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Rioplatense Mathematical Olympiad, Level 3
2012 Rioplatense Mathematical Olympiad, Level 3
2012 Rioplatense Mathematical Olympiad, Level 3
Part of
Rioplatense Mathematical Olympiad, Level 3
Subcontests
(6)
6
1
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board 100x100 with mallest possible number of different residues modulo 33
In each square of a
100
×
100
100 \times 100
100
×
100
board there is written an integer. The allowed operation is to choose four squares that form the figure or any of its reflections or rotations, and add
1
1
1
to each of the four numbers. The aim is, through operations allowed, achieving a board with the smallest possible number of different residues modulo
33
33
33
. What is the minimum number that can be achieved with certainty?
5
1
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one of a^n+1, a^{n+1}+1 , ... , a^{2n-2}+1 does not have a common divisor
Let
a
≥
2
a \ge 2
a
≥
2
and
n
≥
3
n \ge 3
n
≥
3
be integers . Prove that one of the numbers
a
n
+
1
,
a
n
+
1
+
1
,
.
.
.
,
a
2
n
−
2
+
1
a^n+ 1 , a^{n + 1}+ 1 , ... , a^{2 n-2}+ 1
a
n
+
1
,
a
n
+
1
+
1
,
...
,
a
2
n
−
2
+
1
does not share any odd divisor greater than
1
1
1
with any of the other numbers.
3
1
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divide a triangle in n triangles with n parallel bisectors, one in each
Let
T
T
T
be a non-isosceles triangle and
n
≥
4
n \ge 4
n
≥
4
an integer . Prove that you can divide
T
T
T
in
n
n
n
triangles and draw in each of them an inner bisector so that those
n
n
n
bisectors are parallel.
2
1
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rectangle is divided into n^2 smaller by n - 1 horizontal and n-1 vertical
A rectangle is divided into
n
2
n^2
n
2
smaller rectangle by
n
−
1
n - 1
n
−
1
horizontal lines and
n
−
1
n - 1
n
−
1
vertical lines. Between those rectangles there are exactly
5660
5660
5660
which are not congruent. For what minimum value of
n
n
n
is this possible?
4
1
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Sum of floor function.
Find all real numbers
x
x
x
, such that: a)
⌊
x
⌋
+
⌊
2
x
⌋
+
.
.
.
+
⌊
2012
x
⌋
=
2013
\lfloor x \rfloor + \lfloor 2x \rfloor +...+ \lfloor 2012x \rfloor = 2013
⌊
x
⌋
+
⌊
2
x
⌋
+
...
+
⌊
2012
x
⌋
=
2013
b)
⌊
x
⌋
+
⌊
2
x
⌋
+
.
.
.
+
⌊
2013
x
⌋
=
2014
\lfloor x \rfloor + \lfloor 2x \rfloor +...+ \lfloor 2013x \rfloor = 2014
⌊
x
⌋
+
⌊
2
x
⌋
+
...
+
⌊
2013
x
⌋
=
2014
1
1
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Apocalyptic numbers
An integer
n
n
n
is called apocalyptic if the addition of
6
6
6
different positive divisors of
n
n
n
gives
3528
3528
3528
. For example,
2012
2012
2012
is apocalyptic, because it has six divisors,
1
1
1
,
2
2
2
,
4
4
4
,
503
503
503
,
1006
1006
1006
and
2012
2012
2012
, that add up to
3528
3528
3528
.Find the smallest positive apocalyptic number.