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International Contests
Rioplatense Mathematical Olympiad, Level 3
2019 Rioplatense Mathematical Olympiad, Level 3
2019 Rioplatense Mathematical Olympiad, Level 3
Part of
Rioplatense Mathematical Olympiad, Level 3
Subcontests
(6)
6
1
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Sequence of integers
Let
α
>
1
\alpha>1
α
>
1
be a real number such that the sequence
a
n
=
α
⌊
α
n
⌋
−
⌊
α
n
+
1
⌋
a_n=\alpha\lfloor \alpha^n\rfloor- \lfloor \alpha^{n+1}\rfloor
a
n
=
α
⌊
α
n
⌋
−
⌊
α
n
+
1
⌋
, with
n
≥
1
n\geq 1
n
≥
1
, is periodic, that is, there is a positive integer
p
p
p
such that
a
n
+
p
=
a
n
a_{n+p}=a_n
a
n
+
p
=
a
n
for all
n
n
n
. Prove that
α
\alpha
α
is an integer.
5
1
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Rio Geo.
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
<
A
C
AB<AC
A
B
<
A
C
and circuncircle
ω
\omega
ω
. Let
M
M
M
and
N
N
N
be the midpoints of
A
C
AC
A
C
and
A
B
AB
A
B
respectively and
G
G
G
is the centroid of
A
B
C
ABC
A
BC
. Let
P
P
P
be the foot of perpendicular of
A
A
A
to the line
B
C
BC
BC
, and the point
Q
Q
Q
is the intersection of
G
P
GP
GP
and
ω
\omega
ω
(
Q
,
P
,
G
Q,P,G
Q
,
P
,
G
are collinears in this order). The line
Q
M
QM
QM
cuts
ω
\omega
ω
in
M
1
M_1
M
1
and the line
Q
N
QN
QN
cuts
ω
\omega
ω
in
N
1
N_1
N
1
. If
K
K
K
is the intersection of
B
M
1
BM_1
B
M
1
and
C
N
1
CN_1
C
N
1
prove that
P
P
P
,
G
G
G
and
K
K
K
are collinears.
4
1
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Infinite triples
Prove that there are infinite triples
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
of positive integers
a
,
b
,
c
>
1
a,b,c>1
a
,
b
,
c
>
1
,
g
c
d
(
a
,
b
)
=
g
c
d
(
b
,
c
)
=
g
c
d
(
c
,
a
)
=
1
gcd(a,b)=gcd(b,c)=gcd(c,a)=1
g
c
d
(
a
,
b
)
=
g
c
d
(
b
,
c
)
=
g
c
d
(
c
,
a
)
=
1
such that
a
+
b
+
c
a+b+c
a
+
b
+
c
divides
a
b
+
b
c
+
c
a
a^b+b^c+c^a
a
b
+
b
c
+
c
a
.
1
1
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Hexagon and square
Let
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
be a regular hexagon, in the sides
A
B
AB
A
B
,
C
D
CD
C
D
,
D
E
DE
D
E
and
F
A
FA
F
A
we choose four points
P
,
Q
,
R
P,Q,R
P
,
Q
,
R
and
S
S
S
respectively, such that
P
Q
R
S
PQRS
PQRS
is a square. Prove that
P
Q
PQ
PQ
and
B
C
BC
BC
are parallel.
2
1
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Rio function
Find all functions
f
:
R
→
R
f:\mathbb{R}\rightarrow \mathbb{R}
f
:
R
→
R
such that
f
(
f
(
x
)
2
+
f
(
y
2
)
)
=
(
x
−
y
)
f
(
x
−
f
(
y
)
)
f(f(x)^2+f(y^2))=(x-y)f(x-f(y))
f
(
f
(
x
)
2
+
f
(
y
2
))
=
(
x
−
y
)
f
(
x
−
f
(
y
))
3
1
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Dog dictionary
In the dog dictionary the words are any sequence of letters
A
A
A
and
U
U
U
for example
A
A
AA
AA
,
U
A
U
UAU
U
A
U
and
A
U
A
U
AUAU
A
U
A
U
. For each word, your "profundity" will be the quantity of subwords we can obtain by the removal of some letters. For each positive integer
n
n
n
, determine the largest "profundity" of word, in dog dictionary, can have with
n
n
n
letters. Note: The word
A
A
U
U
A
AAUUA
AA
UU
A
has "profundity"
14
14
14
because your subwords are
A
,
U
,
A
U
,
A
A
,
U
U
,
U
A
,
A
U
U
,
U
U
A
,
A
A
U
,
A
U
A
,
A
A
A
,
A
A
U
U
,
A
A
U
A
,
A
U
U
A
A, U, AU, AA, UU, UA, AUU, UUA, AAU, AUA, AAA, AAUU, AAUA, AUUA
A
,
U
,
A
U
,
AA
,
UU
,
U
A
,
A
UU
,
UU
A
,
AA
U
,
A
U
A
,
AAA
,
AA
UU
,
AA
U
A
,
A
UU
A
.