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Problems
Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
1976 Spain Mathematical Olympiad
1976 Spain Mathematical Olympiad
Part of
Spain Mathematical Olympiad
Subcontests
(8)
2
1
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sum of almost all r -tuples with components 1 or -1
Consider the set
C
C
C
of all
r
r
r
-tuple whose components are
1
1
1
or
−
1
-1
−
1
. Calculate the sum of all the components of all the elements of
C
C
C
excluding the
r
r
r
-tuple
(
1
,
1
,
1
,
.
.
.
,
1
)
(1, 1, 1, . . . , 1)
(
1
,
1
,
1
,
...
,
1
)
.
8
1
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y =|x^2 - 4x + 3|
Given the function
y
=
∣
x
2
−
4
x
+
3
∣
.
y =|x^2 - 4x + 3|.
y
=
∣
x
2
−
4
x
+
3∣.
Study its continuity and differentiability at the point of abscissa
1
1
1
. Its graph determines with the
X
X
X
axis a closed figure. Determine the area of said figure.
7
1
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max depreciation of price of diamond
The price of a diamond is proportional to the square of its weight. Show that, breaking it into two parts, there is a depreciation of its value. When is it the maximum depreciation?
6
1
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2 matrices wanted, one symmetric and one antisymmetric, with sum given matrix
Given a square matrix
M
M
M
of order
n
n
n
over the field of numbers real, find, as a function of
M
M
M
, two matrices, one symmetric and one antisymmetric, such that their sum is precisely
M
M
M
.
5
1
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z^4 + 4(i + 1)z + 1 = 0
Show that the equation
z
4
+
4
(
i
+
1
)
z
+
1
=
0
z^4 + 4(i + 1)z + 1 = 0
z
4
+
4
(
i
+
1
)
z
+
1
=
0
has a root in each quadrant of the complex plane.
4
1
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24 divides (n^5 -5n^3 + 4n)/(n + 2)
Show that the expression
n
5
−
5
n
3
+
4
n
n
+
2
\frac{n^5 -5n^3 + 4n}{n + 2}
n
+
2
n
5
−
5
n
3
+
4
n
where n is any integer, it is always divisible by
24
24
24
.
3
1
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product of geometric transformations
Through a lens that inverts the image we look at the rearview mirror of our car. If it reflects the license plate of the car that follows us,
C
S
−
3965
−
E
N
CS-3965-EN
CS
−
3965
−
EN
, draw the image we receive. Also draw the one obtained by permuting previous transformations, that is, reflecting in the mirror the image that the license plate gives the lens. Is the product of both transformations , reflection in the mirror and refraction through the lens, commutative?
1
1
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scent of a classic, square construction through for 4 given points
In a plane there are four fixed points
A
,
B
,
C
,
D
A, B, C, D
A
,
B
,
C
,
D
, no
3
3
3
collinear. Construct a square with sides
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
such that
A
∈
a
A \in a
A
∈
a
,
B
∈
b
B \in b
B
∈
b
,
C
∈
c
C \in c
C
∈
c
,
D
∈
d
D \in d
D
∈
d
.