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Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
1983 Spain Mathematical Olympiad
1983 Spain Mathematical Olympiad
Part of
Spain Mathematical Olympiad
Subcontests
(8)
1
1
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dogs breaks the rope around a cylindrical column
While Theophrastus was talking to Aristotle about the classification of plants, had a dog tied to a perfectly smooth cylindrical column of radius
r
r
r
, with a very fine rope that wrapped around the column and with a loop. The dog had the extreme free from the rope around his neck. In trying to reach Theophrastus, he put the rope tight and it broke. Find out how far from the column the knot was in the time to break the rope.[hide=original wording]Mientras Teofrasto hablaba con Arist´oteles sobre la clasificaci´on de las plantas, ten´ıa un perro atado a una columna cil´ındrica perfectamente lisa de radio r, con una cuerda muy fina que envolv´ıa la columna y con un lazo. El perro ten´ıa el extremo libre de la cuerda cogido a su cuello. Al intentar alcanzar a Teofrasto, puso la cuerda tirante y ´esta se rompi´o. Averiguar a qu´e distancia de la columna estaba el nudo en el momento de romperse la cuerda.
3
1
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arithmetic mean of areas of triangles by n+1 points on semicircle and 2 ends
A semicircle of radius
r
r
r
is divided into
n
+
1
n + 1
n
+
1
equal parts and any point
k
k
k
of the division with the ends of the semicircle forms a triangle
A
k
A_k
A
k
. Calculate the limit, as
n
n
n
tends to infinity, of the arithmetic mean of the areas of the triangles.
7
1
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2 liters of water in a regular tetrahedron of edge 30 cm
A regular tetrahedron with an edge of
30
30
30
cm rests on one of its faces. Assuming it is hollow,
2
2
2
liters of water are poured into it. Find the height of the ''upper'' liquid and the area of the ''free'' surface of the water.
2
1
Hide problems
triangle construction given A. b/c and r
Construct a triangle knowing an angle, the ratio of the sides that form it and the radius of the inscribed circle.
8
1
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3 brothers age problem
In
1960
1960
1960
, the oldest of three brothers has an age that is the sum of the of his younger siblings. A few years later, the sum of the ages of two of brothers is double that of the other. A number of years have now passed since
1960
1960
1960
, which is equal to two thirds of the sum of the ages that the three brothers were at that year, and one of them has reached
21
21
21
years. What is the age of each of the others two?
6
1
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3x2 diophantine system
In a cafeteria, a glass of lemonade, three sandwiches and seven biscuits have cost
1
1
1
shilling and
2
2
2
pence, and a glass of lemonade, four sandwiches and
10
10
10
biscuits they are worth
1
1
1
shilling and
5
5
5
pence. Find the price of: a) a glass of lemonade, a sandwich and a cake; b) two glasses of lemonade, three sandwiches and five biscuits. (
1
1
1
shilling =
12
12
12
pence).
5
1
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coordinates of vertices of square wanted
Find the coordinates of the vertices of a square
A
B
C
D
ABCD
A
BC
D
, knowing that
A
A
A
is on the line
y
−
2
x
−
6
=
0
y -2x -6 = 0
y
−
2
x
−
6
=
0
,
C
C
C
at
x
=
0
x = 0
x
=
0
and
B
B
B
is the point
(
a
,
0
)
(a, 0)
(
a
,
0
)
, being
a
=
log
2
/
3
(
16
/
81
)
a = \log_{2/3}(16/81)
a
=
lo
g
2/3
(
16/81
)
.
4
1
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16x^5 - 20x^3 + 5x + m = 0
Determine the number of real roots of the equation
16
x
5
−
20
x
3
+
5
x
+
m
=
0.
16x^5 - 20x^3 + 5x + m = 0.
16
x
5
−
20
x
3
+
5
x
+
m
=
0.