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Problems
Contests
National and Regional Contests
Chile Contests
Chile National Olympiad
2009 Chile National Olympiad
2009 Chile National Olympiad
Part of
Chile National Olympiad
Subcontests
(6)
6
1
Hide problems
n green points in plane
There are
n
≥
6
n \ge 6
n
≥
6
green points in the plane, such that no
3
3
3
of them are collinear. Suppose further that
6
6
6
of these points are the vertices of a convex hexagon. Prove that there are
5
5
5
green points that form a pentagon that does not contain any other green point inside.
5
1
Hide problems
numbers 1-14 to face and vertices of cube A
Let
A
A
A
and
B
B
B
be two cubes. Numbers
1
,
2
,
.
.
.
,
14
1,2,...,14
1
,
2
,
...
,
14
, are assigned in any order, to the faces and vertices of cube
A
A
A
. Then each edge of cube
A
A
A
is assigned the average of the numbers assigned to the two faces that contain it. Finally assigned to each face of the cube
B
B
B
the sum of the numbers associated with the vertices, the face and the edges on the corresponding face of cube
A
A
A
. If
S
S
S
is the sum of the numbers assigned to the faces of
B
B
B
, find the largest and smallest value that
S
S
S
can take.
4
1
Hide problems
x + 1, x^x + 1, x^{x^x}+1,... are divisible by 2009
Find a positive integer
x
x
x
, with
x
>
1
x> 1
x
>
1
such that all numbers in the sequence
x
+
1
,
x
x
+
1
,
x
x
x
+
1
,
.
.
.
x + 1,x^x + 1,x^{x^x}+1,...
x
+
1
,
x
x
+
1
,
x
x
x
+
1
,
...
are divisible by
2009.
2009.
2009.
3
1
Hide problems
sum of i/a_i = integer
Let
S
=
1
a
1
+
2
a
2
+
.
.
.
+
100
a
100
S = \frac{1}{a_1}+\frac{2}{a_2}+ ... +\frac{100}{a_{100}}
S
=
a
1
1
+
a
2
2
+
...
+
a
100
100
where
a
1
,
a
2
,
.
.
.
,
a
100
a_1, a_2,..., a_{100}
a
1
,
a
2
,
...
,
a
100
are positive integers. What are all the possible integer values that
S
S
S
can take ?
1
1
Hide problems
9 points in the interior of a square of side 1
Consider
9
9
9
points in the interior of a square of side
1
1
1
. Prove that there are three of them that form a triangle with an area less than or equal to
1
8
\frac18
8
1
.
2
1
Hide problems
difference of lenghts of min and max diagonal in regular 9-gon Chile 2009 L2 P2
Consider
P
P
P
a regular
9
9
9
-sided convex polygon with each side of length
1
1
1
. A diagonal at
P
P
P
is any line joining two non-adjacent vertices of
P
P
P
. Calculate the difference between the lengths of the largest and smallest diagonal of
P
P
P
.