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Problems
Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
2018 Spain Mathematical Olympiad
2018 Spain Mathematical Olympiad
Part of
Spain Mathematical Olympiad
Subcontests
(5)
3
1
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Geometric Inequality
Let
A
B
C
ABC
A
BC
be an acute-angled triangle with circumcenter
O
O
O
and let
M
M
M
be a point on
A
B
AB
A
B
. The circumcircle of
A
M
O
AMO
A
MO
intersects
A
C
AC
A
C
a second time on
K
K
K
and the circumcircle of
B
O
M
BOM
BOM
intersects
B
C
BC
BC
a second time on
N
N
N
.Prove that
[
M
N
K
]
≥
[
A
B
C
]
4
\left[MNK\right] \geq \frac{\left[ABC\right]}{4}
[
MN
K
]
≥
4
[
A
BC
]
and determine the equality case.
5
1
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Integers that can be expressed as ax+by
Let
a
,
b
a, b
a
,
b
be coprime positive integers. A positive integer
n
n
n
is said to be weak if there do not exist any nonnegative integers
x
,
y
x, y
x
,
y
such that
a
x
+
b
y
=
n
ax+by=n
a
x
+
b
y
=
n
. Prove that if
n
n
n
is a weak integer and
n
<
a
b
6
n < \frac{ab}{6}
n
<
6
ab
, then there exists an integer
k
≥
2
k \geq 2
k
≥
2
such that
k
n
kn
kn
is weak.
1
1
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2x+1 is a perfect square but the following x+1 integers are not.
Find all positive integers
x
x
x
such that
2
x
+
1
2x+1
2
x
+
1
is a perfect square but none of the integers
2
x
+
2
,
2
x
+
3
,
…
,
3
x
+
2
2x+2, 2x+3, \ldots, 3x+2
2
x
+
2
,
2
x
+
3
,
…
,
3
x
+
2
are perfect squares.
4
1
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Coloring points on a spherical surface.
Points on a spherical surface with radius
4
4
4
are colored in
4
4
4
different colors. Prove that there exist two points with the same color such that the distance between them is either
4
3
4\sqrt{3}
4
3
or
2
6
2\sqrt{6}
2
6
.(Distance is Euclidean, that is, the length of the straight segment between the points)
6
1
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f(x+f(y))=yf(xy+1)
Find all functions such that f: \mathbb{R}^\plus{} \rightarrow \mathbb{R}^\plus{} and f(x\plus{}f(y))\equal{}yf(xy\plus{}1) for every x,y\in \mathbb{R}^\plus{}.