MathDB
Problems
Contests
National and Regional Contests
Brazil Contests
Brazil National Olympiad
2007 Brazil National Olympiad
2007 Brazil National Olympiad
Part of
Brazil National Olympiad
Subcontests
(6)
6
1
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Lots of different differences
Given real numbers
x
1
<
x
2
<
…
<
x
n
x_1 < x_2 < \ldots < x_n
x
1
<
x
2
<
…
<
x
n
such that every real number occurs at most two times among the differences x_j \minus{} x_i,
1
≤
i
<
j
≤
n
1\leq i < j \leq n
1
≤
i
<
j
≤
n
, prove that there exists at least
⌊
n
/
2
⌋
\lfloor n/2\rfloor
⌊
n
/2
⌋
real numbers that occurs exactly one time among such differences.
4
1
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Pulling squares off: find the winning strategy
200
7
2
2007^2
200
7
2
unit squares are arranged forming a
2007
×
2007
2007\times 2007
2007
×
2007
table. Arnold and Bernold play the following game: each move by Arnold consists of taking four unit squares that forms a
2
×
2
2\times 2
2
×
2
square; each move by Bernold consists of taking a single unit square. They play anternatively, Arnold being the first. When Arnold is not able to perform his move, Bernold takes all the remaining unit squares. The person with more unit squares in the end is the winner. Is it possible to Bernold to win the game, no matter how Arnold play?
5
1
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A complete quadrangle problem with bisectors and diagonals
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrangle,
P
P
P
the intersection of lines
A
B
AB
A
B
and
C
D
CD
C
D
,
Q
Q
Q
the intersection of lines
A
D
AD
A
D
and
B
C
BC
BC
and
O
O
O
the intersection of diagonals
A
C
AC
A
C
and
B
D
BD
B
D
. Show that if \angle POQ\equal{} 90^\circ then
P
O
PO
PO
is the bisector of
∠
A
O
D
\angle AOD
∠
A
O
D
and
O
Q
OQ
OQ
is the bisector of
∠
A
O
B
\angle AOB
∠
A
OB
.
3
1
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A lot of distances!
Consider
n
n
n
points in a plane which are vertices of a convex polygon. Prove that the set of the lengths of the sides and the diagonals of the polygon has at least
⌊
n
/
2
⌋
\lfloor n/2\rfloor
⌊
n
/2
⌋
elements.
2
1
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Quadratic residues in a given interval
Find the number of integers
c
c
c
such that \minus{}2007 \leq c \leq 2007 and there exists an integer
x
x
x
such that x^2 \plus{} c is a multiple of
2
2007
2^{2007}
2
2007
.
1
1
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An n-quadratic equation
Let f(x) \equal{} x^2 \plus{} 2007x \plus{} 1. Prove that for every positive integer
n
n
n
, the equation \underbrace{f(f(\ldots(f}_{n\ {\rm times}}(x))\ldots)) \equal{} 0 has at least one real solution.