Fix
b
b
b
. Prove that the set
A
b
=
{
a
∈
R
∣
p
e
r
(
a
;
b
)
≠
∅
}
A_b=\{a\in R \mid per(a;b)\neq \emptyset\}
A
b
=
{
a
∈
R
∣
p
er
(
a
;
b
)
=
∅
}
admits a minimum. Find this minimum.
5
1
Hide problems
Prove that all the terms are integer
Consider the sequence
(
a
n
)
n
∈
N
(a_n)_{n\in \mathbb{N}}
(
a
n
)
n
∈
N
with
a
0
=
a
1
=
a
2
=
a
3
=
1
a_0=a_1=a_2=a_3=1
a
0
=
a
1
=
a
2
=
a
3
=
1
and
a
n
a
n
−
4
=
a
n
−
1
a
n
−
3
+
a
n
−
2
2
a_na_{n-4}=a_{n-1}a_{n-3} + a^2_{n-2}
a
n
a
n
−
4
=
a
n
−
1
a
n
−
3
+
a
n
−
2
2
. Prove that all the terms of this sequence are integer numbers.
4
1
Hide problems
10x10 board
Consider all the ways of writing exactly ten times each of the numbers
0
,
1
,
2
,
…
,
9
0, 1, 2, \ldots , 9
0
,
1
,
2
,
…
,
9
in the squares of a
10
×
10
10 \times 10
10
×
10
board. Find the greatest integer
n
n
n
with the property that there is always a row or a column with
n
n
n
different numbers.
3
1
Hide problems
Integer sequence
Let
x
1
,
x
2
,
.
.
.
,
x
2004
x_1, x_2, ..., x_{2004}
x
1
,
x
2
,
...
,
x
2004
be a sequence of integer numbers such that
x
k
+
3
=
x
k
+
2
+
x
k
x
k
+
1
x_{k+3}=x_{k+2}+x_{k}x_{k+1}
x
k
+
3
=
x
k
+
2
+
x
k
x
k
+
1
,
∀
1
≤
k
≤
2001
\forall 1 \le k \le 2001
∀1
≤
k
≤
2001
. Is it possible that more than half of the elements are negative?
1
1
Hide problems
Prove that ABCD is a rhombus
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral. Prove that the incircles of the triangles
A
B
C
ABC
A
BC
,
B
C
D
BCD
BC
D
,
C
D
A
CDA
C
D
A
and
D
A
B
DAB
D
A
B
have a point in common if, and only if,
A
B
C
D
ABCD
A
BC
D
is a rhombus.