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Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
2006 Spain Mathematical Olympiad
2006 Spain Mathematical Olympiad
Part of
Spain Mathematical Olympiad
Subcontests
(3)
2
2
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finite number of octahedra cutting a painted wooden octahedron
The dimensions of a wooden octahedron are natural numbers. We painted all its surface (the six faces), cut it by planes parallel to the cubed faces of an edge unit and observed that exactly half of the cubes did not have any painted faces. Prove that the number of octahedra with such property is finite.(It may be useful to keep in mind that
1
2
3
=
1
,
79...
<
1
,
8
\sqrt[3]{\frac{1}{2}}=1,79 ... <1,8
3
2
1
=
1
,
79...
<
1
,
8
).[hide=original wording] Las dimensiones de un ortoedro de madera son enteras. Pintamos toda su superficie (las seis caras), lo cortamos mediante planos paralelos a las caras en cubos de una unidad de arista y observamos que exactamente la mitad de los cubos no tienen ninguna cara pintada. Probar que el número de ortoedros con tal propiedad es finito
product of 4 consecutive naturals not square nor cube
Prove that the product of four consecutive natural numbers can not be neither square nor perfect cube.
3
2
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distances of a center in circle from sides of isosceles triangle
A
B
C
ABC
A
BC
is an isosceles triangle with
A
B
=
A
C
AB = AC
A
B
=
A
C
. Let
P
P
P
be any point of a circle tangent to the sides
A
B
AB
A
B
in
B
B
B
and to AC in C. Denote
a
a
a
,
b
b
b
and
c
c
c
to the distances from
P
P
P
to the sides
B
C
,
A
C
BC, AC
BC
,
A
C
and
A
B
AB
A
B
respectively. Prove that:
a
2
=
b
c
a^2=bc
a
2
=
b
c
\sqrt{S_1}+\sqrt{S_2}\le \sqrt{S} in convex ABCD
The diagonals
A
C
AC
A
C
and
B
D
BD
B
D
of a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
intersect at
E
E
E
. Denotes by
S
1
,
S
2
S_1,S_2
S
1
,
S
2
and
S
S
S
the areas of the triangles
A
B
E
ABE
A
BE
,
C
D
E
CDE
C
D
E
and the quadrilateral
A
B
C
D
ABCD
A
BC
D
respectively. Prove that
S
1
+
S
2
≤
S
\sqrt{S_1}+\sqrt{S_2}\le \sqrt{S}
S
1
+
S
2
≤
S
. When equality is reached?
1
2
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integer polynomial , k does not divide P(i), i=1,..., k
Let
P
(
x
)
P(x)
P
(
x
)
be a polynomial with integer coefficients. Prove that if there is an integer
k
k
k
such that none of the integers
P
(
1
)
,
P
(
2
)
,
.
.
.
,
P
(
k
)
P(1),P(2), ..., P(k)
P
(
1
)
,
P
(
2
)
,
...
,
P
(
k
)
is divisible by
k
k
k
, then
P
(
x
)
P(x)
P
(
x
)
does not have integer roots.
f(x)f(y)+f\(\lambda /x})f(\lambda/y)=2f(xy) w f(\lambda)=1
Find all the functions
f
:
(
0
,
+
∞
)
→
R
f:(0,+\infty) \to R
f
:
(
0
,
+
∞
)
→
R
that satisfy the equation
f
(
x
)
f
(
y
)
+
f
(
λ
x
)
f
(
λ
y
)
=
2
f
(
x
y
)
f(x)f(y)+f\big(\frac{\lambda}{x})f(\frac{\lambda}{y})=2f(xy)
f
(
x
)
f
(
y
)
+
f
(
x
λ
)
f
(
y
λ
)
=
2
f
(
x
y
)
for all pairs of
x
,
y
x,y
x
,
y
real and positive numbers, where
λ
\lambda
λ
is a positive real number such that
f
(
λ
)
=
1
f(\lambda )=1
f
(
λ
)
=
1