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National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
2005 Spain Mathematical Olympiad
2005 Spain Mathematical Olympiad
Part of
Spain Mathematical Olympiad
Subcontests
(3)
3
2
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Spanish Mathematical Olympiad 2005, Problem 3
We will say that a triangle is multiplicative if the product of the heights of two of its sides is equal to the length of the third side. Given
A
B
C
…
X
Y
Z
ABC \dots XYZ
A
BC
…
X
Y
Z
is a regular polygon with
n
n
n
sides of length
1
1
1
. The
n
−
3
n-3
n
−
3
diagonals that go out from vertex
A
A
A
divide the triangle
Z
A
B
ZAB
Z
A
B
in
n
−
2
n-2
n
−
2
smaller triangles. Prove that each one of these triangles is multiplicative.
triangle with b+c=2a
In a triangle with sides
a
,
b
,
c
a, b, c
a
,
b
,
c
the side
a
a
a
is the arithmetic mean of
b
b
b
and
c
c
c
. Prove that: a)
0
o
≤
A
≤
6
0
o
0^o \le A \le 60^o
0
o
≤
A
≤
6
0
o
. b) The height relative to side
a
a
a
is three times the inradius
r
r
r
. c) The distance from the circumcenter to side
a
a
a
is
R
−
r
R - r
R
−
r
, where
R
R
R
is the circumradius.
2
2
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min {r-s^2,s-u^2, u-v^2,v-r^2 } <= 1/4
Let
r
,
s
,
u
,
v
r,s,u,v
r
,
s
,
u
,
v
be real numbers. Prove that:
m
i
n
{
r
−
s
2
,
s
−
u
2
,
u
−
v
2
,
v
−
r
2
}
≤
1
4
min\{r-s^2,s-u^2, u-v^2,v-r^2\}\le \frac{1}{4}
min
{
r
−
s
2
,
s
−
u
2
,
u
−
v
2
,
v
−
r
2
}
≤
4
1
Spanish Mathematical Olympiad 2005, Problem 2
Is it possible to color points in the Cartesian Plane
(
x
,
y
)
(x,y)
(
x
,
y
)
with integer coordinates with three colors, such that each color appears infinitely many times on infinitely many lines parallel to the
x
x
x
-axis and that any three points, each of a different color, are not in a line? Justify your answer.
1
2
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Spanish Mathematical Olympiad 2005, Problem 1
Let
a
a
a
and
b
b
b
be integers. Demonstrate that the equation
(
x
−
a
)
(
x
−
b
)
(
x
−
3
)
+
1
=
0
(x-a)(x-b)(x-3) +1 = 0
(
x
−
a
)
(
x
−
b
)
(
x
−
3
)
+
1
=
0
has an integer solution.
decimal expression of 1/n+1/(n+1)+1/(n+2) is periodic
Prove that for every positive integer
n
n
n
, the decimal expression of
1
n
+
1
n
+
1
+
1
n
+
2
\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}
n
1
+
n
+
1
1
+
n
+
2
1
is periodic .