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International Contests
Rioplatense Mathematical Olympiad, Level 3
2001 Rioplatense Mathematical Olympiad, Level 3
2001 Rioplatense Mathematical Olympiad, Level 3
Part of
Rioplatense Mathematical Olympiad, Level 3
Subcontests
(6)
6
1
Hide problems
for every n, exists a number that appears n times in sequence f(m)=m+S(m)
For
m
=
1
,
2
,
3
,
.
.
.
m = 1, 2, 3, ...
m
=
1
,
2
,
3
,
...
denote
S
(
m
)
S(m)
S
(
m
)
the sum of the digits of
m
m
m
, and let
f
(
m
)
=
m
+
S
(
m
)
f(m)=m+S(m)
f
(
m
)
=
m
+
S
(
m
)
. Show that for each positive integer
n
n
n
, there exists a number that appears exactly
n
n
n
times in the sequence
f
(
1
)
,
f
(
2
)
,
.
.
.
,
f
(
m
)
,
.
.
.
f(1),f(2),...,f(m),...
f
(
1
)
,
f
(
2
)
,
...
,
f
(
m
)
,
...
3
1
Hide problems
another sequence with floor function, prove S_{2001}=2S_{2000}+1
For every integer
n
>
1
n > 1
n
>
1
, the sequence
(
S
n
)
\left( {{S}_{n}} \right)
(
S
n
)
is defined by
S
n
=
⌊
2
n
2
+
2
+
.
.
.
+
2
⏟
n
r
a
d
i
c
a
l
s
⌋
{{S}_{n}}=\left\lfloor {{2}^{n}}\underbrace{\sqrt{2+\sqrt{2+...+\sqrt{2}}}}_{n\ radicals} \right\rfloor
S
n
=
2
n
n
r
a
d
i
c
a
l
s
2
+
2
+
...
+
2
where
⌊
x
⌋
\left\lfloor x \right\rfloor
⌊
x
⌋
denotes the floor function of
x
x
x
. Prove that
S
2001
=
2
S
2000
+
1
{{S}_{2001}}=2\,{{S}_{2000}}+1
S
2001
=
2
S
2000
+
1
. .
1
1
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system in integers, a^2+b^2=5mn and m^2+n^2=5ab
Find all integer numbers
a
,
b
,
m
a, b, m
a
,
b
,
m
and
n
n
n
, such that the following two equalities are verified:
a
2
+
b
2
=
5
m
n
a^2+b^2=5mn
a
2
+
b
2
=
5
mn
and
m
2
+
n
2
=
5
a
b
m^2+n^2=5ab
m
2
+
n
2
=
5
ab
4
1
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f(f(x)-y)f(x+f(y))=x^2-y^2 , f: R \to R$
Find all functions
f
:
R
→
R
f: R \to R
f
:
R
→
R
such that, for any
x
,
y
∈
R
x, y \in R
x
,
y
∈
R
:
f
(
f
(
x
)
−
y
)
⋅
f
(
x
+
f
(
y
)
)
=
x
2
−
y
2
f\left( f\left( x \right)-y \right)\cdot f\left( x+f\left( y \right) \right)={{x}^{2}}-{{y}^{2}}
f
(
f
(
x
)
−
y
)
⋅
f
(
x
+
f
(
y
)
)
=
x
2
−
y
2
2
1
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equal segments given and wanted, circumcenters, symmetric point, perpendicular,
Let
A
B
C
ABC
A
BC
be an acute triangle and
A
1
,
B
1
A_1, B_1
A
1
,
B
1
and
C
1
C_1
C
1
, points on the sides
B
C
,
C
A
BC, CA
BC
,
C
A
and
A
B
AB
A
B
, respectively, such that
C
B
1
=
A
1
B
1
CB_1 = A_1B_1
C
B
1
=
A
1
B
1
and
B
C
1
=
A
1
C
1
BC_1 = A_1C_1
B
C
1
=
A
1
C
1
. Let
D
D
D
be the symmetric of
A
1
A_1
A
1
with respect to
B
1
C
1
,
O
B_1C_1, O
B
1
C
1
,
O
and
O
1
O_1
O
1
are the circumcenters of triangles
A
B
C
ABC
A
BC
and
A
1
B
1
C
1
A_1B_1C_1
A
1
B
1
C
1
, respectively. If
A
≠
D
,
O
≠
O
1
A \ne D, O \ne O_1
A
=
D
,
O
=
O
1
and
A
D
AD
A
D
is perpendicular to
O
O
1
OO_1
O
O
1
, prove that
A
B
=
A
C
AB = AC
A
B
=
A
C
.
5
1
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Sum is 180
Let
A
B
C
ABC
A
BC
be a acute-angled triangle with centroid
G
G
G
, the angle bisector of
∠
A
B
C
\angle ABC
∠
A
BC
intersects
A
C
AC
A
C
in
D
D
D
. Let
P
P
P
and
Q
Q
Q
be points in
B
D
BD
B
D
where
∠
P
B
A
=
∠
P
A
B
\angle PBA = \angle PAB
∠
PB
A
=
∠
P
A
B
and
∠
Q
B
C
=
∠
Q
C
B
\angle QBC = \angle QCB
∠
QBC
=
∠
QCB
. Let
M
M
M
be the midpoint of
Q
P
QP
QP
, let
N
N
N
be a point in the line
G
M
GM
GM
such that
G
N
=
2
G
M
GN = 2GM
GN
=
2
GM
(where
G
G
G
is the segment
M
N
MN
MN
), prove that:
∠
A
N
C
+
∠
A
B
C
=
180
\angle ANC + \angle ABC = 180
∠
A
NC
+
∠
A
BC
=
180