MathDB
Problems
Contests
National and Regional Contests
Brazil Contests
Brazil National Olympiad
2002 Brazil National Olympiad
2002 Brazil National Olympiad
Part of
Brazil National Olympiad
Subcontests
(6)
6
1
Hide problems
Binary sequences and error correcting codes
Show that we cannot form more than
4096
4096
4096
binary sequences of length
24
24
24
so that any two differ in at least
8
8
8
positions.
5
1
Hide problems
Cover a square with a bunch of squares
A finite collection of squares has total area
4
4
4
. Show that they can be arranged to cover a square of side
1
1
1
.
4
1
Hide problems
Sum set diameters: try to do it with the least calculation!
For any non-empty subset
A
A
A
of
{
1
,
2
,
…
,
n
}
\{1, 2, \ldots , n\}
{
1
,
2
,
…
,
n
}
define
f
(
A
)
f(A)
f
(
A
)
as the largest element of
A
A
A
minus the smallest element of
A
A
A
. Find
∑
f
(
A
)
\sum f(A)
∑
f
(
A
)
where the sum is taken over all non-empty subsets of
{
1
,
2
,
…
,
n
}
\{1, 2, \ldots , n\}
{
1
,
2
,
…
,
n
}
.
3
1
Hide problems
Draw-the-snake-on-the-board problem
The squares of an
m
×
n
m\times n
m
×
n
board are labeled from
1
1
1
to
m
n
mn
mn
so that the squares labeled
i
i
i
and
i
+
1
i+1
i
+
1
always have a side in common. Show that for some
k
k
k
the squares
k
k
k
and
k
+
3
k+3
k
+
3
have a side in common.
2
1
Hide problems
Same area and same perimeter => two equal sides
A
B
C
D
ABCD
A
BC
D
is a cyclic quadrilateral and
M
M
M
a point on the side
C
D
CD
C
D
such that
A
D
M
ADM
A
D
M
and
A
B
C
M
ABCM
A
BCM
have the same area and the same perimeter. Show that two sides of
A
B
C
D
ABCD
A
BC
D
have the same length.
1
1
Hide problems
The sum is never a perfect power
Show that there is a set of
2002
2002
2002
distinct positive integers such that the sum of one or more elements of the set is never a square, cube, or higher power.