MathDB
Problems
Contests
International Contests
Cono Sur Olympiad
1997 Cono Sur Olympiad
1997 Cono Sur Olympiad
Part of
Cono Sur Olympiad
Subcontests
(5)
4
1
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A construction of a board
Consider a board with
n
n
n
rows and
4
4
4
columns. In the first line are written
4
4
4
zeros (one in each house). Next, each line is then obtained from the previous line by performing the following operation: one of the houses, (that you can choose), is maintained as in the previous line; the other three are changed: * if in the previous line there was a
0
0
0
, then in the down square
1
1
1
is placed; * if in the previous line there was a
1
1
1
, then in the down square
2
2
2
is placed; * if in the previous line there was a
2
2
2
, then in the down square
0
0
0
is placed; Build the largest possible board with all its distinct lines and demonstrate that it is impossible to build a larger board.
6
1
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MNP is congruent to ABC
Let
A
B
C
ABC
A
BC
be a acute-angle triangle and
X
X
X
be point in the plane of this triangle. Let
M
,
N
,
P
M,N,P
M
,
N
,
P
be the orthogonal projections of
X
X
X
in the lines that contains the altitudes of this triangle Determine the positions of the point
X
X
X
such that the triangle
M
N
P
MNP
MNP
is congruent to
A
B
C
ABC
A
BC
2
1
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Show that H,B and R are collinears
Let
C
C
C
be a circunference,
O
O
O
is your circumcenter,
A
B
AB
A
B
is your diameter and
R
R
R
is any point in
C
C
C
(
R
R
R
is different of
A
A
A
and
B
B
B
) Let
P
P
P
be the foot of perpendicular by
O
O
O
to
A
R
AR
A
R
, in the line
O
P
OP
OP
we match a point
Q
Q
Q
, where
Q
P
QP
QP
is
O
P
2
\frac{OP}{2}
2
OP
and the point
Q
Q
Q
isn't in the segment
O
P
OP
OP
. In
Q
Q
Q
, we will do a parallel line to
A
B
AB
A
B
that cut the line
A
R
AR
A
R
in
T
T
T
. Denote
H
H
H
the point of intersections of the line
A
Q
AQ
A
Q
and
O
T
OT
OT
. Show that
H
H
H
,
B
B
B
and
R
R
R
are collinears.
5
1
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Number Theory + Combinat
Let
n
n
n
be a natural number
n
>
3
n>3
n
>
3
. Show that in the multiples of
9
9
9
less than
1
0
n
10^n
1
0
n
, exist more numbers with the sum of your digits equal to
9
(
n
−
2
)
9(n - 2)
9
(
n
−
2
)
than numbers with the sum of your digits equal to
9
(
n
−
1
)
9(n - 1)
9
(
n
−
1
)
.
3
1
Hide problems
"Easy" number theory
Show that, exist infinite triples
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
where
a
,
b
,
c
a, b, c
a
,
b
,
c
are natural numbers, such that:
2
a
2
+
3
b
2
−
5
c
2
=
1997
2a^2 + 3b^2 - 5c^2 = 1997
2
a
2
+
3
b
2
−
5
c
2
=
1997