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Problems
Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
2020 Spain Mathematical Olympiad
2020 Spain Mathematical Olympiad
Part of
Spain Mathematical Olympiad
Subcontests
(6)
6
1
Hide problems
Bounds on the number of elements representable as difference of squares/cubes
Let
S
S
S
be a finite set of integers. We define
d
2
(
S
)
d_2(S)
d
2
(
S
)
and
d
3
(
S
)
d_3(S)
d
3
(
S
)
as:
∙
\bullet
∙
d
2
(
S
)
d_2(S)
d
2
(
S
)
is the number of elements
a
∈
S
a \in S
a
∈
S
such that there exist
x
,
y
∈
Z
x, y \in \mathbb{Z}
x
,
y
∈
Z
such that
x
2
−
y
2
=
a
x^2-y^2 = a
x
2
−
y
2
=
a
∙
\bullet
∙
d
3
(
S
)
d_3(S)
d
3
(
S
)
is the number of elements
a
∈
S
a \in S
a
∈
S
such that there exist
x
,
y
∈
Z
x, y \in \mathbb{Z}
x
,
y
∈
Z
such that
x
3
−
y
3
=
a
x^3-y^3 = a
x
3
−
y
3
=
a
(a) Let
m
m
m
be an integer and
S
=
{
m
,
m
+
1
,
…
,
m
+
2019
}
S = \{m, m+1, \ldots, m+2019\}
S
=
{
m
,
m
+
1
,
…
,
m
+
2019
}
. Prove:
d
2
(
S
)
>
13
7
d
3
(
S
)
d_2(S) > \frac{13}{7} d_3(S)
d
2
(
S
)
>
7
13
d
3
(
S
)
(b) Let
S
n
=
{
1
,
2
,
…
,
n
}
S_n = \{1, 2, \ldots, n\}
S
n
=
{
1
,
2
,
…
,
n
}
with
n
n
n
a positive integer. Prove that there exists a
N
N
N
so that for all
n
>
N
n > N
n
>
N
:
d
2
(
S
n
)
>
4
⋅
d
3
(
S
n
)
d_2(S_n) > 4 \cdot d_3(S_n)
d
2
(
S
n
)
>
4
⋅
d
3
(
S
n
)
5
1
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AC+BC = sqrt(2)AB -> O(BMP) tangent to AC.
In an acute-angled triangle
A
B
C
ABC
A
BC
, let
M
M
M
be the midpoint of
A
B
AB
A
B
and
P
P
P
the foot of the altitude to
B
C
BC
BC
. Prove that if
A
C
+
B
C
=
2
A
B
AC+BC = \sqrt{2}AB
A
C
+
BC
=
2
A
B
, then the circumcircle of triangle
B
M
P
BMP
BMP
is tangent to
A
C
AC
A
C
.
4
1
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Adding game.
Ana and Benito play a game which consists of
2020
2020
2020
turns. Initially, there are
2020
2020
2020
cards on the table, numbered from
1
1
1
to
2020
2020
2020
, and Ana possesses an extra card with number
0
0
0
. In the
k
k
k
-th turn, the player that doesn't possess card
k
−
1
k-1
k
−
1
chooses whether to take the card with number
k
k
k
or to give it to the other player. The number in each card indicates its value in points. At the end of the game whoever has most points wins. Determine whether one player has a winning strategy or whether both players can force a tie, and describe the strategy.
3
1
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Rectangular parallelepiped with all vertices of the same color
To each point of
Z
3
\mathbb{Z}^3
Z
3
we assign one of
p
p
p
colors.Prove that there exists a rectangular parallelepiped with all its vertices in
Z
3
\mathbb{Z}^3
Z
3
and of the same color.
2
1
Hide problems
f(1)=1, f(n)=f(n/2) for n even, f(n) = f(n-1) + (-1)^f(n-1) for n odd.
Consider the succession of integers
{
f
(
n
)
}
n
=
1
∞
\{f(n)\}_{n=1}^{\infty}
{
f
(
n
)
}
n
=
1
∞
defined as:
∙
\bullet
∙
f
(
1
)
=
1
f(1) = 1
f
(
1
)
=
1
.
∙
\bullet
∙
f
(
n
)
=
f
(
n
/
2
)
f(n) = f(n/2)
f
(
n
)
=
f
(
n
/2
)
if
n
n
n
is even.
∙
\bullet
∙
If
n
>
1
n > 1
n
>
1
odd and
f
(
n
−
1
)
f(n-1)
f
(
n
−
1
)
odd, then
f
(
n
)
=
f
(
n
−
1
)
−
1
f(n) = f(n-1)-1
f
(
n
)
=
f
(
n
−
1
)
−
1
.
∙
\bullet
∙
If
n
>
1
n > 1
n
>
1
odd and
f
(
n
−
1
)
f(n-1)
f
(
n
−
1
)
even, then
f
(
n
)
=
f
(
n
−
1
)
+
1
f(n) = f(n-1)+1
f
(
n
)
=
f
(
n
−
1
)
+
1
.a) Compute
f
(
2
2020
−
1
)
f(2^{2020}-1)
f
(
2
2020
−
1
)
.b) Prove that
{
f
(
n
)
}
n
=
1
∞
\{f(n)\}_{n=1}^{\infty}
{
f
(
n
)
}
n
=
1
∞
is not periodical, that is, there do not exist positive integers
t
t
t
and
n
0
n_0
n
0
such that
f
(
n
+
t
)
=
f
(
n
)
f(n+t) = f(n)
f
(
n
+
t
)
=
f
(
n
)
for all
n
≥
n
0
n \geq n_0
n
≥
n
0
.
1
1
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Polynomials with roots in arithmetic progression.
A polynomial
p
(
x
)
p(x)
p
(
x
)
with real coefficients is said to be almeriense if it is of the form:
p
(
x
)
=
x
3
+
a
x
2
+
b
x
+
a
p(x) = x^3+ax^2+bx+a
p
(
x
)
=
x
3
+
a
x
2
+
b
x
+
a
And its three roots are positive real numbers in arithmetic progression. Find all almeriense polynomials such that
p
(
7
4
)
=
0
p\left(\frac{7}{4}\right) = 0
p
(
4
7
)
=
0