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Problems
Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
2012 Spain Mathematical Olympiad
2012 Spain Mathematical Olympiad
Part of
Spain Mathematical Olympiad
Subcontests
(3)
3
2
Hide problems
n-x balls in x+1 boxes
Let
x
x
x
and
n
n
n
be integers such that
1
≤
x
≤
n
1\le x\le n
1
≤
x
≤
n
. We have
x
+
1
x+1
x
+
1
separate boxes and
n
−
x
n-x
n
−
x
identical balls. Define
f
(
n
,
x
)
f(n,x)
f
(
n
,
x
)
as the number of ways that the
n
−
x
n-x
n
−
x
balls can be distributed into the
x
+
1
x+1
x
+
1
boxes. Let
p
p
p
be a prime number. Find the integers
n
n
n
greater than
1
1
1
such that the prime number
p
p
p
is a divisor of
f
(
n
,
x
)
f(n,x)
f
(
n
,
x
)
for all
x
∈
{
1
,
2
,
…
,
n
−
1
}
x\in\{1,2,\ldots ,n-1\}
x
∈
{
1
,
2
,
…
,
n
−
1
}
.
R,P,D and S are concyclic
Let
A
B
C
ABC
A
BC
be an acute-angled triangle. Let
ω
\omega
ω
be the inscribed circle with centre
I
I
I
,
Ω
\Omega
Ω
be the circumscribed circle with centre
O
O
O
and
M
M
M
be the midpoint of the altitude
A
H
AH
A
H
where
H
H
H
lies on
B
C
BC
BC
. The circle
ω
\omega
ω
be tangent to the side
B
C
BC
BC
at the point
D
D
D
. The line
M
D
MD
M
D
cuts
ω
\omega
ω
at a second point
P
P
P
and the perpendicular from
I
I
I
to
M
D
MD
M
D
cuts
B
C
BC
BC
at
N
N
N
. The lines
N
R
NR
NR
and
N
S
NS
NS
are tangent to the circle
Ω
\Omega
Ω
at
R
R
R
and
S
S
S
respectively. Prove that the points
R
,
P
,
D
R,P,D
R
,
P
,
D
and
S
S
S
lie on the same circle.
2
2
Hide problems
Spanish functional equation
Find all functions
f
:
R
→
R
f:\mathbb{R}\to\mathbb{R}
f
:
R
→
R
such that
(
x
−
2
)
f
(
y
)
+
f
(
y
+
2
f
(
x
)
)
=
f
(
x
+
y
f
(
x
)
)
(x-2)f(y)+f(y+2f(x))=f(x+yf(x))
(
x
−
2
)
f
(
y
)
+
f
(
y
+
2
f
(
x
))
=
f
(
x
+
y
f
(
x
))
for all
x
,
y
∈
R
x,y\in\mathbb{R}
x
,
y
∈
R
.
Show every term in sequence is an integer
A sequence
(
a
n
)
n
≥
1
(a_n)_{n\ge 1}
(
a
n
)
n
≥
1
of integers is defined by the recurrence
a
1
=
1
,
a
2
=
5
,
a
n
=
a
n
−
1
2
+
4
a
n
−
2
for
n
≥
2.
a_1=1,\ a_2=5,\ a_n=\frac{a_{n-1}^2+4}{a_{n-2}}\ \text{for}\ n\ge 2.
a
1
=
1
,
a
2
=
5
,
a
n
=
a
n
−
2
a
n
−
1
2
+
4
for
n
≥
2.
Prove that all terms of the sequence are integers and find an explicit formula for
a
n
a_n
a
n
.
1
2
Hide problems
Can 3n^2+2n+2 be a square?
Determine if the number
λ
n
=
3
n
2
+
2
n
+
2
\lambda_n=\sqrt{3n^2+2n+2}
λ
n
=
3
n
2
+
2
n
+
2
is irrational for all non-negative integers
n
n
n
.
Diophantine for n and k
Find all positive integers
n
n
n
and
k
k
k
such that
(
n
+
1
)
n
=
2
n
k
+
3
n
+
1
(n+1)^n=2n^k+3n+1
(
n
+
1
)
n
=
2
n
k
+
3
n
+
1
.