MathDB
Problems
Contests
National and Regional Contests
Mexico Contests
Regional Olympiad of Mexico Southeast
2024 Regional Olympiad of Mexico Southeast
2024 Regional Olympiad of Mexico Southeast
Part of
Regional Olympiad of Mexico Southeast
Subcontests
(4)
4
1
Hide problems
Sum of integer sequence and powers of 2
Let
n
n
n
be a non-negative integer and define
a
n
=
2
n
−
n
a_n = 2^n - n
a
n
=
2
n
−
n
. Determine all non-negative integers
m
m
m
such that
s
m
=
a
0
+
a
1
+
⋯
+
a
m
s_m = a_0 + a_1 + \dots + a_m
s
m
=
a
0
+
a
1
+
⋯
+
a
m
is a power of 2.
3
1
Hide problems
Coloring cubes in a large cube with restrictions
A large cube of size
4
×
4
×
4
4 \times 4 \times 4
4
×
4
×
4
is made up of 64 small unit cubes. Exactly 16 of these small cubes must be colored red, subject to the following condition:In each block of
1
×
1
×
4
1 \times 1 \times 4
1
×
1
×
4
,
1
×
4
×
1
1 \times 4 \times 1
1
×
4
×
1
, and
4
×
1
×
1
4 \times 1 \times 1
4
×
1
×
1
cubes, there must be exactly one red cube.Determine how many different ways it is possible to choose the 16 small cubes to be colored red.Note: Two colorings are considered different even if one can be obtained from the other by rotations or symmetries of the cube.
2
1
Hide problems
Relationship between circumradius and perpendicular in acute triangle
Let
A
B
C
ABC
A
BC
be an acute triangle with circumradius
R
R
R
. Let
D
D
D
be the midpoint of
B
C
BC
BC
and
F
F
F
the midpoint of
A
B
AB
A
B
. The perpendicular to
A
C
AC
A
C
through
F
F
F
and the perpendicular to
B
C
BC
BC
through
B
B
B
intersect at
N
N
N
. Prove that
N
D
=
R
ND = R
N
D
=
R
.
1
1
Hide problems
Primes related to sums of integers
Find all pairs of positive integers
a
,
b
a, b
a
,
b
such that the numbers
a
+
1
a+1
a
+
1
,
b
+
1
b+1
b
+
1
,
2
a
+
1
2a+1
2
a
+
1
,
2
b
+
1
2b+1
2
b
+
1
,
a
+
3
b
a+3b
a
+
3
b
, and
b
+
3
a
b+3a
b
+
3
a
are all prime numbers.