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Problems
Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
2011 Spain Mathematical Olympiad
2011 Spain Mathematical Olympiad
Part of
Spain Mathematical Olympiad
Subcontests
(3)
1
2
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Coloring the edges between 67 vertices
Each pair of vertices of a regular
67
67
67
-gon is joined by a line segment. Suppose
n
n
n
of these segments are selected, and each of them is painted one of ten available colors. Find the minimum possible value of
n
n
n
for which, regardless of which
n
n
n
segments were selected and how they were painted, there will always be a vertex of the polygon that belongs to seven segments of the same color.
Congruent angles at a midpoint of a side in obtuse triangle
In triangle
A
B
C
ABC
A
BC
,
∠
B
=
2
∠
C
\angle B=2\angle C
∠
B
=
2∠
C
and
∠
A
>
9
0
∘
\angle A>90^\circ
∠
A
>
9
0
∘
. Let
D
D
D
be the point on the line
A
B
AB
A
B
such that
C
D
CD
C
D
is perpendicular to
A
C
AC
A
C
, and let
M
M
M
be the midpoint of
B
C
BC
BC
. Prove that
∠
A
M
B
=
∠
D
M
C
\angle AMB=\angle DMC
∠
A
MB
=
∠
D
MC
.
3
2
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Tangent points of a quadrilateral to a sphere are coplanar
Let
A
A
A
,
B
B
B
,
C
C
C
,
D
D
D
be four points in space not all lying on the same plane. The segments
A
B
AB
A
B
,
B
C
BC
BC
,
C
D
CD
C
D
, and
D
A
DA
D
A
are tangent to the same sphere. Prove that their four points of tangency are coplanar.
Sequence where 2011 divides S_{2011a} - S_{a}
The sequence
S
0
,
S
1
,
S
2
,
…
S_0,S_1,S_2,\ldots
S
0
,
S
1
,
S
2
,
…
is defined by[*]
S
n
=
1
S_n=1
S
n
=
1
for
0
≤
n
≤
2011
0\le n\le 2011
0
≤
n
≤
2011
, and [*]
S
n
+
2012
=
S
n
+
2011
+
S
n
S_{n+2012}=S_{n+2011}+S_n
S
n
+
2012
=
S
n
+
2011
+
S
n
for
n
≥
0
n\ge 0
n
≥
0
.Prove that
S
2011
a
−
S
a
S_{2011a}-S_a
S
2011
a
−
S
a
is a multiple of
2011
2011
2011
for all nonnegative integers
a
a
a
.
2
2
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Ineq: √( (ab+bc+ca)/(a^2+b^2+c^2) ) + Σ ( a/(b+c) ) ≥ 5/2
Let
a
a
a
,
b
b
b
,
c
c
c
be positive real numbers. Prove that
a
b
+
c
+
b
c
+
a
+
c
a
+
b
+
a
b
+
b
c
+
c
a
a
2
+
b
2
+
c
2
≥
5
2
\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}+\sqrt{\frac{ab+bc+ca}{a^2+b^2+c^2}}\ge\frac52
b
+
c
a
+
c
+
a
b
+
a
+
b
c
+
a
2
+
b
2
+
c
2
ab
+
b
c
+
c
a
≥
2
5
and determine when equality holds.
Rationals x & y are different colors if xy=1, x+y=0 or x+y=1
Each rational number is painted either white or red. Call such a coloring of the rationals sanferminera if for any distinct rationals numbers
x
x
x
and
y
y
y
satisfying one of the following three conditions: [*]
x
y
=
1
xy=1
x
y
=
1
, [*]
x
+
y
=
0
x+y=0
x
+
y
=
0
, [*]
x
+
y
=
1
x+y=1
x
+
y
=
1
,we have
x
x
x
and
y
y
y
painted different colors. How many sanferminera colorings are there?