MathDB
Problems
Contests
National and Regional Contests
Brazil Contests
Brazil National Olympiad
2008 Brazil National Olympiad
2008 Brazil National Olympiad
Part of
Brazil National Olympiad
Subcontests
(3)
3
2
Hide problems
Find the minimum (try not to use Calculus)
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
real numbers such that x \plus{} y \plus{} z \equal{} xy \plus{} yz \plus{} zx. Find the minimum value of {x \over x^2 \plus{} 1} \plus{} {y\over y^2 \plus{} 1} \plus{} {z\over z^2 \plus{} 1}
Count prophetic words!
The venusian prophet Zabruberson sent to his pupils a
10000
10000
10000
-letter word, each letter being
A
A
A
or
E
E
E
: the Zabrubic word. Their pupils consider then that for
1
≤
k
≤
10000
1 \leq k \leq 10000
1
≤
k
≤
10000
, each word comprised of
k
k
k
consecutive letters of the Zabrubic word is a prophetic word of length
k
k
k
. It is known that there are at most
7
7
7
prophetic words of lenght
3
3
3
. Find the maximum number of prophetic words of length
10
10
10
.
2
2
Hide problems
6n colored points in a line
Let
S
S
S
be a set of
6
n
6n
6
n
points in a line. Choose randomly
4
n
4n
4
n
of these points and paint them blue; the other
2
n
2n
2
n
points are painted green. Prove that there exists a line segment that contains exactly
3
n
3n
3
n
points from
S
S
S
,
2
n
2n
2
n
of them blue and
n
n
n
of them green.
A functional equation in positive integers
Prove that for all integers
a
>
1
a > 1
a
>
1
and
b
>
1
b > 1
b
>
1
there exists a function
f
f
f
from the positive integers to the positive integers such that f(a\cdot f(n)) \equal{} b\cdot n for all
n
n
n
positive integer.
1
2
Hide problems
Multiples beginning with 2008
A positive integer is dapper if at least one of its multiples begins with
2008
2008
2008
. For example,
7
7
7
is dapper because
200858
200858
200858
is a multiple of
7
7
7
and begins with
2008
2008
2008
. Observe that 200858 \equal{} 28694\times 7. Prove that every positive integer is dapper.
Cyclic quadrilateral and reflections wrt bisectors
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral and
r
r
r
and
s
s
s
the lines obtained reflecting
A
B
AB
A
B
with respect to the internal bisectors of
∠
C
A
D
\angle CAD
∠
C
A
D
and
∠
C
B
D
\angle CBD
∠
CB
D
, respectively. If
P
P
P
is the intersection of
r
r
r
and
s
s
s
and
O
O
O
is the center of the circumscribed circle of
A
B
C
D
ABCD
A
BC
D
, prove that
O
P
OP
OP
is perpendicular to
C
D
CD
C
D
.