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Problems
Contests
National and Regional Contests
Brazil Contests
Brazil National Olympiad
1989 Brazil National Olympiad
1989 Brazil National Olympiad
Part of
Brazil National Olympiad
Subcontests
(5)
4
1
Hide problems
Chips in two-row board
A game is played by two contestants A and B, each one having ten chips numbered from 1 to 10. The board of game consists of two numbered rows, from 1 to 1492 on the first row and from 1 to 1989 on the second.At the
n
n
n
-th turn,
n
=
1
,
2
,
…
,
10
n=1,2,\ldots,10
n
=
1
,
2
,
…
,
10
, A puts his chip numbered
n
n
n
in any empty cell, and B puts his chip numbered
n
n
n
in any empty cell on the row not containing the chip numbered
n
n
n
from A.B wins the game if, after the 10th turn, both rows show the numbers of the chips in the same relative order. Otherwise, A wins. [*] Which player has a winning strategy? [*] Suppose now both players has
k
k
k
chips numbered 1 to
k
k
k
. Which player has a winning strategy? [*] Suppose further the rows are the set
Q
\mathbb{Q}
Q
of rationals and the set
Z
\mathbb{Z}
Z
of integers. Which player has a winning strategy?
5
1
Hide problems
Circumcenter inside tetrahedron implies big side
A tetrahedron is such that the center of the its circumscribed sphere is inside the tetrahedron.Show that at least one of its edges has a size larger than or equal to the size of the edge of a regular tetrahedron inscribed in this same sphere.
3
1
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Integer to integer functions
A function
f
f
f
, defined for the set of integers, is such that
f
(
x
)
=
x
−
10
f(x)=x-10
f
(
x
)
=
x
−
10
if
x
>
100
x>100
x
>
100
and
f
(
x
)
=
f
(
f
(
x
+
11
)
)
f(x)=f(f(x+11))
f
(
x
)
=
f
(
f
(
x
+
11
))
if
x
≤
100
x \leq 100
x
≤
100
.Determine, justifying your answer, the set of all possible values for
f
f
f
.
2
1
Hide problems
k(k+1)/3 is square iff k is square
Let
k
k
k
a positive integer number such that
k
(
k
+
1
)
3
\frac{k(k+1)}{3}
3
k
(
k
+
1
)
is a perfect square. Show that
k
3
\frac{k}{3}
3
k
and
k
+
1
k+1
k
+
1
are both perfect squares.
1
1
Hide problems
A mirror triangle reflecting a triangle
The sides of a triangle
T
T
T
, with vertices
(
0
,
0
)
(0,0)
(
0
,
0
)
,
(
3
,
0
)
(3,0)
(
3
,
0
)
and
(
0
,
3
)
(0,3)
(
0
,
3
)
are mirrors. Show that one of the images of the triagle
T
1
T_1
T
1
with vertices
(
0
,
0
)
(0,0)
(
0
,
0
)
,
(
0
,
1
)
(0,1)
(
0
,
1
)
and
(
2
,
0
)
(2,0)
(
2
,
0
)
is the triangle with vertices
(
24
,
36
)
(24,36)
(
24
,
36
)
,
(
24
,
37
)
(24,37)
(
24
,
37
)
and
(
26
,
36
)
(26,36)
(
26
,
36
)
.