MathDB
Problems
Contests
International Contests
Cono Sur Olympiad
2007 Cono Sur Olympiad
2007 Cono Sur Olympiad
Part of
Cono Sur Olympiad
Subcontests
(3)
3
2
Hide problems
Parallel lines thru points on the altitudes of a triangle
Let
A
B
C
ABC
A
BC
be an acute triangle with altitudes
A
D
AD
A
D
,
B
E
BE
BE
,
C
F
CF
CF
where
D
D
D
,
E
E
E
,
F
F
F
lie on
B
C
BC
BC
,
A
C
AC
A
C
,
A
B
AB
A
B
, respectively. Let
M
M
M
be the midpoint of
B
C
BC
BC
. The circumcircle of triangle
A
E
F
AEF
A
EF
cuts the line
A
M
AM
A
M
at
A
A
A
and
X
X
X
. The line
A
M
AM
A
M
cuts the line
C
F
CF
CF
at
Y
Y
Y
. Let
Z
Z
Z
be the point of intersection of
A
D
AD
A
D
and
B
X
BX
BX
. Show that the lines
Y
Z
YZ
Y
Z
and
B
C
BC
BC
are parallel.
Each multiple k,2k,...,nk contains all the digits 0,1,..,9
Show that for each positive integer
n
n
n
, there is a positive integer
k
k
k
such that the decimal representation of each of the numbers
k
,
2
k
,
…
,
n
k
k, 2k,\ldots, nk
k
,
2
k
,
…
,
nk
contains all of the digits
0
,
1
,
2
,
…
,
9
0, 1, 2,\ldots, 9
0
,
1
,
2
,
…
,
9
.
2
2
Hide problems
100 positive integers whose sum & product are equal
Given are
100
100
100
positive integers whose sum equals their product. Determine the minimum number of
1
1
1
s that may occur among the
100
100
100
numbers.
Pentagon with integer sides & inscribed circle
Let
A
B
C
D
E
ABCDE
A
BC
D
E
be a convex pentagon that satisfies all of the following: [*]There is a circle
Γ
\Gamma
Γ
tangent to each of the sides. [*]The lengths of the sides are all positive integers. [*]At least one of the sides of the pentagon has length
1
1
1
. [*]The side
A
B
AB
A
B
has length
2
2
2
. Let
P
P
P
be the point of tangency of
Γ
\Gamma
Γ
with
A
B
AB
A
B
. (a) Determine the lengths of the segments
A
P
AP
A
P
and
B
P
BP
BP
. (b) Give an example of a pentagon satisfying the given conditions.
1
2
Hide problems
Diophantine equation x^3y + x + y = xy + 2xy^2
Find all pairs
(
x
,
y
)
(x,y)
(
x
,
y
)
of nonnegative integers that satisfy
x
3
y
+
x
+
y
=
x
y
+
2
x
y
2
.
x^3y+x+y=xy+2xy^2.
x
3
y
+
x
+
y
=
x
y
+
2
x
y
2
.
Almost completely coloring a chessboard
Some cells of a
2007
×
2007
2007\times 2007
2007
×
2007
table are colored. The table is charrua if none of the rows and none of the columns are completely colored.(a) What is the maximum number
k
k
k
of colored cells that a charrua table can have? (b) For such
k
k
k
, calculate the number of distinct charrua tables that exist.