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Problems
Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
1986 Spain Mathematical Olympiad
1986 Spain Mathematical Olympiad
Part of
Spain Mathematical Olympiad
Subcontests
(6)
1
1
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distance between real x,y by d(x,y) =\sqrt{([x]-[y])^2+(\{x\}-\{y\})^2}
Define the distance between real numbers
x
x
x
and
y
y
y
by
d
(
x
,
y
)
=
(
[
x
]
−
[
y
]
)
2
+
(
{
x
}
−
{
y
}
)
2
d(x,y) =\sqrt{([x]-[y])^2+(\{x\}-\{y\})^2}
d
(
x
,
y
)
=
([
x
]
−
[
y
]
)
2
+
({
x
}
−
{
y
}
)
2
. Determine (as a union of intervals) the set of real numbers whose distance from
3
/
2
3/2
3/2
is less than
202
/
100
202/100
202/100
.
2
1
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a segment d divides a segment s if there is a natural number n so s=nd ...
A segment
d
d
d
is said to divide a segment
s
s
s
if there is a natural number
n
n
n
such that
s
=
n
d
=
d
+
d
+
.
.
.
+
d
s = nd = d+d+ ...+d
s
=
n
d
=
d
+
d
+
...
+
d
(
n
n
n
times). (a) Prove that if a segment
d
d
d
divides segments
s
s
s
and
s
′
s'
s
′
with
s
<
s
′
s < s'
s
<
s
′
, then it also divides their difference
s
′
−
s
s'-s
s
′
−
s
. (b) Prove that no segment divides the side
s
s
s
and the diagonal
s
′
s'
s
′
of a regular pentagon (consider the pentagon formed by the diagonals of the given pentagon without explicitly computing the ratios).
5
1
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y^2 = x^3 +bx+b^2, incribe a triangle with vertices of rational coordinates
Consider the curve
Γ
\Gamma
Γ
defined by the equation
y
2
=
x
3
+
b
x
+
b
2
y^2 = x^3 +bx+b^2
y
2
=
x
3
+
b
x
+
b
2
, where
b
b
b
is a nonzero rational constant. Inscribe in the curve
Γ
\Gamma
Γ
a triangle whose vertices have rational coordinates.
6
1
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\prod_{k=1}^{14} cos (k\pi /15)
Evaluate
∏
k
=
1
14
c
o
s
(
k
π
15
)
\prod_{k=1}^{14} cos \big(\frac{k\pi}{15}\big)
k
=
1
∏
14
cos
(
15
kπ
)
4
1
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compare (a+b)/2 with \mu (a,b) i.e. mean value of a and b with respect to g
Denote by
m
(
a
,
b
)
m(a,b)
m
(
a
,
b
)
the arithmetic mean of positive real numbers
a
,
b
a,b
a
,
b
. Given a positive real function
g
g
g
having positive derivatives of the first and second order, define
μ
(
a
,
b
)
\mu (a,b)
μ
(
a
,
b
)
the mean value of
a
a
a
and
b
b
b
with respect to
g
g
g
by
2
g
(
μ
(
a
,
b
)
)
=
g
(
a
)
+
g
(
b
)
2g(\mu (a,b)) = g(a)+g(b)
2
g
(
μ
(
a
,
b
))
=
g
(
a
)
+
g
(
b
)
. Decide which of the two mean values
m
m
m
and
μ
\mu
μ
is larger.
3
1
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Anybody... help me solve the question
Find all natural numbers
n
n
n
such that
5
n
+
3
5^n+3
5
n
+
3
is a power of
2
2
2