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Rioplatense Mathematical Olympiad, Level 3
1995 Rioplatense Mathematical Olympiad, Level 3
1995 Rioplatense Mathematical Olympiad, Level 3
Part of
Rioplatense Mathematical Olympiad, Level 3
Subcontests
(6)
4
1
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1/b = sum 1/ n_k
Given the natural numbers
a
a
a
and
b
b
b
, with
1
≤
a
<
b
1 \le a <b
1
≤
a
<
b
, prove that there exist natural numbers
n
1
<
n
2
<
.
.
.
<
n
k
n_1<n_2< ...<n_k
n
1
<
n
2
<
...
<
n
k
, with
k
≤
a
k \le a
k
≤
a
such that
a
b
=
1
n
1
+
1
n
2
+
.
.
.
+
1
n
k
\frac{a}{b}=\frac{1}{n_1}+\frac{1}{n_2}+...+\frac{1}{n_k}
b
a
=
n
1
1
+
n
2
1
+
...
+
n
k
1
1
1
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n + 1 is the sum of p perfect squares if pn + 1 is a perfect square
Let
n
n
n
and
p
p
p
be two integers with
p
p
p
positive prime, such that
p
n
+
1
pn + 1
p
n
+
1
is a perfect square. Show that
n
+
1
n + 1
n
+
1
is the sum of
p
p
p
perfect squares, not necessarily distinct.
6
1
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2n-sided rhombic polygon splits into 666 rhombic quadrilaterals
A convex polygon with
2
n
2n
2
n
sides is called rhombic if its sides are equal and all pairs of opposite sides are parallel. A rhombic polygon can be partitioned into rhombic quadrilaterals. For what value of
n
n
n
, a
2
n
2n
2
n
-sided rhombic polygon splits into
666
666
666
rhombic quadrilaterals?
5
1
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game with sum of distances betewen 2n points
Consider
2
n
2n
2
n
points in the plane. Two players
A
A
A
and
B
B
B
alternately choose a point on each move. After
2
n
2n
2
n
moves, there are no points left to choose from and the game ends. Add up all the distances between the points chosen by
A
A
A
and add up all the distances between the points chosen by
B
B
B
. The one with the highest sum wins. If
A
A
A
starts the game, describe the winner's strategy.Clarification: Consider that all the partial sums of distances between points give different numbers.
3
1
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n + 1 points at each edge of a tetrahedron
Given a regular tetrahedron with edge
a
a
a
, its edges are divided into
n
n
n
equal segments, thus obtaining
n
+
1
n + 1
n
+
1
points:
2
2
2
at the ends and
n
−
1
n - 1
n
−
1
inside. The following set of planes is considered:
∙
\bullet
∙
those that contain the faces of the tetrahedron, and
∙
\bullet
∙
each of the planes parallel to a face of the tetrahedron and containing at least one of the points determined above. Now all those points
P
P
P
that belong (simultaneously) to four planes of that set are considered. Determine the smallest positive natural
n
n
n
so that among those points
P
P
P
the eight vertices of a square-based rectangular parallelepiped can be chosen.
2
1
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triangle with max area, sides related to segments of a given triangle
In a circle of center
O
O
O
and radius
r
r
r
, a triangle
A
B
C
ABC
A
BC
of orthocenter
H
H
H
is inscribed. It is considered a triangle
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
whose sides have by length the measurements of the segments
A
B
,
C
H
AB, CH
A
B
,
C
H
and
2
r
2r
2
r
. Determine the triangle
A
B
C
ABC
A
BC
so that the area of the triangle
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
is maximum.