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Contests
National and Regional Contests
Chile Contests
Chile National Olympiad
2016 Chile National Olympiad
2016 Chile National Olympiad
Part of
Chile National Olympiad
Subcontests
(6)
5
1
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3x3 system x^2 - y = (z -1)^2 (Chile 2016 L2 P5)
Determine all triples
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
of nonnegative real numbers that verify the following system of equations:
x
2
−
y
=
(
z
−
1
)
2
x^2 - y = (z -1)^2
x
2
−
y
=
(
z
−
1
)
2
y
2
−
z
=
(
x
−
1
)
2
y^2 - z = (x -1)^2
y
2
−
z
=
(
x
−
1
)
2
z
2
−
x
=
(
y
−
1
)
2
z^2 - x = (y - 1)^2
z
2
−
x
=
(
y
−
1
)
2
4
1
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last digit of denominator of product k/ 2^k (Chile 2016 L1 L2 P4)
The product
1
2
⋅
2
4
⋅
3
8
⋅
4
16
⋅
.
.
.
⋅
99
2
99
⋅
100
2
100
\frac12 \cdot \frac24 \cdot \frac38 \cdot \frac{4}{16} \cdot ... \cdot \frac{99}{2^{99}} \cdot \frac{100}{2^{100}}
2
1
⋅
4
2
⋅
8
3
⋅
16
4
⋅
...
⋅
2
99
99
⋅
2
100
100
is written in its most simplified form. What is the last digit of the denominator?
3
1
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giraffe chess piece
The giraffe is a chess piece that moves
4
4
4
squares in one direction and then a box in a perpendicular direction. What is the smallest value of
n
n
n
such that the giraffe that starts from a corner on an
n
×
n
n \times n
n
×
n
board can visit all the squares of said board?
1
1
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a_n=1234567891011 ...n is not divisible by 3 (Chile 2016 L2 P1)
The natural number
a
n
a_n
a
n
is obtained by writing together and ordered, in decimal notation , all natural numbers between
1
1
1
and
n
n
n
. So we have for example that
a
1
=
1
a_1 = 1
a
1
=
1
,
a
2
=
12
a_2 = 12
a
2
=
12
,
a
3
=
123
a_3 = 123
a
3
=
123
,
.
.
.
. . .
...
,
a
11
=
1234567891011
a_{11} = 1234567891011
a
11
=
1234567891011
,
.
.
.
...
...
. Find all values of
n
n
n
for which
a
n
a_n
a
n
is not divisible by
3
3
3
.
2
1
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point that each line passing bisects areas in triangle (Chile 2016 L2 P2)
For a triangle
△
A
B
C
\vartriangle ABC
△
A
BC
, determine whether or not there exists a point
P
P
P
on the interior of
△
A
B
C
\vartriangle ABC
△
A
BC
in such a way that every straight line through
P
P
P
divides the triangle
△
A
B
C
\vartriangle ABC
△
A
BC
in two polygons of equal area.
6
1
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locus of points such 3 points are collinear, in space (Chile 2016 L2 P6)
Let
P
1
P_1
P
1
and
P
2
P_2
P
2
be two non-parallel planes in space, and
A
A
A
a point that does not It is in none of them. For each point
X
X
X
, let
X
1
X_1
X
1
denote its reflection with respect to
P
1
P_1
P
1
, and
X
2
X_2
X
2
its reflection with respect to
P
2
P_2
P
2
. Determine the locus of points
X
X
X
for the which
X
1
,
X
2
X_1, X_2
X
1
,
X
2
and
A
A
A
are collinear.