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National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
2007 Spain Mathematical Olympiad
2007 Spain Mathematical Olympiad
Part of
Spain Mathematical Olympiad
Subcontests
(6)
Problem 6
1
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Spain Mathematical Olympiad 2007, Problem 6
Given a halfcircle of diameter
A
B
=
2
R
AB = 2R
A
B
=
2
R
, consider a chord
C
D
CD
C
D
of length
c
c
c
. Let
E
E
E
be the intersection of
A
C
AC
A
C
with
B
D
BD
B
D
and
F
F
F
the inersection of
A
D
AD
A
D
with
B
C
BC
BC
. Prove that the segment
E
F
EF
EF
has a constant length and direction when varying the chord
C
D
CD
C
D
about the halfcircle.
Problem 5
1
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Spain Mathematical Olympiad 2007, Problem 5
Let
a
≠
1
a \neq 1
a
=
1
and be a real positive number and
n
n
n
be an integer greater than
1.
1.
1.
Demonstrate that
n
2
<
(
a
n
+
a
−
n
−
2
)
(
a
+
a
−
1
−
2
)
.
n^2 < \frac{(a^n + a^{-n} -2)}{(a + a^{-1} -2)}.
n
2
<
(
a
+
a
−
1
−
2
)
(
a
n
+
a
−
n
−
2
)
.
Problem 4
1
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Spain Mathematical Olympiad 2007, Problem 4
What are the positive integer numbers that we are able to obtain in
2007
2007
2007
distinct ways, when the sum is at least out of two positive consecutive integers? What is the smallest of all of them? Example: the number 9 is written in exactly two such distinct ways:
9
=
4
+
5
9 = 4 + 5
9
=
4
+
5
9
=
2
+
3
+
4.
9 = 2 + 3 + 4.
9
=
2
+
3
+
4.
Problem 3
1
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Spain Mathematical Olympiad 2007, Problem 3
O
O
O
is the circumcenter of triangle
A
B
C
ABC
A
BC
. The bisector from
A
A
A
intersects the opposite side in point
P
P
P
. Prove that the following is satisfied:
A
P
2
+
O
A
2
−
O
P
2
=
b
c
.
AP^2 + OA^2 - OP^2 = bc.
A
P
2
+
O
A
2
−
O
P
2
=
b
c
.
Problem 2
1
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2007 Spain Mathematical Olympiad, Problem 2
Determine all the possible non-negative integer values that are able to satisfy the expression:
(
m
2
+
m
n
+
n
2
)
(
m
n
−
1
)
\frac{(m^2+mn+n^2)}{(mn-1)}
(
mn
−
1
)
(
m
2
+
mn
+
n
2
)
if
m
m
m
and
n
n
n
are non-negative integers such that
m
n
≠
1
mn \neq 1
mn
=
1
.
Problem 1
1
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2007 Spain Mathematical Olympiad, Problem 1
Let
a
0
,
a
1
,
a
2
,
a
3
,
a
4
a_0, a_1, a_2, a_3, a_4
a
0
,
a
1
,
a
2
,
a
3
,
a
4
be five positive numbers in the arithmetic progression with a difference
d
d
d
. Prove that
a
2
3
≤
1
10
(
a
0
3
+
4
a
1
3
+
4
a
3
3
+
a
4
3
)
.
a^3_2 \leq \frac{1}{10}(a^3_0 + 4a^3_1 + 4a^3_3 + a^3_4).
a
2
3
≤
10
1
(
a
0
3
+
4
a
1
3
+
4
a
3
3
+
a
4
3
)
.