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Problems
Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
2003 Spain Mathematical Olympiad
2003 Spain Mathematical Olympiad
Part of
Spain Mathematical Olympiad
Subcontests
(6)
Problem 6
1
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2003 National Spain Olympiad, Problem 6
We string
2
n
2n
2
n
white balls and
2
n
2n
2
n
black balls, forming a continuous chain. Demonstrate that, in whatever order the balls are placed, it is always possible to cut a segment of the chain to contain exactly
n
n
n
white balls and
n
n
n
black balls.
Problem 5
1
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2003 National Spain Olympiad, Problem 5
How many possible areas are there in a convex hexagon with all of its angles being equal and its sides having lengths
1
,
2
,
3
,
4
,
5
1, 2, 3, 4, 5
1
,
2
,
3
,
4
,
5
and
6
,
6,
6
,
in any order?
Problem 4
1
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2003 National Spain Olympiad, Problem 4
Let
x
{x}
x
be a real number such that
x
3
+
2
x
2
+
10
x
=
20.
{x^3 + 2x^2 + 10x = 20.}
x
3
+
2
x
2
+
10
x
=
20.
Demonstrate that both
x
{x}
x
and
x
2
{x^2}
x
2
are irrational.
Problem 3
1
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2003 National Spain Olympiad, Problem 3
The altitudes of the triangle
A
B
C
{ABC}
A
BC
meet in the point
H
{H}
H
. You know that
A
B
=
C
H
{AB = CH}
A
B
=
C
H
. Determine the value of the angle
B
C
A
^
\widehat{BCA}
BC
A
.
Problem 2
1
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2003 National Spain Olympiad, Problem 2
Does there exist such a finite set of real numbers
M
{M}
M
that has at least two distinct elements and has the property that for two numbers,
a
{a}
a
,
b
{b}
b
, belonging to
M
{M}
M
, the number
2
a
−
b
2
{2a - b^2}
2
a
−
b
2
is also an element in
M
{M}
M
?
Problem 1
1
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2003 National Spain Olympiad, Problem 1
Prove that for any prime
p
{p}
p
, different than
2
{2}
2
and
5
{5}
5
, there exists such a multiple of
p
{p}
p
whose digits are all nines. For example, if
p
=
13
{p = 13}
p
=
13
, such a multiple is
999999
=
13
∗
76923
{999999 = 13 * 76923}
999999
=
13
∗
76923
.