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Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
2010 Spain Mathematical Olympiad
2010 Spain Mathematical Olympiad
Part of
Spain Mathematical Olympiad
Subcontests
(3)
3
2
Hide problems
Find greatest constant k for which inequality holds
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral.
A
C
AC
A
C
and
B
D
BD
B
D
meet at
P
P
P
, with
∠
A
P
D
=
6
0
∘
\angle APD=60^{\circ}
∠
A
P
D
=
6
0
∘
. Let
E
,
F
,
G
E,F,G
E
,
F
,
G
, and
H
H
H
be the midpoints of
A
B
,
B
C
,
C
D
AB,BC,CD
A
B
,
BC
,
C
D
and
D
A
DA
D
A
respectively. Find the greatest positive real number
k
k
k
for which
E
G
+
3
H
F
≥
k
d
+
(
1
−
k
)
s
EG+3HF\ge kd+(1-k)s
EG
+
3
H
F
≥
k
d
+
(
1
−
k
)
s
where
s
s
s
is the semi-perimeter of the quadrilateral
A
B
C
D
ABCD
A
BC
D
and
d
d
d
is the sum of the lengths of its diagonals. When does the equality hold?
f_A(N) is the number of different solutions to linear diop.
Let
p
p
p
be a prime number and
A
A
A
an infinite subset of the natural numbers. Let
f
A
(
n
)
f_A(n)
f
A
(
n
)
be the number of different solutions of
x
1
+
x
2
+
…
+
x
p
=
n
x_1+x_2+\ldots +x_p=n
x
1
+
x
2
+
…
+
x
p
=
n
, with
x
1
,
x
2
,
…
x
p
∈
A
x_1,x_2,\ldots x_p\in A
x
1
,
x
2
,
…
x
p
∈
A
. Does there exist a number
N
N
N
for which
f
A
(
n
)
f_A(n)
f
A
(
n
)
is constant for all
n
<
N
n<N
n
<
N
?
2
2
Hide problems
Spanish functional equation, find n for f(n)=2010
Let
N
0
\mathbb{N}_0
N
0
and
Z
\mathbb{Z}
Z
be the set of all non-negative integers and the set of all integers, respectively. Let
f
:
N
0
→
Z
f:\mathbb{N}_0\rightarrow\mathbb{Z}
f
:
N
0
→
Z
be a function defined as
f
(
n
)
=
−
f
(
⌊
n
3
⌋
)
−
3
{
n
3
}
f(n)=-f\left(\left\lfloor\frac{n}{3}\right\rfloor \right)-3\left\{\frac{n}{3}\right\}
f
(
n
)
=
−
f
(
⌊
3
n
⌋
)
−
3
{
3
n
}
where
⌊
x
⌋
\lfloor x \rfloor
⌊
x
⌋
is the greatest integer smaller than or equal to
x
x
x
and
{
x
}
=
x
−
⌊
x
⌋
\{ x\}=x-\lfloor x \rfloor
{
x
}
=
x
−
⌊
x
⌋
. Find the smallest integer
n
n
n
such that
f
(
n
)
=
2010
f(n)=2010
f
(
n
)
=
2010
.
PA' and B'C' intersect on A-median
In a triangle
A
B
C
ABC
A
BC
, let
P
P
P
be a point on the bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
and let
A
′
,
B
′
A',B'
A
′
,
B
′
and
C
′
C'
C
′
be points on lines
B
C
,
C
A
BC,CA
BC
,
C
A
and
A
B
AB
A
B
respectively such that
P
A
′
PA'
P
A
′
is perpendicular to
B
C
,
P
B
′
⊥
A
C
BC,PB'\perp AC
BC
,
P
B
′
⊥
A
C
, and
P
C
′
⊥
A
B
PC'\perp AB
P
C
′
⊥
A
B
. Prove that
P
A
′
PA'
P
A
′
and
B
′
C
′
B'C'
B
′
C
′
intersect on the median
A
M
AM
A
M
, where
M
M
M
is the midpoint of
B
C
BC
BC
.
1
2
Hide problems
'pucelana' sequence of 16 consecutive odd integers
A pucelana sequence is an increasing sequence of
16
16
16
consecutive odd numbers whose sum is a perfect cube. How many pucelana sequences are there with
3
3
3
-digit numbers only?
Spain MO 4 - Inequality
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be three positive real numbers. Show that
a
+
b
+
3
c
3
a
+
3
b
+
2
c
+
a
+
3
b
+
c
3
a
+
2
b
+
3
c
+
3
a
+
b
+
c
2
a
+
3
b
+
3
c
≥
15
8
\frac {a+b+3c}{3a+3b+2c}+\frac {a+3b+c}{3a+2b+3c}+\frac {3a+b+c}{2a+3b+3c} \ge \frac {15}{8}
3
a
+
3
b
+
2
c
a
+
b
+
3
c
+
3
a
+
2
b
+
3
c
a
+
3
b
+
c
+
2
a
+
3
b
+
3
c
3
a
+
b
+
c
≥
8
15