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International Contests
Rioplatense Mathematical Olympiad, Level 3
2017 Rioplatense Mathematical Olympiad, Level 3
2017 Rioplatense Mathematical Olympiad, Level 3
Part of
Rioplatense Mathematical Olympiad, Level 3
Subcontests
(6)
6
1
Hide problems
equivalent sequences
For each fixed positiver integer
n
n
n
,
n
≥
4
n\geq 4
n
≥
4
and
P
P
P
an integer, let
(
P
)
n
∈
[
1
,
n
]
(P)_n \in [1, n]
(
P
)
n
∈
[
1
,
n
]
be the smallest positive residue of
P
P
P
modulo
n
n
n
. Two sequences
a
1
,
a
2
,
…
,
a
k
a_1, a_2, \dots, a_k
a
1
,
a
2
,
…
,
a
k
and
b
1
,
b
2
,
…
,
b
k
b_1, b_2, \dots, b_k
b
1
,
b
2
,
…
,
b
k
with the terms in
[
1
,
n
]
[1, n]
[
1
,
n
]
are defined as equivalent, if there is
t
t
t
positive integer, gcd
(
t
,
n
)
=
1
(t,n)=1
(
t
,
n
)
=
1
, such that the sequence
(
t
a
1
)
n
,
…
,
(
t
a
k
)
n
(ta_1)_n, \dots, (ta_k)_n
(
t
a
1
)
n
,
…
,
(
t
a
k
)
n
is a permutation of
b
1
,
b
2
,
…
,
b
k
b_1, b_2, \dots, b_k
b
1
,
b
2
,
…
,
b
k
. Let
α
\alpha
α
a sequence of size
n
n
n
and your terms are in
[
1
,
n
]
[1, n]
[
1
,
n
]
, such that each term appears
h
h
h
times in the sequence
α
\alpha
α
and
2
h
≥
n
2h\geq n
2
h
≥
n
. Show that
α
\alpha
α
is equivalent to some sequence
β
\beta
β
which contains a subsequence such that your size is(at most) equal to
h
h
h
and your sum is exactly equal to
n
n
n
.
4
1
Hide problems
n = sum of 2017 perfect squares and (with at least) 2017 distinct ways
Is there a number
n
n
n
such that one can write
n
n
n
as the sum of
2017
2017
2017
perfect squares and (with at least)
2017
2017
2017
distinct ways?
3
1
Hide problems
m | n^{2016}+n^{2015}+...+n^2+n+1, n |m^{2016}+m^{2015} +...+m^2+m+1
Show that there are infinitely many pairs of positive integers
(
m
,
n
)
(m,n)
(
m
,
n
)
, with
m
<
n
m<n
m
<
n
, such that
m
m
m
divides
n
2016
+
n
2015
+
⋯
+
n
2
+
n
+
1
n^{2016}+n^{2015}+\dots+n^2+n+1
n
2016
+
n
2015
+
⋯
+
n
2
+
n
+
1
and
n
n
n
divides
m
2016
+
m
2015
+
⋯
+
m
2
+
m
+
1
m^{2016}+m^{2015} +\dots+m^2+m+1
m
2016
+
m
2015
+
⋯
+
m
2
+
m
+
1
.
1
1
Hide problems
(x+a)(x+b)=x+a+b diophantine
Let
a
a
a
be a fixed positive integer. Find the largest integer
b
b
b
such that
(
x
+
a
)
(
x
+
b
)
=
x
+
a
+
b
(x+a)(x+b)=x+a+b
(
x
+
a
)
(
x
+
b
)
=
x
+
a
+
b
, for some integer
x
x
x
.
5
1
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QP bisects <AQC
Let
A
B
C
ABC
A
BC
be a triangle and
I
I
I
is your incenter, let
P
P
P
be a point in
A
C
AC
A
C
such that
P
I
PI
P
I
is perpendicular to
A
C
AC
A
C
, and let
D
D
D
be the reflection of
B
B
B
wrt circumcenter of
△
A
B
C
\triangle ABC
△
A
BC
. The line
D
I
DI
D
I
intersects again the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
in the point
Q
Q
Q
. Prove that
Q
P
QP
QP
is the angle bisector of the angle
∠
A
Q
C
\angle AQC
∠
A
QC
.
2
1
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n distinct circles , k lines
One have
n
n
n
distinct circles(with the same radius) such that for any
k
+
1
k+1
k
+
1
circles there are (at least) two circles that intersects in two points. Show that for each line
l
l
l
one can make
k
k
k
lines, each one parallel with
l
l
l
, such that each circle has (at least) one point of intersection with some of these lines.