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Problems
Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
1985 Spain Mathematical Olympiad
1985 Spain Mathematical Olympiad
Part of
Spain Mathematical Olympiad
Subcontests
(8)
6
1
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intersecting rays, antiparallels, angle bisectors, find point
Let
O
X
OX
OX
and
O
Y
OY
O
Y
be non-collinear rays. Through a point
A
A
A
on
O
X
OX
OX
, draw two lines
r
1
r_1
r
1
and
r
2
r_2
r
2
that are antiparallel with respect to
∠
X
O
Y
\angle XOY
∠
XO
Y
. Let
r
1
r_1
r
1
cut
O
Y
OY
O
Y
at
M
M
M
and
r
2
r_2
r
2
cut
O
Y
OY
O
Y
at
N
N
N
. (Thus,
∠
O
A
M
=
∠
O
N
A
\angle OAM = \angle ONA
∠
O
A
M
=
∠
ON
A
). The bisectors of
∠
A
M
Y
\angle AMY
∠
A
M
Y
and
∠
A
N
Y
\angle ANY
∠
A
N
Y
meet at
P
P
P
. Determine the location of
P
P
P
.
8
1
Hide problems
exists a 3x3 matrix with constant sums and constant products also ?
A square matrix is sum-magic if the sum of all elements in each row, column and major diagonal is constant. Similarly, a square matrix is product-magic if the product of all elements in each row, column and major diagonal is constant. Determine if there exist
3
×
3
3\times 3
3
×
3
matrices of real numbers which are both sum-magic and product-magic.
2
1
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exists a subset E of Z \to Z so that ... ?
Determine if there exists a subset
E
E
E
of
Z
×
Z
Z \times Z
Z
×
Z
with the properties: (i)
E
E
E
is closed under addition, (ii)
E
E
E
contains
(
0
,
0
)
,
(0,0),
(
0
,
0
)
,
(iii) For every
(
a
,
b
)
≠
(
0
,
0
)
,
E
(a,b) \ne (0,0), E
(
a
,
b
)
=
(
0
,
0
)
,
E
contains exactly one of
(
a
,
b
)
(a,b)
(
a
,
b
)
and
−
(
a
,
b
)
-(a,b)
−
(
a
,
b
)
.Remark: We define
(
a
,
b
)
+
(
a
′
,
b
′
)
=
(
a
+
a
′
,
b
+
b
′
)
(a,b)+(a',b') = (a+a',b+b')
(
a
,
b
)
+
(
a
′
,
b
′
)
=
(
a
+
a
′
,
b
+
b
′
)
and
−
(
a
,
b
)
=
(
−
a
,
−
b
)
-(a,b) = (-a,-b)
−
(
a
,
b
)
=
(
−
a
,
−
b
)
.
1
1
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bijective map where images of lines are lines, and preserve parallels
Let
f
:
P
→
P
f : P\to P
f
:
P
→
P
be a bijective map from a plane
P
P
P
to itself such that: (i)
f
(
r
)
f (r)
f
(
r
)
is a line for every line
r
r
r
, (ii)
f
(
r
)
f (r)
f
(
r
)
is parallel to
r
r
r
for every line
r
r
r
. What possible transformations can
f
f
f
be?
4
1
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if k so that abc = k(a+b+c) then a^3+b^3+c^3 not prime, in positive integers
Prove that for each positive integer
k
k
k
there exists a triple
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
of positive integers such that
a
b
c
=
k
(
a
+
b
+
c
)
abc = k(a+b+c)
ab
c
=
k
(
a
+
b
+
c
)
. In all such cases prove that
a
3
+
b
3
+
c
3
a^3+b^3+c^3
a
3
+
b
3
+
c
3
is not a prime.
7
1
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2 roots of x^5-px-1 = 0, are roots of x^2-ax+b= 0$ for some integers a,b
Find the values of
p
p
p
for which the equation
x
5
−
p
x
−
1
=
0
x^5 - px-1 = 0
x
5
−
p
x
−
1
=
0
has two roots
r
r
r
and
s
s
s
which are the roots of equation
x
2
−
a
x
+
b
=
0
x^2-ax+b= 0
x
2
−
a
x
+
b
=
0
for some integers
a
,
b
a,b
a
,
b
.
5
1
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circumcircle of triangle of roots of cybic in C, z^3 +(-1+i)z^2+(1-i)z+i= 0
Find the equation of the circle in the complex plane determined by the roots of the equation
z
3
+
(
−
1
+
i
)
z
2
+
(
1
−
i
)
z
+
i
=
0
z^3 +(-1+i)z^2+(1-i)z+i= 0
z
3
+
(
−
1
+
i
)
z
2
+
(
1
−
i
)
z
+
i
=
0
.
3
1
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solve tan^2 2x+2 tan2x tan3x = 1
Solve the equation
t
a
n
2
2
x
+
2
t
a
n
2
x
t
a
n
3
x
=
1
tan^2 2x+2 tan2x tan3x = 1
t
a
n
2
2
x
+
2
t
an
2
x
t
an
3
x
=
1