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Chile Junior Math Olympiad
2024 Chile Junior Math Olympiad
2024 Chile Junior Math Olympiad
Part of
Chile Junior Math Olympiad
Subcontests
(5)
3
1
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Stackexchange again....
Determine all triples
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
of positive integers such that:
a
+
b
+
c
=
a
b
c
.
a + b + c = abc.
a
+
b
+
c
=
ab
c
.
2
1
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The famous powers of two
Emilia and Julieta have a pile of 2024 cards and play the following game: they take turns, and each player removes a number of cards that must be a power of two, i.e.,
1
,
2
,
4
,
8
,
…
1, 2, 4, 8, \dots
1
,
2
,
4
,
8
,
…
. The player who removes the last card wins. Julieta starts the game. Prove that there exists a strategy for Julieta that guarantees her victory, no matter how Emilia plays.
1
1
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3D geo :O in Chile
A plastic ball with a radius of 45 mm has a circular hole made in it. The hole is made to fit a ball with a radius of 35 mm, in such a way that the distance between their centers is 60 mm. Calculate the radius of the hole.
6
1
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Comeback to 2022...
In a regular polygon with 100 vertices, 10 vertices are painted blue, and 10 other vertices are painted red.1. Prove that there exist two distinct blue vertices
A
1
A_1
A
1
and
A
2
A_2
A
2
, and two distinct red vertices
R
1
R_1
R
1
and
R
2
R_2
R
2
, such that the distance between
A
1
A_1
A
1
and
R
1
R_1
R
1
is equal to the distance between
A
2
A_2
A
2
and
R
2
R_2
R
2
.2. Prove that there exist two distinct blue vertices
A
1
A_1
A
1
and
A
2
A_2
A
2
, and two distinct red vertices
R
1
R_1
R
1
and
R
2
R_2
R
2
, such that the distance between
A
1
A_1
A
1
and
A
2
A_2
A
2
is equal to the distance between
R
1
R_1
R
1
and
R
2
R_2
R
2
.
4
1
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Ravi kill it?
Consider a triangle with sides of length
a
a
a
,
b
b
b
, and
c
c
c
that satisfy the following conditions: a + b = c + 3 c^2 + 9 = 2ab Find the area of the triangle.