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Problems
Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
1975 Spain Mathematical Olympiad
1975 Spain Mathematical Olympiad
Part of
Spain Mathematical Olympiad
Subcontests
(8)
1
1
Hide problems
lim 1/n (sum x^k/n^k)
Calculate the limit
lim
n
→
∞
1
n
(
1
n
k
+
2
k
n
k
+
.
.
.
.
+
(
n
−
1
)
k
n
k
+
n
k
n
k
)
.
\lim_{n \to \infty} \frac{1}{n} \left(\frac{1}{n^k} +\frac{2^k}{n^k} +....+\frac{(n-1)^k}{n^k} +\frac{n^k}{n^k}\right).
n
→
∞
lim
n
1
(
n
k
1
+
n
k
2
k
+
....
+
n
k
(
n
−
1
)
k
+
n
k
n
k
)
.
(For the calculation of the limit, the integral construction procedure can be followed).
7
1
Hide problems
f(x) =1/(|x + 3| + |x + 1| + |x - 2| + |x -5|)
Consider the real function defined by
f
(
x
)
=
1
∣
x
+
3
∣
+
∣
x
+
1
∣
+
∣
x
−
2
∣
+
∣
x
−
5
∣
f(x) =\frac{1}{|x + 3| + |x + 1| + |x - 2| + |x -5|}
f
(
x
)
=
∣
x
+
3∣
+
∣
x
+
1∣
+
∣
x
−
2∣
+
∣
x
−
5∣
1
for all
x
∈
R
x \in R
x
∈
R
. a) Determine its maximum. b) Graphic representation.
6
1
Hide problems
x_{n+2} = x_{n+1} + 2x_n, y_{n+2} = 2y_{n+1} + 3y_n
Let
{
x
n
}
\{x_n\}
{
x
n
}
and
{
y
n
}
\{y_n\}
{
y
n
}
be two sequences of natural numbers defined as follow:
x
1
=
1
,
x
2
=
1
,
x
n
+
2
=
x
n
+
1
+
2
x
n
x_1 = 1, \,\,\, x_2 = 1, \,\,\, x_{n+2} = x_{n+1} + 2x_n
x
1
=
1
,
x
2
=
1
,
x
n
+
2
=
x
n
+
1
+
2
x
n
for
n
=
1
,
2
,
3
,
.
.
.
n = 1, 2, 3, ...
n
=
1
,
2
,
3
,
...
y
1
=
1
,
y
2
=
7
,
y
n
+
2
=
2
y
n
+
1
+
3
y
n
y_1 = 1, \,\,\, y_2 = 7, \,\,\, y_{n+2} = 2y_{n+1} + 3y_n
y
1
=
1
,
y
2
=
7
,
y
n
+
2
=
2
y
n
+
1
+
3
y
n
for
n
=
1
,
2
,
3
,
.
.
.
n = 1, 2, 3, ...
n
=
1
,
2
,
3
,
...
Prove that, except for the case
x
1
=
y
1
=
1
x_1 = y_1 = 1
x
1
=
y
1
=
1
, there is no natural value that occurs in the two sequences.
8
1
Hide problems
two real numbers between 0 and 1 are randomly chosen
Two real numbers between
0
0
0
and
1
1
1
are randomly chosen. Calculate the probability that any one of them is less than the square of the other.
5
1
Hide problems
point construction M on line r such that <(r,AM)=2<(r,BM) fixed points A,B
In the plane we have a line
r
r
r
and two points
A
A
A
and
B
B
B
outside the line and in the same half plane. Determine a point
M
M
M
on the line such that the angle of
r
r
r
with
A
M
AM
A
M
is double that of
r
r
r
with
B
M
BM
BM
. (Consider the smaller angle of two lines of the angles they form).
2
1
Hide problems
(1 +1/x)^x
Study the real function
f
(
x
)
=
(
1
+
1
x
)
x
f(x) = \left(1 +\frac{1}{x}\right)^x
f
(
x
)
=
(
1
+
x
1
)
x
defined for
x
∈
R
−
{
−
1
,
0
}
x \in R - \{-1, 0\}
x
∈
R
−
{
−
1
,
0
}
. Graphic representation.
4
1
Hide problems
sum X_i >= n if their product =1 for x_i>0
Prove that if the product of
n
n
n
real and positive numbers is equal to
1
1
1
, its sum is greater than or equal to
n
n
n
.
3
1
Hide problems
Z_{(5)} , a subset of Q
We will designate by
Z
(
5
)
Z_{(5)}
Z
(
5
)
a certain subset of the set
Q
Q
Q
of the rational numbers . A rational belongs to
Z
(
5
)
Z_{(5)}
Z
(
5
)
if and only if there exist equal fraction to this rational such that
5
5
5
is not a divisor of its denominator. (For example, the rational number
13
/
10
13/10
13/10
does not belong to
Z
(
5
)
Z_{(5)}
Z
(
5
)
, since the denominator of all fractions equal to
13
/
10
13/10
13/10
is a multiple of
5
5
5
. On the other hand, the rational
75
/
10
75/10
75/10
belongs to
Z
(
5
)
Z_{(5)}
Z
(
5
)
since that
75
/
10
=
15
/
12
75/10 = 15/12
75/10
=
15/12
). Reasonably answer the following questions: a) What algebraic structure (semigroup, group, etc.) does
Z
(
5
)
Z_{(5)}
Z
(
5
)
have with respect to the sum? b) And regarding the product? c) Is
Z
(
5
)
Z_{(5)}
Z
(
5
)
a subring of
Q
Q
Q
? d) Is
Z
(
5
)
Z_{(5)}
Z
(
5
)
a vector space?