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Problems
Contests
National and Regional Contests
Chile Contests
Chile National Olympiad
2014 Chile National Olympiad
2014 Chile National Olympiad
Part of
Chile National Olympiad
Subcontests
(6)
6
1
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2n lines in the plane
Prove that for every set of
2
n
2n
2
n
lines in the plane, such that there are no two parallel lines, there are two lines that divide the plane into four quadrants such that in each quadrant the number of unbounded regions is equal to
n
n
n
.[asy] unitsize(1cm);pair[] A, B; pair P, Q, R, S;A[1] = (0,5.2); B[1] = (6.1,0); A[2] = (1.5,5.5); B[2] = (3.5,0); A[3] = (6.8,5.5); B[3] = (1,0); A[4] = (7,4.5); B[4] = (0,4); P = extension(A[2],B[2],A[4],B[4]); Q = extension(A[3],B[3],A[4],B[4]); R = extension(A[1],B[1],A[2],B[2]); S = extension(A[1],B[1],A[3],B[3]);fill(P--Q--S--R--cycle, palered); fill(A[4]--(7,0)--B[1]--S--Q--cycle, paleblue); draw(A[1]--B[1]); draw(A[2]--B[2]); draw(A[3]--B[3]); draw(A[4]--B[4]);label("Bounded region", (3.5,3.7), fontsize(8)); label("Unbounded region", (5.4,2.5), fontsize(8)); [/asy]
4
1
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n^3-9n + 27 is not divisible by 81
Prove that for every integer
n
n
n
the expression
n
3
−
9
n
+
27
n^3-9n + 27
n
3
−
9
n
+
27
is not divisible by
81
81
81
.
3
1
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2014 points in plane
In the plane there are
2014
2014
2014
plotted points, such that no
3
3
3
are collinear. For each pair of plotted points, draw the line that passes through them. prove that for every three of marked points there are always two that are separated by an amount odd number of lines.
1
1
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one of ab(1-c)^2 , bc(1-a)^2 , ca(1- b)^2 <= 1/16 if 0<a,b,c<=1
Let
a
,
b
,
c
a, b,c
a
,
b
,
c
real numbers that are greater than
0
0
0
and less than
1
1
1
. Show that there is at least one of these three values
a
b
(
1
−
c
)
2
ab(1-c)^2
ab
(
1
−
c
)
2
,
b
c
(
1
−
a
)
2
bc(1-a)^2
b
c
(
1
−
a
)
2
,
c
a
(
1
−
b
)
2
ca(1- b)^2
c
a
(
1
−
b
)
2
which is less than or equal to
1
16
\frac{1}{16}
16
1
.
5
1
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any quad that can be cut into finite no of #s is # 2014 Chile L1 P6
Prove that if a quadrilateral
A
B
C
D
ABCD
A
BC
D
can be cut into a finite number of parallelograms, then
A
B
C
D
ABCD
A
BC
D
is a parallelogram.
2
1
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area of quadrilateral with vertices centroids (Chile 2014 L2 P2 )
Consider an
A
B
C
D
ABCD
A
BC
D
parallelogram of area
1
1
1
. Let
E
E
E
be the center of gravity of the triangle
A
B
C
,
F
ABC, F
A
BC
,
F
the center of gravity of the triangle
B
C
D
,
G
BCD, G
BC
D
,
G
the center of gravity of the triangle
C
D
A
CDA
C
D
A
and
H
H
H
the center of gravity of the triangle
D
A
B
DAB
D
A
B
. Calculate the area of quadrilateral
E
F
G
H
EFGH
EFG
H
.