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Problems
Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
1996 Spain Mathematical Olympiad
1996 Spain Mathematical Olympiad
Part of
Spain Mathematical Olympiad
Subcontests
(6)
6
1
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regular pentagons on each side of regular pentagon, folding paper, volume
A regular pentagon is constructed externally on each side of a regular pentagon of side
1
1
1
. The figure is then folded and the two edges of the external pentagons meeting at each vertex of the original pentagon are glued together. Find the volume of water that can be poured into the obtained container.
5
1
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16 secret agents spying each other
At Port Aventura there are
16
16
16
secret agents, each of whom is watching one or more other agents. It is known that if agent
A
A
A
is watching agent
B
B
B
, then
B
B
B
is not watching
A
A
A
. Moreover, any
10
10
10
agents can be ordered so that the first is watching the second, the second is watching the third, etc, the last is watching the first. Show that any
11
11
11
agents can also be so ordered.
4
1
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solve \sqrt{x^2 - p}+2\sqrt{x^2-1} = x in real, where p is real also
For each real value of
p
p
p
, find all real solutions of the equation
x
2
−
p
+
2
x
2
−
1
=
x
\sqrt{x^2 - p}+2\sqrt{x^2-1} = x
x
2
−
p
+
2
x
2
−
1
=
x
.
1
1
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if \frac{a+1}{b}+ \frac{b+1}{a} is integer then gcd(a,b)<=\sqrt{a+b}
The natural numbers
a
a
a
and
b
b
b
are such that
a
+
1
b
+
b
+
1
a
\frac{a+1}{b}+ \frac{b+1}{a}
b
a
+
1
+
a
b
+
1
is an integer. Show that the greatest common divisor of a and b is not greater than
a
+
b
\sqrt{a+b}
a
+
b
.
2
1
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if for G centroid, AB+GC = AC+GB, then the triangle is isosceles
Let
G
G
G
be the centroid of a triangle
A
B
C
ABC
A
BC
. Prove that if
A
B
+
G
C
=
A
C
+
G
B
AB+GC = AC+GB
A
B
+
GC
=
A
C
+
GB
, then the triangle is isosceles
3
1
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1996 spain
Consider the functions
f
(
x
)
=
a
x
2
+
b
x
+
c
f(x) = ax^{2} + bx + c
f
(
x
)
=
a
x
2
+
b
x
+
c
,
g
(
x
)
=
c
x
2
+
b
x
+
a
g(x) = cx^{2} + bx + a
g
(
x
)
=
c
x
2
+
b
x
+
a
, where a, b, c are real numbers. Given that |f(-1)| \leq 1 , |f(0)| \leq 1 , |f(1)| \leq 1 , prove that
∣
f
(
x
)
∣
≤
5
4
|f(x)| \leq \frac{5}{4}
∣
f
(
x
)
∣
≤
4
5
and |g(x)| \leq 2 for -1 \leq x \leq 1 .