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Problems
Contests
National and Regional Contests
Mexico Contests
Regional Olympiad of Mexico Southeast
2020 Regional Olympiad of Mexico Southeast
2020 Regional Olympiad of Mexico Southeast
Part of
Regional Olympiad of Mexico Southeast
Subcontests
(6)
3
1
Hide problems
minimizing some number
Bokos tribus have
2021
2021
2021
closed chests, we know that every chest have some amount of rupias and some amount of diamonts. They are going to do a deal with Link, that consits that Link will stay with a amount of chests and Bokos with the rest. Before opening the chests, Link has to say the amount of chest that he will stay with. After this the chests open and Link has to choose the chests with the amount that he previously said. Link doesn´t want to make Bokos angry so he wants to say the smallest number of chest that he will stay with, but guaranteeing that he stay with at least with the half of diamonts, and at least the half of the rupias. What number does Link needs to say?
4
1
Hide problems
cross with +1´s
Consider a cross like in the figure but with size
2021
2021
2021
. Every square have a
+
1
+1
+
1
. Every minute we select a sub-cross of size
3
3
3
and multiply their squares by
−
1
-1
−
1
. It´s posible achieve that all the squares of the cross with size
2021
2021
2021
have a
−
1
-1
−
1
?
5
1
Hide problems
Geometry inequality
Let
A
B
C
ABC
A
BC
an acute triangle with
∠
B
A
C
≥
6
0
∘
\angle BAC\geq 60^\circ
∠
B
A
C
≥
6
0
∘
and
Γ
\Gamma
Γ
it´s circumcircule. Let
P
P
P
the intersection of the tangents to
Γ
\Gamma
Γ
from
B
B
B
and
C
C
C
. Let
Ω
\Omega
Ω
the circumcircle of the triangle
B
P
C
BPC
BPC
. The bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
intersect
Γ
\Gamma
Γ
again in
E
E
E
and
Ω
\Omega
Ω
in
D
D
D
, in the way that
E
E
E
is between
A
A
A
and
D
D
D
. Prove that
A
E
E
D
≤
2
\frac{AE}{ED}\leq 2
E
D
A
E
≤
2
and determine when equality holds.
6
1
Hide problems
Prove in a sequence exist some non prime number
Prove that for all
a
,
b
a, b
a
,
b
and
x
0
x_0
x
0
positive integers, in the sequence
x
1
,
x
2
,
x
3
,
⋯
x_1, x_2, x_3, \cdots
x
1
,
x
2
,
x
3
,
⋯
defined by
x
n
+
1
=
a
x
n
+
b
,
n
≥
0
x_{n+1}=ax_n+b, n\geq 0
x
n
+
1
=
a
x
n
+
b
,
n
≥
0
Exist an
x
i
x_i
x
i
that is not prime for some
i
≥
1
i\geq 1
i
≥
1
2
1
Hide problems
Prove a quadrilateral is cyclic
Let
A
B
C
ABC
A
BC
a triangle with
A
B
<
A
C
AB<AC
A
B
<
A
C
and let
I
I
I
it´s incenter. Let
Γ
\Gamma
Γ
the circumcircle of
△
B
I
C
\triangle BIC
△
B
I
C
.
A
I
AI
A
I
intersect
Γ
\Gamma
Γ
again in
P
P
P
. Let
Q
Q
Q
a point in side
A
C
AC
A
C
such that
A
B
=
A
Q
AB=AQ
A
B
=
A
Q
and let
R
R
R
a point in
A
B
AB
A
B
with
B
B
B
between
A
A
A
and
R
R
R
such that
A
R
=
A
C
AR=AC
A
R
=
A
C
. Prove that
I
Q
P
R
IQPR
I
QPR
is cyclic.
1
1
Hide problems
3 divides a number
Find all natural numbers
n
n
n
, such that
3
3
3
divides the number
n
⋅
2
n
+
1
n\cdot 2^n+1
n
⋅
2
n
+
1
.