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Problems
Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
1981 Spain Mathematical Olympiad
1981 Spain Mathematical Olympiad
Part of
Spain Mathematical Olympiad
Subcontests
(8)
5
1
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integral and limit f_n(x)= n^2x or 3/n
Given a nonzero natural number
n
n
n
, let
f
n
f_n
f
n
be the function of the closed interval
[
0
,
1
]
[0, 1]
[
0
,
1
]
in
R
R
R
defined like this:
f
n
(
x
)
=
{
n
2
x
,
i
f
0
≤
x
<
1
/
n
3
/
n
,
i
f
1
/
n
≤
x
≤
1
f_n(x) = \begin{cases}n^2x, \,\,\, if \,\,\, 0 \le x < 1/n\\ 3/n, \,\,\,if \,\,\,1/n \le x \le 1 \end{cases}
f
n
(
x
)
=
{
n
2
x
,
i
f
0
≤
x
<
1/
n
3/
n
,
i
f
1/
n
≤
x
≤
1
a) Represent the function graphically. b) Calculate
A
n
=
∫
0
1
f
n
(
x
)
d
x
A_n =\int_0^1 f_n(x) dx
A
n
=
∫
0
1
f
n
(
x
)
d
x
. c) Find, if it exists,
lim
n
→
∞
A
n
\lim_{n\to \infty} A_n
lim
n
→
∞
A
n
.
6
1
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composition of center and axis symmetry
Prove that the transformation product of the symmetry of center
(
0
,
0
)
(0, 0)
(
0
,
0
)
with the symmetry of the axis, with the line of equation
x
=
y
+
1
x = y + 1
x
=
y
+
1
, can be expressed as a product of an axis symmetry the line
e
e
e
by a translation of vector
v
→
\overrightarrow{v}
v
, with
e
e
e
parallel to
v
→
\overrightarrow{v}
v
, . Determine a line
e
e
e
and a vector
v
→
\overrightarrow{v}
v
, that meet the indicated conditions. have to be unique
e
e
e
and
v
→
\overrightarrow{v}
v
,?
4
1
Hide problems
\int \frac{dx}{sin (x - 1) sin (x - 2)}
Calculate the integral
∫
d
x
sin
(
x
−
1
)
sin
(
x
−
2
)
.
\int \frac{dx}{\sin (x - 1) \sin (x - 2)} .
∫
sin
(
x
−
1
)
sin
(
x
−
2
)
d
x
.
Hint: Change
tan
x
=
t
\tan x = t
tan
x
=
t
.
2
1
Hide problems
min path on surface of cylindrical glass
A cylindrical glass beaker is
8
8
8
cm high and its circumference rim is
12
12
12
cm wide . Inside,
3
3
3
cm from the edge, there is a tiny drop of honey. In a point on its outer surface, belonging to the plane passing through the axis of the cylinder and for the drop of honey, and located
1
1
1
cm from the base (or bottom) of the glass, there is a fly. What is the shortest path that the fly must travel, walking on the surface from the glass, to the drop of honey, and how long is said path?[hide=original wording]Un vaso de vidrio cil´ındrico tiene 8 cm de altura y su borde 12 cm de circunferencia. En su interior, a 3 cm del borde, hay una diminuta gota de miel. En un punto de su superficie exterior, perteneciente al plano que pasa por el eje del cilindro y por la gota de miel, y situado a 1 cm de la base (o fondo) del vaso, hay una mosca. ¿Cu´al es el camino m´as corto que la mosca debe recorrer, andando sobre la superficie del vaso, hasta la gota de miel, y qu´e longitud tiene dicho camino?
7
1
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4 machines proude balls at factory, percentages
In a tennis ball factory there are
4
4
4
machines
m
1
,
m
2
,
m
3
,
m
4
m_1 , m_2 , m_3 , m_4
m
1
,
m
2
,
m
3
,
m
4
, which produce, respectively,
10
%
10\%
10%
,
20
%
20\%
20%
,
30
%
30\%
30%
and
40
%
40\%
40%
of the balls that come out of the factory. The machine
m
1
m_1
m
1
introduces defects in
1
%
1\%
1%
of the balls it manufactures, the machine
m
2
m_2
m
2
in
2
%
2\%
2%
,
m
3
m_3
m
3
in
4
%
4\%
4%
and
m
4
m_4
m
4
in
15
%
15\%
15%
. Of the balls manufactured In one day, one is chosen at random and it turns out to be defective. What is the probability that Has this ball been made by the machine
m
3
m_3
m
3
?
8
1
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a^4 + 4a^3 + 11a^2 + 6a+ 2 a sum of 3 squares and divisible by 4 for odd a
If
a
a
a
is an odd number, show that
a
4
+
4
a
3
+
11
a
2
+
6
a
+
2
a^4 + 4a^3 + 11a^2 + 6a+ 2
a
4
+
4
a
3
+
11
a
2
+
6
a
+
2
is a sum of three squares and is divisible by
4
4
4
.
1
1
Hide problems
7 + 77 + 777 +...+ 7... 7
Calculate the sum of
n
n
n
addends
7
+
77
+
777
+
.
.
.
+
7...7.
7 + 77 + 777 +...+ 7... 7.
7
+
77
+
777
+
...
+
7...7.
3
1
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angle of intersecting lines wanted such that symmetric to be coplanar
Given the intersecting lines
r
r
r
and
s
s
s
, consider the lines
u
u
u
and
v
v
v
as such what: a)
u
u
u
is symmetric to
r
r
r
with respect to
s
s
s
, b)
v
v
v
is symmetric to
s
s
s
with respect to
r
r
r
. Determine the angle that the given lines must form such that
u
u
u
and
v
v
v
to be coplanar.