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Problems
Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
1972 Spain Mathematical Olympiad
1972 Spain Mathematical Olympiad
Part of
Spain Mathematical Olympiad
Subcontests
(8)
5
1
Hide problems
equilateral construction 2 vertices on parallel lines
Given two parallel lines
r
r
r
and
r
′
r'
r
′
and a point
P
P
P
on the plane that contains them and that is not on them, determine an equilateral triangle whose vertex is point
P
P
P
, and the other two, one on each of the two lines. https://cdn.artofproblemsolving.com/attachments/9/3/1d475eb3e9a8a48f4a85a2a311e1bda978e740.png
2
1
Hide problems
min or max absolute sum of distances of a point from vertices
A point moves on the sides of the triangle
A
B
C
ABC
A
BC
, defined by the vertices
A
(
−
1.8
,
0
)
A(-1.8, 0)
A
(
−
1.8
,
0
)
,
B
(
3.2
,
0
)
B(3.2, 0)
B
(
3.2
,
0
)
,
C
(
0
,
2.4
)
C(0, 2.4)
C
(
0
,
2.4
)
. Determine the positions of said point, in which the sum of their distance to the three vertices is absolute maximum or minimum. https://cdn.artofproblemsolving.com/attachments/2/5/9e5bb48cbeefaa5f4c069532bf5605b9c1f5ea.png
8
1
Hide problems
an equivalence relation in a vector subspace of R^3
We know that
R
3
=
{
(
x
1
,
x
2
,
x
3
)
∣
x
i
∈
R
,
i
=
1
,
2
,
3
}
R^3 = \{(x_1, x_2, x_3) | x_i \in R, i = 1, 2, 3\}
R
3
=
{(
x
1
,
x
2
,
x
3
)
∣
x
i
∈
R
,
i
=
1
,
2
,
3
}
is a vector space regarding the laws of composition
(
x
1
,
x
2
,
x
3
)
+
(
y
1
,
y
2
,
y
3
)
=
(
x
1
+
y
1
,
x
2
+
y
2
,
x
3
+
y
3
)
(x_1, x_2, x_3) + (y_1, y_2, y_3) = (x_1 + y_1, x_2 + y_2, x_3 + y_3)
(
x
1
,
x
2
,
x
3
)
+
(
y
1
,
y
2
,
y
3
)
=
(
x
1
+
y
1
,
x
2
+
y
2
,
x
3
+
y
3
)
,
λ
(
x
1
,
x
2
,
x
3
)
=
(
λ
x
1
,
λ
x
2
,
λ
x
3
)
\lambda (x_1, x_2, x_3) = (\lambda x_1, \lambda x_2, \lambda x_3)
λ
(
x
1
,
x
2
,
x
3
)
=
(
λ
x
1
,
λ
x
2
,
λ
x
3
)
,
λ
∈
R
\lambda \in R
λ
∈
R
. We consider the following subset of
R
3
R^3
R
3
:
L
=
{
(
x
1
,
x
2
,
x
3
)
∈
R
3
∣
x
1
+
x
2
+
x
3
=
0
}
L =\{(x_1, x2, x_3) \in R^3 | x_1 + x_2 + x_3 = 0\}
L
=
{(
x
1
,
x
2
,
x
3
)
∈
R
3
∣
x
1
+
x
2
+
x
3
=
0
}
. a) Prove that
L
L
L
is a vector subspace of
R
3
R^3
R
3
. b) In
R
3
R^3
R
3
the following relation is defined
x
‾
R
y
‾
⇔
x
‾
−
y
‾
∈
L
,
x
‾
,
y
‾
∈
R
3
\overline{x} R \overline{y} \Leftrightarrow \overline{x} -\overline{y} \in L, \overline{x} , \overline{y} \in R^3
x
R
y
⇔
x
−
y
∈
L
,
x
,
y
∈
R
3
. Prove that it is an equivalence relation. c) Find two vectors of
R
3
R^3
R
3
that belong to the same class as the vector
(
−
1
,
3
,
2
)
(-1, 3, 2)
(
−
1
,
3
,
2
)
.
6
1
Hide problems
inradii of triangle by touchpoints of 3 ext. tangent circles 1972 Spanish P6
Given three circumferences of radii
r
r
r
,
r
′
r'
r
′
and
r
′
′
r''
r
′′
, each tangent externally to the other two, calculate the radius of the circle inscribed in the triangle whose vertices are their three centers.
4
1
Hide problems
projections on sets on complex plane
The following sets of points are considered in the plane:
A
=
{
A=\{
A
=
{
affixes of complexes
z
z
z
such that arg
(
z
−
(
2
+
3
i
)
)
=
π
/
4
}
(z - (2 + 3i))=\pi /4\}
(
z
−
(
2
+
3
i
))
=
π
/4
}
,
B
=
{
B =\{
B
=
{
affixes of complexes
z
z
z
such that mod
(
z
−
(
2
+
i
)
<
2
}
( z- (2 + i)<2\}
(
z
−
(
2
+
i
)
<
2
}
. Determine the orthogonal projection on the
X
X
X
axis of
A
∩
B
A \cap B
A
∩
B
.
3
1
Hide problems
min length across all lateral faces of regular hexagonal prism
Given a regular hexagonal prism. Find a polygonal line that, starting from a vertex of the base, runs through all the lateral faces and ends at the vertex of the face top, located on the same edge as the starting vertex, and has a minimum length.
7
1
Hide problems
5^n + 2 x 3^{n-1} + 1 is multiple of 8
Prove that for every positive integer
n
n
n
, the number
A
n
=
5
n
+
2
⋅
3
n
−
1
+
1
A_n = 5^n + 2 \cdot 3^{n-1} + 1
A
n
=
5
n
+
2
⋅
3
n
−
1
+
1
is a multiple of
8
8
8
.
1
1
Hide problems
2x2 matrices from elements of a ring K
Let
K
K
K
be a ring with unit and
M
M
M
the set of
2
×
2
2 \times 2
2
×
2
matrices constituted with elements of
K
K
K
. An addition and a multiplication are defined in
M
M
M
in the usual way between arrays. It is requested to: a) Check that
M
M
M
is a ring with unit and not commutative with respect to the laws of defined composition. b) Check that if
K
K
K
is a commutative field, the elements of
M
M
M
that have inverse they are characterized by the condition
a
d
−
b
c
≠
0
ad - bc \ne 0
a
d
−
b
c
=
0
. c) Prove that the subset of
M
M
M
formed by the elements that have inverse is a multiplicative group.