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Problems
Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
2013 Spain Mathematical Olympiad
2013 Spain Mathematical Olympiad
Part of
Spain Mathematical Olympiad
Subcontests
(6)
6
1
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Characterize the convex quadrilateral
Let
A
B
C
D
ABCD
A
BC
D
a convex quadrilateral where:
∣
A
B
∣
+
∣
C
D
∣
=
2
∣
A
C
∣
|AB|+|CD|=\sqrt{2} |AC|
∣
A
B
∣
+
∣
C
D
∣
=
2
∣
A
C
∣
and
∣
B
C
∣
+
∣
D
A
∣
=
2
∣
B
D
∣
|BC|+|DA|=\sqrt{2} |BD|
∣
BC
∣
+
∣
D
A
∣
=
2
∣
B
D
∣
What form does the quadrilateral have?
5
1
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Any number can be written as the sum of two terms in the seq
Study if it there exist an strictly increasing sequence of integers
0
=
a
0
<
a
1
<
a
2
<
.
.
.
0=a_0<a_1<a_2<...
0
=
a
0
<
a
1
<
a
2
<
...
satisfying the following conditions
i
)
i)
i
)
Any natural number can be written as the sum of two terms of the sequence (not necessarily distinct).
i
i
)
ii)
ii
)
For any positive integer
n
n
n
we have
a
n
>
n
2
16
a_n > \frac{n^2}{16}
a
n
>
16
n
2
4
1
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Expressing n as a^3+b^5+c^7+d^9+e^11
Are there infinitely many positive integers
n
n
n
that can not be represented as
n
=
a
3
+
b
5
+
c
7
+
d
9
+
e
11
n = a^3+b^5+c^7+d^9+e^{11}
n
=
a
3
+
b
5
+
c
7
+
d
9
+
e
11
, where
a
,
b
,
c
,
d
,
e
a,b,c,d,e
a
,
b
,
c
,
d
,
e
are positive integers? Explain why.
3
1
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Combinatorics.
Let
k
,
n
k,n
k
,
n
be positive integers with
n
≥
k
≥
3
n \geq k \geq 3
n
≥
k
≥
3
. We consider
n
+
1
n+1
n
+
1
points on the real plane with none three of them on the same line. We colour any segment between the points with one of
k
k
k
possibilities. We say that an angle is a "bicolour angle" iff its vertex is one of the
n
+
1
n+1
n
+
1
points and the two segments that define it are of different colours. Show that there is always a way to colour the segments that makes more than
n
⌊
n
k
⌋
2
k
(
k
−
1
)
2
n \Big\lfloor{\frac{n}{k}}\Big\rfloor^2 \frac{k(k-1)}{2}
n
⌊
k
n
⌋
2
2
k
(
k
−
1
)
bicolour angles.
2
1
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x^n+y^n+z^n is constant
Find all the possible values of a positive integer
n
n
n
for which the expression
S
n
=
x
n
+
y
n
+
z
n
S_n=x^n+y^n+z^n
S
n
=
x
n
+
y
n
+
z
n
is constant for all real
x
,
y
,
z
x,y,z
x
,
y
,
z
with
x
y
z
=
1
xyz=1
x
yz
=
1
and
x
+
y
+
z
=
0
x+y+z=0
x
+
y
+
z
=
0
.
1
1
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Inequality
Let
a
,
b
,
n
a,b,n
a
,
b
,
n
positive integers with
a
>
b
a>b
a
>
b
and
a
b
−
1
=
n
2
ab-1=n^2
ab
−
1
=
n
2
. Prove that
a
−
b
≥
4
n
−
3
a-b \geq \sqrt{4n-3}
a
−
b
≥
4
n
−
3
and study the cases where the equality holds.