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Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
2001 Spain Mathematical Olympiad
2001 Spain Mathematical Olympiad
Part of
Spain Mathematical Olympiad
Subcontests
(6)
Problem 6
1
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2001 Spain Mathematical Olympiad, Problem 6
Define the function
f
:
N
→
N
f: \mathbb{N} \rightarrow \mathbb{N}
f
:
N
→
N
which satisfies, for any
s
,
n
∈
N
s, n \in \mathbb{N}
s
,
n
∈
N
, the following conditions:
f
(
1
)
=
f
(
2
s
)
f(1) = f(2^s)
f
(
1
)
=
f
(
2
s
)
and if
n
<
2
s
n < 2^s
n
<
2
s
, then
f
(
2
s
+
n
)
=
f
(
n
)
+
1.
f(2^s + n) = f(n) + 1.
f
(
2
s
+
n
)
=
f
(
n
)
+
1.
Calculate the maximum value of
f
(
n
)
f(n)
f
(
n
)
when
n
≤
2001
n \leq 2001
n
≤
2001
and find the smallest natural number
n
n
n
such that
f
(
n
)
=
2001.
f(n) = 2001.
f
(
n
)
=
2001.
Problem 5
1
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2001 Spain Mathematical Olympiad, Problem 5
A quadrilateral
A
B
C
D
ABCD
A
BC
D
is inscribed in a circle of radius 1 whose diameter is
A
B
AB
A
B
. If the quadrilateral
A
B
C
D
ABCD
A
BC
D
has an incircle, prove that
C
D
≤
2
5
−
2
CD \leq 2 \sqrt{5} - 2
C
D
≤
2
5
−
2
.
Problem 4
1
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2001 Spanish Mathematical Olympiad, Problem 4
The integers between
1
1
1
and
9
9
9
inclusive are distributed in the units of a
3
3
3
x
3
3
3
table. You sum six numbers of three digits: three that are read in the rows from left to right, and three that are read in the columns from top to bottom. Is there any such distribution for which the value of this sum is equal to
2001
2001
2001
?
Problem 3
1
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2001 Spanish Mathematical Olympiad, Problem 3
You have five segments of lengths
a
1
,
a
2
,
a
3
,
a
4
,
a_1, a_2, a_3, a_4,
a
1
,
a
2
,
a
3
,
a
4
,
and
a
5
a_5
a
5
such that it is possible to form a triangle with any three of them. Demonstrate that at least one of those triangles has angles that are all acute.
Problem 2
1
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2001 Spanish Mathematical Olympiad, Problem 2
Let
P
P
P
be a point on the interior of triangle
A
B
C
ABC
A
BC
, such that the triangle
A
B
P
ABP
A
BP
satisfies
A
P
=
B
P
AP = BP
A
P
=
BP
. On each of the other sides of
A
B
C
ABC
A
BC
, build triangles
B
Q
C
BQC
BQC
and
C
R
A
CRA
CR
A
exteriorly, both similar to triangle
A
B
P
ABP
A
BP
satisfying:
B
Q
=
Q
C
BQ = QC
BQ
=
QC
and
C
R
=
R
A
.
CR = RA.
CR
=
R
A
.
Prove that the point
P
,
Q
,
C
,
P,Q,C,
P
,
Q
,
C
,
and
R
R
R
are collinear or are the vertices of a parallelogram.
Problem 1
1
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2001 Spanish Mathematical Olympiad, Problem 1
Prove that the graph of the polynomial
P
(
x
)
P(x)
P
(
x
)
is symmetric in respect to point
A
(
a
,
b
)
A(a,b)
A
(
a
,
b
)
if and only if there exists a polynomial
Q
(
x
)
Q(x)
Q
(
x
)
such that:
P
(
x
)
=
b
+
(
x
−
a
)
Q
(
(
x
−
a
)
2
)
)
.
P(x) = b + (x-a)Q((x-a)^2)).
P
(
x
)
=
b
+
(
x
−
a
)
Q
((
x
−
a
)
2
))
.