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Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
2016 Spain Mathematical Olympiad
2016 Spain Mathematical Olympiad
Part of
Spain Mathematical Olympiad
Subcontests
(6)
4
1
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Integers between m^2 and m^2+m
Let
m
m
m
be a positive integer and
a
a
a
and
b
b
b
be distinct positive integers strictly greater than
m
2
m^2
m
2
and strictly less than
m
2
+
m
m^2+m
m
2
+
m
. Find all integers
d
d
d
such that
m
2
<
d
<
m
2
+
m
m^2 < d < m^2+m
m
2
<
d
<
m
2
+
m
and
d
d
d
divides
a
b
ab
ab
.
5
1
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Spanish Mathematical Olympiad 2016 Day 2 Problem 5
From all possible permutations from
(
a
1
,
a
2
,
.
.
.
,
a
n
)
(a_1,a_2,...,a_n)
(
a
1
,
a
2
,
...
,
a
n
)
from the set
{
1
,
2
,
.
.
,
n
}
\{1,2,..,n\}
{
1
,
2
,
..
,
n
}
,
n
≥
1
n\geq 1
n
≥
1
, consider the sets that satisfies the
2
(
a
1
+
a
2
+
.
.
.
+
a
m
)
2(a_1+a_2+...+a_m)
2
(
a
1
+
a
2
+
...
+
a
m
)
is divisible by
m
m
m
, for every
m
=
1
,
2
,
.
.
.
,
n
m=1,2,...,n
m
=
1
,
2
,
...
,
n
. Find the total number of permutations.
6
1
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Spanish Mathematical Olympiad Day 2 Problem 6
Let
n
≥
2
n\geq 2
n
≥
2
an integer. Find the least value of
γ
\gamma
γ
such that for any positive real numbers
x
1
,
x
2
,
.
.
.
,
x
n
x_1,x_2,...,x_n
x
1
,
x
2
,
...
,
x
n
with
x
1
+
x
2
+
.
.
.
+
x
n
=
1
x_1+x_2+...+x_n=1
x
1
+
x
2
+
...
+
x
n
=
1
and any real
y
1
+
y
2
+
.
.
.
+
y
n
=
1
y_1+y_2+...+y_n=1
y
1
+
y
2
+
...
+
y
n
=
1
and
0
≤
y
1
,
y
2
,
.
.
.
,
y
n
≤
1
2
0\leq y_1,y_2,...,y_n\leq \frac{1}{2}
0
≤
y
1
,
y
2
,
...
,
y
n
≤
2
1
the following inequality holds:
x
1
x
2
.
.
.
x
n
≤
γ
(
x
1
y
1
+
x
2
y
2
+
.
.
.
+
x
n
y
n
)
x_1x_2...x_n\leq \gamma \left(x_1y_1+x_2y_2+...+x_ny_n\right)
x
1
x
2
...
x
n
≤
γ
(
x
1
y
1
+
x
2
y
2
+
...
+
x
n
y
n
)
2
1
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Spain National Mathematical Olympiad 2016 Day 1 Problem 2
Given a positive prime number
p
p
p
. Prove that there exist a positive integer
α
\alpha
α
such that
p
∣
α
(
α
−
1
)
+
3
p|\alpha(\alpha-1)+3
p
∣
α
(
α
−
1
)
+
3
, if and only if there exist a positive integer
β
\beta
β
such that
p
∣
β
(
β
−
1
)
+
25
p|\beta(\beta-1)+25
p
∣
β
(
β
−
1
)
+
25
.
1
1
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Spain National Mathematical Olympiad 2016 Day 1 Problem 1
Two real number sequences are guiven, one arithmetic
(
a
n
)
n
∈
N
\left(a_n\right)_{n\in \mathbb {N}}
(
a
n
)
n
∈
N
and another geometric sequence
(
g
n
)
n
∈
N
\left(g_n\right)_{n\in \mathbb {N}}
(
g
n
)
n
∈
N
none of them constant. Those sequences verifies
a
1
=
g
1
≠
0
a_1=g_1\neq 0
a
1
=
g
1
=
0
,
a
2
=
g
2
a_2=g_2
a
2
=
g
2
and
a
10
=
g
3
a_{10}=g_3
a
10
=
g
3
. Find with proof that, for every positive integer
p
p
p
, there is a positive integer
m
m
m
, such that
g
p
=
a
m
g_p=a_m
g
p
=
a
m
.
3
1
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Spain National Mathematical Olympiad 2016 Day 1 Problem 3
In the circumscircle of a triangle
A
B
C
ABC
A
BC
, let
A
1
A_1
A
1
be the point diametrically opposed to the vertex
A
A
A
. Let
A
′
A'
A
′
the intersection point of
A
A
′
AA'
A
A
′
and
B
C
BC
BC
. The perpendicular to the line
A
A
′
AA'
A
A
′
from
A
′
A'
A
′
meets the sides
A
B
AB
A
B
and
A
C
AC
A
C
at
M
M
M
and
N
N
N
, respectively. Prove that the points
A
,
M
,
A
1
A,M,A_1
A
,
M
,
A
1
and
N
N
N
lie on a circle which has the center on the height from
A
A
A
of the triangle
A
B
C
ABC
A
BC
.