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Problems
Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
1987 Spain Mathematical Olympiad
1987 Spain Mathematical Olympiad
Part of
Spain Mathematical Olympiad
Subcontests
(6)
3
1
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a given triangle is divided into n triangles ... tiling triangles related
A given triangle is divided into
n
n
n
triangles in such a way that any line segment which is a side of a tiling triangle is either a side of another tiling triangle or a side of the given triangle. Let
s
s
s
be the total number of sides and
v
v
v
be the total number of vertices of the tiling triangles (counted without multiplicity). (a) Show that if
n
n
n
is odd then such divisions are possible, but each of them has the same number
v
v
v
of vertices and the same number
s
s
s
of sides. Express
v
v
v
and
s
s
s
as functions of
n
n
n
. (b) Show that, for
n
n
n
even, no such tiling is possible
4
1
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systems x+y=1 & (ax+by)^2<=a^2x+b^2y, x+y=1 & (ax+by)^4<=a^4x+b^4y
If
a
a
a
and
b
b
b
are distinct real numbers, solve the systems (a)
{
x
+
y
=
1
(
a
x
+
b
y
)
2
≤
a
2
x
+
b
2
y
\begin{cases} x+y = 1 \\ (ax+by)^2 \le a^2x+b^2y \end{cases}
{
x
+
y
=
1
(
a
x
+
b
y
)
2
≤
a
2
x
+
b
2
y
and (b)
{
x
+
y
=
1
(
a
x
+
b
y
)
4
≤
a
4
x
+
b
4
y
\begin{cases} x+y = 1 \\ (ax+by)^4 \le a^4x+b^4y \end{cases}
{
x
+
y
=
1
(
a
x
+
b
y
)
4
≤
a
4
x
+
b
4
y
6
1
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P_n(x) = x^{n+2}-2x+1, one root c_n in (0,1), lim_{n \to \infty}c_n
For all natural numbers
n
n
n
, consider the polynomial
P
n
(
x
)
=
x
n
+
2
−
2
x
+
1
P_n(x) = x^{n+2}-2x+1
P
n
(
x
)
=
x
n
+
2
−
2
x
+
1
. (a) Show that the equation
P
n
(
x
)
=
0
P_n(x)=0
P
n
(
x
)
=
0
has exactly one root
c
n
c_n
c
n
in the open interval
(
0
,
1
)
(0,1)
(
0
,
1
)
. (b) Find
l
i
m
n
→
∞
c
n
lim_{n \to \infty}c_n
l
i
m
n
→
∞
c
n
.
2
1
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&Sigma;&kappa; \sqrt{{n \choose k}} <\sqrt{2^{n-1}n^3}, from k=1 to n
Show that for each natural number
n
>
1
n > 1
n
>
1
1
⋅
(
n
1
)
+
2
⋅
(
n
2
)
+
.
.
.
+
n
⋅
(
n
n
)
<
2
n
−
1
n
3
1 \cdot \sqrt{{n \choose 1}}+ 2 \cdot \sqrt{{n \choose 2}}+...+n \cdot \sqrt{{n \choose n}} <\sqrt{2^{n-1}n^3}
1
⋅
(
1
n
)
+
2
⋅
(
2
n
)
+
...
+
n
⋅
(
n
n
)
<
2
n
−
1
n
3
1
1
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equilateral with vertices on 3 concentric circles, radii sidelengths of scalene
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be the side lengths of a scalene triangle and let
O
a
,
O
b
O_a, O_b
O
a
,
O
b
and
O
c
O_c
O
c
be three concentric circles with radii
a
,
b
a, b
a
,
b
and
c
c
c
respectively. (a) How many equilateral triangles with different areas can be constructed such that the lines containing the sides are tangent to the circles? (b) Find the possible areas of such triangles.
5
1
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spanish angle chasing
In a triangle
A
B
C
,
D
ABC, D
A
BC
,
D
lies on
A
B
,
E
AB, E
A
B
,
E
lies on
A
C
AC
A
C
and
∠
A
B
E
=
3
0
o
,
∠
E
B
C
=
5
0
o
,
∠
A
C
D
=
2
0
o
\angle ABE = 30^o, \angle EBC = 50^o, \angle ACD = 20^o
∠
A
BE
=
3
0
o
,
∠
EBC
=
5
0
o
,
∠
A
C
D
=
2
0
o
,
∠
D
C
B
=
6
0
o
\angle DCB = 60^o
∠
D
CB
=
6
0
o
. Find
∠
E
D
C
\angle EDC
∠
E
D
C
.