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Problems
Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
2008 Spain Mathematical Olympiad
2008 Spain Mathematical Olympiad
Part of
Spain Mathematical Olympiad
Subcontests
(3)
3
2
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Colouring every point in the plane one of 7 colours
Every point in the plane is coloured one of seven distinct colours. Is there an inscribed trapezoid whose vertices are all of the same colour?
Maximum number of regions after p-secting sides of triangle
Let
p
≥
3
p\ge 3
p
≥
3
be a prime number. Each side of a triangle is divided into
p
p
p
equal parts, and we draw a line from each division point to the opposite vertex. Find the maximum number of regions, every two of them disjoint, that are formed inside the triangle.
2
2
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Spanish inequality for 0<a,b<1
Let
a
a
a
and
b
b
b
be two real numbers, with
0
<
a
,
b
<
1
0<a,b<1
0
<
a
,
b
<
1
. Prove that
a
b
2
+
a
2
b
+
(
1
−
a
)
(
1
−
b
)
2
+
(
1
−
a
)
2
(
1
−
b
)
<
2
\sqrt{ab^2+a^2b}+\sqrt{(1-a)(1-b)^2+(1-a)^2(1-b)}<\sqrt{2}
a
b
2
+
a
2
b
+
(
1
−
a
)
(
1
−
b
)
2
+
(
1
−
a
)
2
(
1
−
b
)
<
2
Two fixed points no r such that (CM)(DN) is constant
Given a circle, two fixed points
A
A
A
and
B
B
B
and a variable point
P
P
P
, all of them on the circle, and a line
r
r
r
,
P
A
PA
P
A
and
P
B
PB
PB
intersect
r
r
r
at
C
C
C
and
D
D
D
, respectively. Find two fixed points on
r
r
r
,
M
M
M
and
N
N
N
, such that
C
M
⋅
D
N
CM\cdot DN
CM
⋅
D
N
is constant for all
P
P
P
.
1
2
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Arithmetic mean of the divisors of n is an integer.
Let
p
p
p
and
q
q
q
be two different prime numbers. Prove that there are two positive integers,
a
a
a
and
b
b
b
, such that the arithmetic mean of the divisors of
n
=
p
a
q
b
n=p^aq^b
n
=
p
a
q
b
is an integer.
Find (a,b) if given a+b and lcm(a,b)
Find two positive integers
a
a
a
and
b
b
b
, when their sum and their least common multiple is given. Find the numbers when the sum is
3972
3972
3972
and the least common multiple is
985928
985928
985928
.