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Problems
Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
1990 Spain Mathematical Olympiad
1990 Spain Mathematical Olympiad
Part of
Spain Mathematical Olympiad
Subcontests
(6)
6
1
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n points so that no 2 pairs are equidistant, segment connects nearest points
There are
n
n
n
points in the plane so that no two pairs are equidistant. Each point is connected to the nearest point by a segment. Show that no point is connected to more than five points.
5
1
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points defined by equal ratios on sides of triangle, min area of triangle, locus
On the sides
B
C
,
C
A
BC,CA
BC
,
C
A
and
A
B
AB
A
B
of a triangle
A
B
C
ABC
A
BC
of area
S
S
S
are taken points
A
′
,
B
′
,
C
′
A' ,B' ,C'
A
′
,
B
′
,
C
′
respectively such that
A
C
′
/
A
B
=
B
A
′
/
B
C
=
C
B
′
/
C
A
=
p
AC' /AB = BA' /BC = CB' /CA = p
A
C
′
/
A
B
=
B
A
′
/
BC
=
C
B
′
/
C
A
=
p
, where
0
<
p
<
1
0 < p < 1
0
<
p
<
1
is variable. (a) Find the area of triangle
A
′
B
′
C
′
A' B' C'
A
′
B
′
C
′
in terms of
p
p
p
. (b) Find the value of
p
p
p
which minimizes this area. (c) Find the locus of the intersection point
P
P
P
of the lines through
A
′
A'
A
′
and
C
′
C'
C
′
parallel to
A
B
AB
A
B
and
A
C
AC
A
C
respectively.
4
1
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\sqrt[3]{\frac{a+1}{2}+\frac{a+3}{6}\sqrt{ \frac{4a+3}{3}}}+ ... indepedent of a
Prove that the sum
a
+
1
2
+
a
+
3
6
4
a
+
3
3
3
+
a
+
1
2
−
a
+
3
6
4
a
+
3
3
3
\sqrt[3]{\frac{a+1}{2}+\frac{a+3}{6}\sqrt{ \frac{4a+3}{3}}} +\sqrt[3]{\frac{a+1}{2}-\frac{a+3}{6}\sqrt{ \frac{4a+3}{3}}}
3
2
a
+
1
+
6
a
+
3
3
4
a
+
3
+
3
2
a
+
1
−
6
a
+
3
3
4
a
+
3
is independent of
a
a
a
for
a
≥
−
3
4
a \ge - \frac{3}{4}
a
≥
−
4
3
and evaluate it.
1
1
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compare \sqrt{3}+\sqrt{10+2\sqrt{3}} with \sqrt{5+\sqrt{22}}+\sqrt{8-\sqrt{22}}}
Prove that
x
+
y
+
x
y
\sqrt{x}+\sqrt{y}+\sqrt{xy}
x
+
y
+
x
y
is equal to
x
+
y
+
x
y
+
2
y
x
\sqrt{x}+\sqrt{y+xy+2y\sqrt{x}}
x
+
y
+
x
y
+
2
y
x
and compare the numbers
3
+
10
+
2
3
\sqrt{3}+\sqrt{10+2\sqrt{3}}
3
+
10
+
2
3
and
5
+
22
+
8
−
22
+
2
15
−
3
22
\sqrt{5+\sqrt{22}}+\sqrt{8- \sqrt{22}+2\sqrt{15-3\sqrt{22}}}
5
+
22
+
8
−
22
+
2
15
−
3
22
.
2
1
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every point of the plane is painted with one of three colors ....
Every point of the plane is painted with one of three colors. Can we always find two points a distance
1
1
1
cm apart which are of the same color?
3
1
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spain 1990
Prove that
⌊
(
4
+
1
1
)
n
⌋
\lfloor{(4+\sqrt11)^{n}}\rfloor
⌊
(
4
+
1
1
)
n
⌋
is odd for every natural number n.