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Problems
Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
1973 Spain Mathematical Olympiad
1973 Spain Mathematical Olympiad
Part of
Spain Mathematical Olympiad
Subcontests
(8)
8
1
Hide problems
P(t) = (1-t, 2-3t, 2t-1), Q(t) = (1-t^2, 2-3t^2, 2t^2 -1)
In a three-dimensional Euclidean space, by
u
1
→
\overrightarrow{u_1}
u
1
,
u
2
→
\overrightarrow{u_2}
u
2
,
u
3
→
\overrightarrow{u_3}
u
3
are denoted the three orthogonal unit vectors on the
x
,
y
x, y
x
,
y
, and
z
z
z
axes, respectively. a) Prove that the point
P
(
t
)
=
(
1
−
t
)
u
1
→
+
(
2
−
3
t
)
u
2
→
+
(
2
t
−
1
)
u
3
→
P(t) = (1-t)\overrightarrow{u_1} +(2-3t)\overrightarrow{u_2} +(2t-1)\overrightarrow{u_3}
P
(
t
)
=
(
1
−
t
)
u
1
+
(
2
−
3
t
)
u
2
+
(
2
t
−
1
)
u
3
, where
t
t
t
takes all real values, describes a straight line (which we will denote by
L
L
L
). b) What describes the point
Q
(
t
)
=
(
1
−
t
2
)
u
1
→
+
(
2
−
3
t
2
)
u
2
→
+
(
2
t
2
−
1
)
u
3
→
Q(t) = (1-t^2)\overrightarrow{u_1} +(2-3t^2)\overrightarrow{u_2} +(2t^2 -1)\overrightarrow{u_3}
Q
(
t
)
=
(
1
−
t
2
)
u
1
+
(
2
−
3
t
2
)
u
2
+
(
2
t
2
−
1
)
u
3
if
t
t
t
takes all the real values? c) Find a vector parallel to
L
L
L
. d) For what values of
t
t
t
is the point
P
(
t
)
P(t)
P
(
t
)
on the plane
2
x
+
3
y
+
2
z
+
1
=
0
2x+ 3y + 2z +1 = 0
2
x
+
3
y
+
2
z
+
1
=
0
? e) Find the Cartesian equation of the plane parallel to the previous one and containing the point
Q
(
3
)
Q(3)
Q
(
3
)
. f) Find the Cartesian equation of the plane perpendicular to
L
L
L
that contains the point
Q
(
2
)
Q(2)
Q
(
2
)
.
7
1
Hide problems
min length from (8, 2) to (5, 11) under conditions
The two points
P
(
8
,
2
)
P(8, 2)
P
(
8
,
2
)
and
Q
(
5
,
11
)
Q(5, 11)
Q
(
5
,
11
)
are considered in the plane. A mobile moves from
P
P
P
to
Q
Q
Q
according to a path that has to fulfill the following conditions: The moving part of
P
P
P
and arrives at a point on the
x
x
x
-axis, along which it travels a segment of length
1
1
1
, then it departs from this axis and goes towards a point on the
y
y
y
axis, on which travels a segment of length
2
2
2
, separates from the
y
y
y
axis finally and goes towards the point
Q
Q
Q
. Among all the possible paths, determine the one with the minimum length, thus like this same length.
3
1
Hide problems
a_n= a_{n-1} +1/n (cos 45^o + i sin 45^o )^n
The sequence
(
a
n
)
(a_n)
(
a
n
)
of complex numbers is considered in the complex plane, in which is:
a
0
=
1
,
a
n
=
a
n
−
1
+
1
n
(
cos
4
5
o
+
i
sin
4
5
o
)
n
.
a_0 = 1, \,\,\, a_n = a_{n-1} +\frac{1}{n}(\cos 45^o + i \sin 45^o )^n.
a
0
=
1
,
a
n
=
a
n
−
1
+
n
1
(
cos
4
5
o
+
i
sin
4
5
o
)
n
.
Prove that the sequence of the real parts of the terms of
(
a
n
)
(a_n)
(
a
n
)
is convergent and its limit is a number between
0.85
0.85
0.85
and
1.15
1.15
1.15
.
5
1
Hide problems
polynomials of degree<=4 with rational coefficients as vector space
Consider the set of all polynomials of degree less than or equal to
4
4
4
with rational coefficients. a) Prove that it has a vector space structure over the field of numbers rational. b) Prove that the polynomials
1
,
x
−
2
,
(
x
−
2
)
2
,
(
x
−
2
)
3
1, x - 2, (x -2)^2, (x - 2)^3
1
,
x
−
2
,
(
x
−
2
)
2
,
(
x
−
2
)
3
and
(
x
−
2
)
4
(x -2)^4
(
x
−
2
)
4
form a base of this space. c) Express the polynomial
7
+
2
x
−
45
x
2
+
3
x
4
7 + 2x - 45x^2 + 3x^4
7
+
2
x
−
45
x
2
+
3
x
4
in the previous base.
2
1
Hide problems
4x4 system of 3 equalities and 1 inequality
Determine all solutions of the system
{
2
x
−
5
y
+
11
z
−
6
=
0
−
x
+
3
y
−
16
z
+
8
=
0
4
x
−
5
y
−
83
z
+
38
=
0
3
x
+
11
y
−
z
+
9
>
0
\begin{cases} 2x - 5y + 11z - 6 = 0 \\ -x + 3y - 16z + 8 = 0 \\ 4x - 5y - 83z + 38 = 0 \\ 3x + 11y - z + 9 > 0 \end{cases}
⎩
⎨
⎧
2
x
−
5
y
+
11
z
−
6
=
0
−
x
+
3
y
−
16
z
+
8
=
0
4
x
−
5
y
−
83
z
+
38
=
0
3
x
+
11
y
−
z
+
9
>
0
in which the first three are equations and the last one is a linear inequality.
1
1
Hide problems
min term of a_n = 1/4 n^4 - 10n^2(n - 1)
Given the sequence
(
a
n
)
(a_n)
(
a
n
)
, in which
a
n
=
1
4
n
4
−
10
n
2
(
n
−
1
)
a_n =\frac14 n^4 - 10n^2(n - 1)
a
n
=
4
1
n
4
−
10
n
2
(
n
−
1
)
, with
n
=
0
,
1
,
2
,
.
.
.
n = 0, 1, 2,...
n
=
0
,
1
,
2
,
...
Determine the smallest term of the sequence.
6
1
Hide problems
distances from sides of an equialteral triangle related
An equilateral triangle of altitude
1
1
1
is considered. For every point
P
P
P
on the interior of the triangle, denote by
x
,
y
,
z
x, y , z
x
,
y
,
z
the distances from the point
P
P
P
to the sides of the triangle. a) Prove that for every point
P
P
P
inside the triangle it is true that
x
+
y
+
z
=
1
x + y + z = 1
x
+
y
+
z
=
1
. b) For which points of the triangle does it hold that the distance to one side is greater than the sum of the distances to the other two? c) We have a bar of length
1
1
1
and we break it into three pieces. find the probability that with these pieces a triangle can be formed.
4
1
Hide problems
8 circles each tangent to 2, in an annulus between concentric circles
Let
C
C
C
and
C
′
C'
C
′
be two concentric circles of radii
r
r
r
and
r
′
r'
r
′
respectively. Determine how much the quotient
r
′
/
r
r'/r
r
′
/
r
must be worth so that in the limited crown (annulus) through
C
C
C
and
C
′
C'
C
′
there are eight circles
C
i
C_i
C
i
,
i
=
1
,
.
.
.
,
8
i = 1, . . . , 8
i
=
1
,
...
,
8
, which are tangent to
C
C
C
and to
C
′
C'
C
′
, and also that
C
i
C_i
C
i
is tangent to
C
i
+
1
C_{i+1}
C
i
+
1
for
i
=
1
,
.
.
.
,
7
i = 1, . . . ,7
i
=
1
,
...
,
7
and
C
8
C_8
C
8
tangent to
C
1
C_1
C
1
.