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Problems
Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
1964 Spain Mathematical Olympiad
1964 Spain Mathematical Olympiad
Part of
Spain Mathematical Olympiad
Subcontests
(8)
8
1
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Last problem in first Spanish Olympiad
The points
A
A
A
and
B
B
B
lie on a horizontal line over a vertical plane. We consider the semicircumference passing through
A
A
A
and
B
B
B
that lies under the horizontal line. A segment of length
a
a
a
, with the same diameter that the semicircumference, moves in a way that always contains the point
A
A
A
and one of its extremes lies always on the semicircumference. Determine the value of the cosine of the angle between this segment and the horizontal line that makes the medium point of the segment to be as down as possible.
7
1
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Changing an order
A table with 1000 cards on a line, numbered from 1 to 1000, is considered. The cards are ordered in the usual way. Now, we proceed in the following way.The first card (which is 1) is put just before the last card (between 999 and 1000) and, after, the new first card (which is 2) is put after the last card (which was 1000). Show that after 1000 movements, the cards are ordered again in the usual way. Show that the analogous result (
n
n
n
movements for
n
n
n
cards) does not hold when
n
n
n
is odd.
6
1
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Graphical representation in 1st Spanish Math Olympiad
Make a graphical representation of the function
y
=
∣
∣
∣
x
−
1
∣
−
2
∣
−
3
∣
y=\vert \vert \vert x-1 \vert -2 \vert -3 \vert
y
=
∣∣∣
x
−
1∣
−
2∣
−
3∣
on the interval
−
8
≤
x
≤
8
-8 \leq x \leq 8
−
8
≤
x
≤
8
.
5
1
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Triangles in a pentagon
Given a regular pentagon, its five diagonals are drawn. How many triangles do appear in the figure? Classify the set of triangles in classes of equal triangles.
4
1
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Maximizing a length
We are given an equilateral triangle
A
B
C
ABC
A
BC
, of side
a
a
a
, inside its circumscribed circle. We consider the smallest of the two portions of circle limited between
A
B
AB
A
B
and the circumference. If we consider parallel lines to
B
C
BC
BC
, some of them cut the portion of circle in a segment. Which is the maximum possible length for one of the segments?
3
1
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About diagonals
A convex polygon of
n
n
n
sides is considered. All its diagonals are drawn and we suppose that any three of them can only intersect on a vertex and that there is no pair of parallel diagonals. Under these conditions, we wish to computea) The total number of intersection points of these diagonals, excluding the vertices.b) How many points, of these intersections, lie inside the polygon and how many lie outside.
2
1
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Problem about a tax
The RTP tax is a function
f
(
x
)
f(x)
f
(
x
)
, where
x
x
x
is the total of the annual profits (in pesetas). Knowing that:a)
f
(
x
)
f(x)
f
(
x
)
is a continuous function b) The derivative
d
f
(
x
)
d
x
\frac{df(x)}{dx}
d
x
df
(
x
)
on the interval
0
≤
6000
0 \leq 6000
0
≤
6000
is constant and equals zero; in the interval
6000
<
x
<
P
6000< x < P
6000
<
x
<
P
is constant and equals
1
1
1
; and when
x
>
P
x>P
x
>
P
is constant and equal 0.14. c)
f
(
0
)
=
0
f(0)=0
f
(
0
)
=
0
and
f
(
140000
)
=
14000
f(140000)=14000
f
(
140000
)
=
14000
.Determine the value of the amount
P
P
P
(in pesetas) and represent graphically the function
y
=
f
(
x
)
y=f(x)
y
=
f
(
x
)
.
1
1
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The first problem of the first Spanish Math Olympiad (1963-)
Given the equation
x
2
+
a
x
+
1
=
0
x^2+ax+1=0
x
2
+
a
x
+
1
=
0
, determine:a) The interval of possible values for
a
a
a
where the solutions to the previous equation are not real. b) The loci of the roots of the polynomial, when
a
a
a
is in the previous interval.