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National and Regional Contests
Chile Contests
Chile National Olympiad
2022 Chile National Olympiad
2022 Chile National Olympiad
Part of
Chile National Olympiad
Subcontests
(6)
2
1
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ID =IE if <ACB=60^o, 2 angle bisectors and incenter
Let
A
B
C
ABC
A
BC
be a triangle such that
∠
C
A
B
=
6
0
o
\angle CAB = 60^o
∠
C
A
B
=
6
0
o
. Consider
D
,
E
D, E
D
,
E
points on sides
A
C
AC
A
C
and
A
B
AB
A
B
respectively such that
B
D
BD
B
D
bisects angle
∠
A
B
C
\angle ABC
∠
A
BC
,
C
E
CE
CE
bisects angle
∠
B
C
A
\angle BCA
∠
BC
A
and let
I
I
I
be the intersection of them. Prove that
∣
I
D
∣
=
∣
I
E
∣
|ID| =|IE|
∣
I
D
∣
=
∣
I
E
∣
.
4
1
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red ant, black ant and termite , shortest possible path, right circular cone
In a right circular cone of wood, the radius of the circumference
T
T
T
of the base circle measures
10
10
10
cm, while every point on said circumference is
20
20
20
cm away. from the apex of the cone. A red ant and a termite are located at antipodal points of
T
T
T
. A black ant is located at the midpoint of the segment that joins the vertex with the position of the termite. If the red ant moves to the black ant's position by the shortest possible path, how far does it travel?
5
1
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divide 21-gon into 2022 triangles , no 3 vetrices are collinear
Is it possible to divide a polygon with
21
21
21
sides into
2022
2022
2022
triangles in such a way that among all the vertices there are not three collinear?
3
1
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can 57- digit no by juxtaposing numbers 472-490 be prime?
The
19
19
19
numbers
472
472
472
,
473
473
473
,
.
.
.
...
...
,
490
490
490
are juxtaposed in some order to form a
57
57
57
-digit number. Can any of the numbers thus obtained be prime?
1
1
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3x3 system \sqrt{x^3 - y} = z - 1 2022 Chile NMO L1 L2 P1
Find all real numbers
x
,
y
,
z
x, y, z
x
,
y
,
z
that satisfy the following system
x
3
−
y
=
z
−
1
\sqrt{x^3 - y} = z - 1
x
3
−
y
=
z
−
1
y
3
−
z
=
x
−
1
\sqrt{y^3 - z} = x - 1
y
3
−
z
=
x
−
1
z
3
−
x
=
y
−
1
\sqrt{z^3 - x} = y - 1
z
3
−
x
=
y
−
1
6
1
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Chile, National Math Olympics
Determine if there is a power of 5 that begins with 2022.