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National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
2017 Spain Mathematical Olympiad
2017 Spain Mathematical Olympiad
Part of
Spain Mathematical Olympiad
Subcontests
(6)
6
1
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Problem 6 Spain 2017
In the triangle
A
B
C
ABC
A
BC
, the respective mid points of the sides
B
C
BC
BC
,
A
B
AB
A
B
and
A
C
AC
A
C
are
D
D
D
,
E
E
E
and
F
F
F
. Let
M
M
M
be the point where the internal bisector of the angle
∠
A
D
B
\angle ADB
∠
A
D
B
intersects the side
A
B
AB
A
B
, and
N
N
N
the point where the internal bisector of the angle
∠
A
D
C
\angle ADC
∠
A
D
C
intersects the side
A
C
AC
A
C
. Also, let
O
O
O
be the intersection point of
A
D
AD
A
D
and
M
N
MN
MN
,
P
P
P
the intersection point of
A
B
AB
A
B
and
F
O
FO
FO
, and
R
R
R
the intersection point of
A
C
AC
A
C
and
E
O
EO
EO
. Prove that
P
R
=
A
D
PR=AD
PR
=
A
D
.
5
1
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Problem 5 Spain 2017
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers so that
a
+
b
+
c
=
1
3
a+b+c = \frac{1}{\sqrt{3}}
a
+
b
+
c
=
3
1
. Find the maximum value of
27
a
b
c
+
a
a
2
+
2
b
c
+
b
b
2
+
2
c
a
+
c
c
2
+
2
a
b
.
27abc+a\sqrt{a^2+2bc}+b\sqrt{b^2+2ca}+c\sqrt{c^2+2ab}.
27
ab
c
+
a
a
2
+
2
b
c
+
b
b
2
+
2
c
a
+
c
c
2
+
2
ab
.
4
1
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Problem 4 Spain 2017
You are given a row made by
2018
2018
2018
squares, numbered consecutively from
0
0
0
to
2017
2017
2017
. Initially, there is a coin in the square
0
0
0
. Two players
A
A
A
and
B
B
B
play alternatively, starting with
A
A
A
, on the following way: In his turn, each player can either make his coin advance
53
53
53
squares or make the coin go back
2
2
2
squares. On each move the coin can never go to a number less than
0
0
0
or greater than
2017
2017
2017
. The player who puts the coin on the square
2017
2017
2017
wins. ¿Who is the one with the wining strategy and how should he play to win?
3
1
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Problem 3 Spain 2017
Let
p
p
p
be an odd prime and
S
q
=
1
2
∗
3
∗
4
+
1
5
∗
6
∗
7
+
.
.
.
+
1
q
(
q
+
1
)
(
q
+
2
)
S_{q} = \frac{1}{2*3*4} + \frac{1}{5*6*7} + ... + \frac{1}{q(q+1)(q+2)}
S
q
=
2
∗
3
∗
4
1
+
5
∗
6
∗
7
1
+
...
+
q
(
q
+
1
)
(
q
+
2
)
1
, where
q
=
3
p
−
5
2
q = \frac{3p-5}{2}
q
=
2
3
p
−
5
. We write
1
2
−
2
S
q
\frac{1}{2}-2S_{q}
2
1
−
2
S
q
in the form
m
n
\frac{m}{n}
n
m
, where
m
m
m
and
n
n
n
are integers. Prove that
m
≡
n
(
m
o
d
p
)
m \equiv n (mod p)
m
≡
n
(
m
o
d
p
)
2
1
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Problem 2 Spain 2017
A midpoint plotter is an instrument which draws the exact mid point of two point previously drawn. Starting off two points
1
1
1
unit of distance apart and using only the midpoint plotter, you have to get two point which are strictly at a distance between
1
2017
\frac{1}{2017}
2017
1
and
1
2016
\frac{1}{2016}
2016
1
units, drawing the minimum amount of points. ¿Which is the minimum number of times you will need to use the midpoint plotter and what strategy should you follow to achieve it?
1
1
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Problem 1 Spain 2017
Find the amount of different values given by the following expression:
n
2
−
2
n
2
−
n
+
2
\frac{n^2-2}{n^2-n+2}
n
2
−
n
+
2
n
2
−
2
where
n
∈
{
1
,
2
,
3
,
.
.
,
100
}
n \in \{1,2,3,..,100\}
n
∈
{
1
,
2
,
3
,
..
,
100
}