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Contests
National and Regional Contests
Chile Contests
Chile National Olympiad
2023 Chile National Olympiad
2023 Chile National Olympiad
Part of
Chile National Olympiad
Subcontests
(6)
6
1
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angle bisector wanted, 135-30-15 triangle
Let
△
A
B
C
\vartriangle ABC
△
A
BC
be a triangle such that
∠
A
B
C
=
3
0
o
\angle ABC = 30^o
∠
A
BC
=
3
0
o
,
∠
A
C
B
=
1
5
o
\angle ACB = 15^o
∠
A
CB
=
1
5
o
. Let
M
M
M
be midpoint of segment
B
C
BC
BC
and point
N
N
N
lies on segment
M
C
MC
MC
, such that the length of
N
C
NC
NC
is equal to length of
A
B
AB
A
B
. Proce that
A
N
AN
A
N
is the bisector of the angle
∠
M
A
C
\angle MAC
∠
M
A
C
. https://cdn.artofproblemsolving.com/attachments/2/7/4c554b53f03288ee69931fdd2c6fbd3e27ab13.png
4
1
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121 points in square of side 60 - 2023 Chile NMO L2 P4
Inside a square with side
60
60
60
,
121
121
121
points are drawn. Prove them are three points that are vertices of a triangle of area not exceeding
30
30
30
.
5
1
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min n, 225|n, with only digits 0,1 - 2023 Chile NMO L1 P6, L2 P5
What is the smallest positive integer that is divisible by
225
225
225
and that has ony the numbers one and zero as digits?
3
1
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2023 triangles of equal area, by 2022 points, equilateral
Let
△
A
B
C
\vartriangle ABC
△
A
BC
be an equilateral triangle with side
1
1
1
.
1011
1011
1011
points
P
1
P_1
P
1
,
P
2
P_2
P
2
,
P
3
P_3
P
3
,
.
.
.
...
...
,
P
1011
P_{1011}
P
1011
on the side
A
C
AC
A
C
and
1011
1011
1011
points
Q
1
Q_1
Q
1
,
Q
2
Q_2
Q
2
,
Q
3
Q_3
Q
3
,
.
.
.
...
...
,
Q
1011
Q_{1011}
Q
1011
on side AB (see figure) in such a way as to generate
2023
2023
2023
triangles of equal area. Find the length of the segment
A
P
1011
AP_{1011}
A
P
1011
. https://cdn.artofproblemsolving.com/attachments/f/6/fea495c16a0b626e0c3882df66d66011a1a3af.png PS. Harder version of [url=https://artofproblemsolving.com/community/c4h3323135p30741470]2023 Chile NMO L1 P3
2
1
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visible lattice points on 3D from the origin x_i = 1,...,30
In Cartesian space, let
Ω
=
{
(
a
,
b
,
c
)
:
a
,
b
,
c
\Omega = \{(a, b, c) : a, b, c
Ω
=
{(
a
,
b
,
c
)
:
a
,
b
,
c
are integers between
1
1
1
and
30
}
30\}
30
}
. A point of
Ω
\Omega
Ω
is said to be visible from the origin if the segment that joins said point with the origin does not contain any other elements of
Ω
\Omega
Ω
. Find the number of points of
Ω
\Omega
Ω
that are visible from the origin.
1
1
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max k, 11^k |n, n! |2023 - 2023 Chile NMO L1 P2 - L2 P1
Let
n
n
n
be a natural number such that
n
!
n!
n
!
is a multiple of
2023
2023
2023
and is not divisible by
37
37
37
. Find the largest power of
11
11
11
that divides
n
!
n!
n
!
.