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International Contests
Rioplatense Mathematical Olympiad, Level 3
1998 Rioplatense Mathematical Olympiad, Level 3
1998 Rioplatense Mathematical Olympiad, Level 3
Part of
Rioplatense Mathematical Olympiad, Level 3
Subcontests
(6)
4
1
Hide problems
a + b is a power of 2 for subset of {1,2,..., 1998} with 1000 elements
Let
M
M
M
be a subset of
{
1
,
2
,
.
.
.
,
1998
}
\{1,2,..., 1998\}
{
1
,
2
,
...
,
1998
}
with
1000
1000
1000
elements. Prove that it is always possible to find two elements
a
a
a
and
b
b
b
in
M
M
M
, not necessarily distinct, such that
a
+
b
a + b
a
+
b
is a power of
2
2
2
.
3
1
Hide problems
e \in B, e divides an odd number of elements of B, iff e \in A
Let
X
X
X
be a finite set of positive integers. Prove that for every subset
A
A
A
of
X
X
X
, there is a subset
B
B
B
of
X
X
X
, with the following property: For each element
e
e
e
of
X
X
X
,
e
e
e
divides an odd number of elements of
B
B
B
, if and only if
e
e
e
is an element of
A
A
A
.
2
1
Hide problems
min max of x_1^2+x_2^2+...+x_n^2 if x_1+ 2x_2 + ... + nx_n = 1
Given an integer
n
>
2
n > 2
n
>
2
, consider all sequences
x
1
,
x
2
,
.
.
.
,
x
n
x_1,x_2,...,x_n
x
1
,
x
2
,
...
,
x
n
of nonnegative real numbers such that
x
1
+
2
x
2
+
.
.
.
+
n
x
n
=
1.
x_1+ 2x_2 + ... + nx_n = 1.
x
1
+
2
x
2
+
...
+
n
x
n
=
1.
Find the maximum value and the minimum value of
x
1
2
+
x
2
2
+
.
.
.
+
x
n
2
x_1^2+x_2^2+...+x_n^2
x
1
2
+
x
2
2
+
...
+
x
n
2
and determine all the sequences
x
1
,
x
2
,
.
.
.
,
x
n
x_1,x_2,...,x_n
x
1
,
x
2
,
...
,
x
n
for which these values are obtained.
6
1
Hide problems
kn points of plane, k given colors, n points of each color
Let
k
k
k
be a fixed positive integer. For each
n
=
1
,
2
,
.
.
.
,
n = 1, 2,...,
n
=
1
,
2
,
...
,
we will call configuration of order
n
n
n
any set of
k
n
kn
kn
points of the plane, which does not contain
3
3
3
collinear, colored with
k
k
k
given colors, so that there are
n
n
n
points of each color. Determine all positive integers
n
n
n
with the following property: in each configuration of order
n
n
n
, it is possible to select three points of each color, such that the
k
k
k
triangles with vertices of the same color that are determined are disjoint in pairs.
5
1
Hide problems
concurrency by joining midpoints of open polygonals XYZ (midpoint of XY+YZ)
We say that
M
M
M
is the midpoint of the open polygonal
X
Y
Z
XYZ
X
Y
Z
, formed by the segments
X
Y
,
Y
Z
XY, YZ
X
Y
,
Y
Z
, if
M
M
M
belongs to the polygonal and divides its length by half. Let
A
B
C
ABC
A
BC
be a acute triangle with orthocenter
H
H
H
. Let
A
1
,
B
1
,
C
1
,
A
2
,
B
2
,
C
2
A_1, B_1,C_1,A_2, B_2,C_2
A
1
,
B
1
,
C
1
,
A
2
,
B
2
,
C
2
be the midpoints of the open polygonal
C
A
B
,
A
B
C
,
B
C
A
,
B
H
C
,
C
H
A
,
A
H
B
CAB, ABC, BCA, BHC, CHA, AHB
C
A
B
,
A
BC
,
BC
A
,
B
H
C
,
C
H
A
,
A
H
B
, respectively. Show that the lines
A
1
A
2
,
B
1
B
2
A_1 A_2, B_1B_2
A
1
A
2
,
B
1
B
2
and
C
1
C
2
C_1C_2
C
1
C
2
are concurrent.
1
1
Hide problems
common chord of two intersecting circles passes through a fixed point
Consider an arc
A
B
AB
A
B
of a circle
C
C
C
and a point
P
P
P
variable in that arc
A
B
AB
A
B
. Let
D
D
D
be the midpoint of the arc
A
P
AP
A
P
that doeas not contain
B
B
B
and let
E
E
E
be the midpoint of the arc
B
P
BP
BP
that does not contain
A
A
A
. Let
C
1
C_1
C
1
be the circle with center
D
D
D
passing through
A
A
A
and
C
2
C_2
C
2
be the circle with center
E
E
E
passing through
B
.
B.
B
.
Prove that the line that contains the intersection points of
C
1
C_1
C
1
and
C
2
C_2
C
2
passes through a fixed point.