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Problems
Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
2014 Spain Mathematical Olympiad
2014 Spain Mathematical Olympiad
Part of
Spain Mathematical Olympiad
Subcontests
(3)
3
2
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Locus on a Circle
Let
B
B
B
and
C
C
C
be two fixed points on a circle centered at
O
O
O
that are not diametrically opposed. Let
A
A
A
be a variable point on the circle distinct from
B
B
B
and
C
C
C
and not belonging to the perpendicular bisector of
B
C
BC
BC
. Let
H
H
H
be the orthocenter of
△
A
B
C
\triangle ABC
△
A
BC
, and
M
M
M
and
N
N
N
be the midpoints of the segments
B
C
BC
BC
and
A
H
AH
A
H
, respectively. The line
A
M
AM
A
M
intersects the circle again at
D
D
D
, and finally,
N
M
NM
NM
and
O
D
OD
O
D
intersect at
P
P
P
. Determine the locus of points
P
P
P
as
A
A
A
moves around the circle.
60 Points in a Unit Circle
60
60
60
points are on the interior of a unit circle (a circle with radius
1
1
1
). Show that there exists a point
V
V
V
on the circumference of the circle such that the sum of the distances from
V
V
V
to the
60
60
60
points is less than or equal to
80
80
80
.
2
2
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Rational Square Root
Given the rational numbers
r
r
r
,
q
q
q
, and
n
n
n
, such that
1
r
+
q
n
+
1
q
+
r
n
=
1
r
+
q
\displaystyle\frac1{r+qn}+\frac1{q+rn}=\frac1{r+q}
r
+
q
n
1
+
q
+
r
n
1
=
r
+
q
1
, prove that
n
−
3
n
+
1
\displaystyle\sqrt{\frac{n-3}{n+1}}
n
+
1
n
−
3
is a rational number.
a^2+13b^2
Let
M
M
M
be the set of all integers in the form of
a
2
+
13
b
2
a^2+13b^2
a
2
+
13
b
2
, where
a
a
a
and
b
b
b
are distinct itnegers.i) Prove that the product of any two elements of
M
M
M
is also an element of
M
M
M
.ii) Determine, reasonably, if there exist infinite pairs of integers
(
x
,
y
)
(x,y)
(
x
,
y
)
so that
x
+
y
∉
M
x+y\not\in M
x
+
y
∈
M
but
x
13
+
y
13
∈
M
x^{13}+y^{13}\in M
x
13
+
y
13
∈
M
.
1
2
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Consecutive 3 Numbers on a Circle
Is it possible to place the numbers
0
,
1
,
2
,
…
,
9
0,1,2,\dots,9
0
,
1
,
2
,
…
,
9
on a circle so that the sum of any three consecutive numbers is a) 13, b) 14, c) 15?
Sequence with cubes
Let
(
x
n
)
(x_n)
(
x
n
)
be a sequence of positive integers defined by
x
1
=
2
x_1=2
x
1
=
2
and
x
n
+
1
=
2
x
n
3
+
x
n
x_{n+1}=2x_n^3+x_n
x
n
+
1
=
2
x
n
3
+
x
n
for all integers
n
≥
1
n\ge1
n
≥
1
. Determine the largest power of
5
5
5
that divides
x
2014
2
+
1
x_{2014}^2+1
x
2014
2
+
1
.