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Problems
Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
1994 Spain Mathematical Olympiad
1994 Spain Mathematical Olympiad
Part of
Spain Mathematical Olympiad
Subcontests
(6)
6
1
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dissecting a convex n-gon is into m triangles
A convex
n
n
n
-gon is dissected into
m
m
m
triangles such that each side of each triangle is either a side of another triangle or a side of the polygon. Prove that
m
+
n
m+n
m
+
n
is even. Find the number of sides of the triangles inside the square and the number of vertices inside the square in terms of
m
m
m
and
n
n
n
.
5
1
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21 pieces, bw, on the squares of a 3x7 rectangle
Let
21
21
21
pieces, some white and some black, be placed on the squares of a
3
×
7
3\times 7
3
×
7
rectangle. Prove that there always exist four pieces of the same color standing at the vertices of a rectangle.
4
1
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A = 36^o, AB = AC, C-bisector intersects AB at D, find BC without trig
In a triangle
A
B
C
ABC
A
BC
with
∠
A
=
3
6
o
\angle A = 36^o
∠
A
=
3
6
o
and
A
B
=
A
C
AB = AC
A
B
=
A
C
, the bisector of the angle at
C
C
C
meets the oposite side at
D
D
D
. Compute the angles of
△
B
C
D
\triangle BCD
△
BC
D
. Express the length of side
B
C
BC
BC
in terms of the length
b
b
b
of side
A
C
AC
A
C
without using trigonometric functions.
3
1
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sunny and rainy days in a year in each of six regions, in a table
A tourist office was investigating the numbers of sunny and rainy days in a year in each of six regions. The results are partly shown in the following table:Region , sunny or rainy , unclassified
A
336
29
A \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 336 \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,29
A
336
29
B
321
44
B \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 321 \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,44
B
321
44
C
335
30
C \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 335 \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,30
C
335
30
D
343
22
D \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 343 \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,22
D
343
22
E
329
36
E \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 329 \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,36
E
329
36
F
330
35
F \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 330 \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,35
F
330
35
Looking at the detailed data, an officer observed that if one region is excluded, then the total number of rainy days in the other regions equals one third of the total number of sunny days in these regions. Determine which region is excluded.
2
1
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locus of projection of a point on plane, locus of circumcenter of sphere
Let
O
x
y
z
Oxyz
O
x
yz
be a trihedron whose edges
x
,
y
,
z
x,y, z
x
,
y
,
z
are mutually perpendicular. Let
C
C
C
be the point on the ray
z
z
z
with
O
C
=
c
OC = c
OC
=
c
. Points
P
P
P
and
Q
Q
Q
vary on the rays
x
x
x
and
y
y
y
respectively in such a way that
O
P
+
O
Q
=
k
OP+OQ = k
OP
+
OQ
=
k
is constant. For every
P
P
P
and
Q
Q
Q
, the circumcenter of the sphere through
O
,
C
,
P
,
Q
O,C,P,Q
O
,
C
,
P
,
Q
is denoted by
W
W
W
. Find the locus of the projection of
W
W
W
on the plane O
x
y
xy
x
y
. Also find the locus of points
W
W
W
.
1
1
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arithmetic progression with 1 perfect square => infinitely many squares
Prove that if an arithmetic progression contains a perfect square, then it contains infinitely many perfect squares.