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Problems
Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
1982 Spain Mathematical Olympiad
1982 Spain Mathematical Olympiad
Part of
Spain Mathematical Olympiad
Subcontests
(8)
8
1
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locus of equidistant points
Given a set
C
C
C
of points in the plane, it is called the distance of a point
P
P
P
from the plane to the set
C
C
C
at the smallest of the distances from
P
P
P
to each of the points of
C
C
C
. Let the sets be
C
=
{
A
,
B
}
C = \{A,B\}
C
=
{
A
,
B
}
, with
A
=
(
1
,
0
)
A = (1, 0)
A
=
(
1
,
0
)
and
B
=
(
2
,
0
)
B = (2, 0)
B
=
(
2
,
0
)
; and
C
′
=
{
A
′
,
B
′
}
C'= \{A',B'\}
C
′
=
{
A
′
,
B
′
}
with
A
′
=
(
0
,
1
)
A' = (0, 1)
A
′
=
(
0
,
1
)
and
B
′
=
(
0
,
7
)
B' = (0, 7)
B
′
=
(
0
,
7
)
, in an orthogonal reference system. Find and draw the set
M
M
M
of points in the plane that are equidistant from
C
C
C
and
C
′
C'
C
′
. Study whether the function whose graph is the set
M
M
M
previously obtained is derivable.
7
1
Hide problems
set of 1/b where a is any integer and b an odd integer
Let
S
S
S
be the subset of rational numbers that can be written in the form
a
/
b
a/b
a
/
b
, where
a
a
a
is any integer and
b
b
b
is an odd integer. Does the sum of two of its elements belong to the
S
S
S
? And the product? Are there elements in
S
S
S
whose inverse belongs to
S
S
S
?
6
1
Hide problems
u^a v^b <= au + bv if u,v>=0 a,b>0 with a+b=1
Prove that if
u
,
v
u, v
u
,
v
are any nonnegative real numbers, and
a
,
b
a,b
a
,
b
positive real numbers such that
a
+
b
=
1
a + b = 1
a
+
b
=
1
, then
u
a
v
b
≤
a
u
+
b
v
.
u^a v^b \le au + bv.
u
a
v
b
≤
a
u
+
b
v
.
4
1
Hide problems
p(|z|) <= x^4 + y^4 where z=x + iy , p(0) = 0
Determine a polynomial of non-negative real coefficients that satisfies the following two conditions:
p
(
0
)
=
0
,
p
(
∣
z
∣
)
≤
x
4
+
y
4
,
p(0) = 0, p(|z|) \le x^4 + y^4,
p
(
0
)
=
0
,
p
(
∣
z
∣
)
≤
x
4
+
y
4
,
being
∣
z
∣
|z|
∣
z
∣
the module of the complex number
z
=
x
+
i
y
z = x + iy
z
=
x
+
i
y
.
3
1
Hide problems
lauched rocket, recovers and loses ...
A rocket is launched and reaches
120
120
120
m in height; in the fall he loses
60
60
60
m, then it recovers
40
40
40
m, loses
30
30
30
again, gains
24
24
24
, loses
20
20
20
, etc. If the process continues indefinitely, at what height does it tend to stabilize?
1
1
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2x2 diophantine system
On the puzzle page of a newspaper this problem is proposed: “Two children, Antonio and José, have
160
160
160
comics. Antonio counts his by
7
7
7
by
7
7
7
and there are
4
4
4
left over. José counts his
8
8
8
by
8
8
8
and he also has
4
4
4
left over. How many comics does he have each?" In the next issue of the newspaper this solution is given: “Antonio has
60
60
60
comics and José has
100
100
100
.” Analyze this solution and indicate what a mathematician would do with this problem.
2
1
Hide problems
composition of axis symmetry with right angle rotation
By composing a symmetry of axis
r
r
r
with a right angle rotation around from a point
P
P
P
that does not belong to the line, another movement
M
M
M
results. Is
M
M
M
an axis symmetry? Is there any line invariant through
M
M
M
?
5
1
Hide problems
square construction knowing sum of diagonal and side
Construct a square knowing the sum of the diagonal and the side.