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Problems
Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
1965 Spain Mathematical Olympiad
1965 Spain Mathematical Olympiad
Part of
Spain Mathematical Olympiad
Subcontests
(8)
8
1
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Infinite circles
Let be
γ
1
\gamma_1
γ
1
a circumference of radius
r
r
r
and
P
P
P
an exterior point that is at distance
a
a
a
from the centre of
γ
1
\gamma_1
γ
1
. We build two tangent lines
r
,
s
r,s
r
,
s
to
γ
1
\gamma_1
γ
1
from
P
P
P
and
γ
2
\gamma_2
γ
2
is constructed as a smaller circumference, tangent to both lines and, also, tangent to
γ
1
\gamma_1
γ
1
. We construct inductively
γ
n
+
1
\gamma_{n+1}
γ
n
+
1
as a tangent circumference to
γ
n
\gamma_{n}
γ
n
and, also, tangent to
r
r
r
and
s
s
s
. Determine:a) The radius of
γ
2
\gamma_2
γ
2
. b) The radius of
γ
n
\gamma_n
γ
n
. c) The sum of the lengths of
γ
1
,
γ
2
,
γ
3
,
.
.
.
\gamma_1, \gamma_2, \gamma_3, ...
γ
1
,
γ
2
,
γ
3
,
...
.
7
1
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Maximizing a mass
A truncated cone has the bigger base of radius
r
r
r
centimetres and the generatrix makes an angle, with that base, whose tangent equals
m
m
m
. The truncated cone is constructed of a material of density
d
d
d
(g/cm
3
^3
3
) and the smaller base is covered by a special material of density
p
p
p
(g/cm
2
^2
2
). Which is the height of the truncated cone that maximizes the total mass?
6
1
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A triangle that fits on a sphere
We have an empty equilateral triangle with length of a side
l
l
l
. We put the triangle, horizontally, over a sphere of radius
r
r
r
. Clearly, if the triangle is small enough, the triangle is held by the sphere. Which is the distance between any vertex of the triangle and the centre of the sphere (as a function of
l
l
l
and
r
r
r
)?
5
1
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A wrong step?
It is well-known that if
p
q
=
r
s
\frac{p}{q}=\frac{r}{s}
q
p
=
s
r
, both of the expressions are also equal to
p
−
r
q
−
s
\frac{p-r}{q-s}
q
−
s
p
−
r
. Now we write the equality
3
x
−
b
3
x
−
5
b
=
3
a
−
4
b
3
a
−
8
b
.
\frac{3x-b}{3x-5b}=\frac{3a-4b}{3a-8b}.
3
x
−
5
b
3
x
−
b
=
3
a
−
8
b
3
a
−
4
b
.
The previous property shows that both fractions should be equal to
3
x
−
b
−
3
a
+
4
b
3
x
−
5
b
−
3
a
+
8
b
=
3
x
−
3
a
+
3
b
3
x
−
3
a
+
3
b
=
1.
\frac{3x-b-3a+4b}{3x-5b-3a+8b}=\frac{3x-3a+3b}{3x-3a+3b}=1.
3
x
−
5
b
−
3
a
+
8
b
3
x
−
b
−
3
a
+
4
b
=
3
x
−
3
a
+
3
b
3
x
−
3
a
+
3
b
=
1.
However, the initial fractions given may not be equal to
1
1
1
. Explain what is going on.
4
1
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For which $x$ does an inequality hold?
Find all the intervals
I
I
I
where any element of the interval
x
∈
I
x \in I
x
∈
I
satisfies
cos
x
+
sin
x
>
1.
\cos x +\sin x >1.
cos
x
+
sin
x
>
1.
Do the same computation when
x
x
x
satisfies
cos
x
+
∣
sin
x
∣
>
1.
\cos x +\vert \sin x \vert>1.
cos
x
+
∣
sin
x
∣
>
1.
3
1
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Length of an spiral
A disk in a record turntable makes
100
100
100
revolutions per minute and it plays during
24
24
24
minutes and
30
30
30
seconds. The recorded line over the disk is a spiral with a diameter that decreases uniformly from
29
29
29
cm to
11.5
11.5
11.5
cm. Compute the length of the recorded line.
2
1
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How many numbers of three digits...
How many numbers of
3
3
3
digits have their central digit greater than any of the other two? How many of them have also three different digits?
1
1
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Common area between triangles
We consider an equilateral triangle with its circumscribed circle, of center
O
O
O
, and radius
4
4
4
cm. We rotate the triangle
90
º
90º
90º
around
O
O
O
. Compute the common area that was covered by the previous position of the triangle and is also covered by the new one.