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Problems
Contests
National and Regional Contests
Chile Contests
Chile National Olympiad
2002 Chile National Olympiad
2002 Chile National Olympiad
Part of
Chile National Olympiad
Subcontests
(7)
2
1
Hide problems
rectangle 15 x n from specific pieces - Chile 2002 L2 P2
Determine all natural numbers
n
n
n
for which it is possible to construct a rectangle of sides
15
15
15
and
n
n
n
, with pieces congruent to:[asy] unitsize(0.6 cm);draw((0,0)--(3,0)); draw((0,1)--(3,1)); draw((0,2)--(1,2)); draw((2,2)--(3,2)); draw((0,0)--(0,2)); draw((1,0)--(1,2)); draw((2,0)--(2,2)); draw((3,0)--(3,2));draw((5,-0.5)--(6,-0.5)); draw((4,0.5)--(7,0.5)); draw((4,1.5)--(7,1.5)); draw((5,2.5)--(6,2.5)); draw((4,0.5)--(4,1.5)); draw((5,-0.5)--(5,2.5)); draw((6,-0.5)--(6,2.5)); draw((7,0.5)--(7,1.5)); [/asy]The squares of the pieces have side
1
1
1
and the pieces cannot overlap or leave free spaces
7
1
Hide problems
convex ordered polygons - Chile 2002 L2 P7
A convex polygon of sides
ℓ
1
,
ℓ
2
,
.
.
.
,
ℓ
n
\ell_1, \ell_2, ..., \ell_n
ℓ
1
,
ℓ
2
,
...
,
ℓ
n
is called ordered if for all reordering
(
σ
(
1
)
,
σ
(
2
)
,
.
.
.
,
σ
(
n
)
)
( \sigma (1), \sigma (2), ..., \sigma (n))
(
σ
(
1
)
,
σ
(
2
)
,
...
,
σ
(
n
))
of the set
(
1
,
2
,
.
.
.
,
n
)
(1, 2,..., n)
(
1
,
2
,
...
,
n
)
there exists a point
P
P
P
inside the polygon such that
d
σ
(
1
)
<
σ
(
2
)
<
.
.
.
<
d
σ
(
n
)
d_{\sigma (1)} < _{\sigma (2)} <...< d_{\sigma (n)}
d
σ
(
1
)
<
σ
(
2
)
<
...
<
d
σ
(
n
)
, where
d
i
d_i
d
i
represents the distance between
P
P
P
and side
ℓ
i
\ell_i
ℓ
i
. Find all the convex ordered polygons.
5
1
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area of right triangle A(T) = E/2+I -1 - Chile 2002 L2 P5
Given a right triangle
T
T
T
, where the coordinates of its vertices are integers, let
E
E
E
be the number of points of integer coordinates that belong to the edge of the triangle
T
T
T
,
I
I
I
the number of points of integer coordinates that belong to the interior of the triangle
T
T
T
. Show that the area
A
(
T
)
A(T)
A
(
T
)
of triangle
T
T
T
is given by:
A
(
T
)
=
E
2
+
I
−
1
A(T) = \frac{E}{2}+I -1
A
(
T
)
=
2
E
+
I
−
1
.
6
1
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3-digit numbers wanted - Chile 2002 L2 P6
Determine all three-digit numbers
N
N
N
such that the average of the six numbers that can be formed by permutation of its three digits is equal to
N
N
N
.
4
1
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1+-2+-3+-...+-2002 =0 ? Chile 2002 L2 P4
All naturals from
1
1
1
to
2002
2002
2002
are placed in a row. Can the signs:
+
+
+
and
−
-
−
be placed between each consecutive pair of them so that the corresponding algebraic sum is
0
0
0
?
3
1
Hide problems
fixed point, cirrcumcircles of 2 squares - Chile 2002 L2 P3
Given the line
A
B
AB
A
B
, let
M
M
M
be a point on it. Towards the same side of the plane and with bases
A
M
AM
A
M
and
M
B
MB
MB
, squares
A
M
C
D
AMCD
A
MC
D
and
M
B
E
F
MBEF
MBEF
are constructed. Let
N
N
N
be the point (different from
M
M
M
) where the circumcircles circumscribed to both squares intersect and let
N
1
N_1
N
1
be the point where the lines
B
C
BC
BC
and
A
F
AF
A
F
intersect. Prove that the points
N
N
N
and
N
1
N_1
N
1
coincide. Prove that as the point
M
M
M
moves on the line
A
B
AB
A
B
, the line
M
N
MN
MN
moves always passing through a fixed point.
1
1
Hide problems
sum of lucky numbers divisible by 2002 - Chile 2002 L2 P1
A Metro ticket, which has six digits, is considered a "lucky number" if its six digits are different and their first three digits add up to the same as the last three (A number such as
026134
026134
026134
is "lucky number"). Show that the sum of all the "lucky numbers" is divisible by
2002
2002
2002
.