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Problems
Contests
National and Regional Contests
Mexico Contests
Regional Olympiad of Mexico Southeast
2023 Regional Olympiad of Mexico Southeast
2023 Regional Olympiad of Mexico Southeast
Part of
Regional Olympiad of Mexico Southeast
Subcontests
(4)
4
1
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solve a equation with fibonacci coefficients
Given the Fibonacci sequence with
f
0
=
f
1
=
1
f_0=f_1=1
f
0
=
f
1
=
1
and for
n
≥
1
,
f
n
+
1
=
f
n
+
f
n
−
1
n\geq 1, f_{n+1}=f_n+f_{n-1}
n
≥
1
,
f
n
+
1
=
f
n
+
f
n
−
1
, find all real solutions to the equation:
x
2024
=
f
2023
x
+
f
2022
.
x^{2024}=f_{2023}x+f_{2022}.
x
2024
=
f
2023
x
+
f
2022
.
3
1
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Connecting Black Cells
Let
n
n
n
be a positive integer. A grid of
n
×
n
n\times n
n
×
n
has some black-colored cells. Drini can color a cell if at least three cells that share a side with it are also colored black. Drini discovers that by repeating this process, all the cells in the grid can be colored. Prove that if there are initially
k
k
k
colored cells, then
k
≥
n
2
+
2
n
3
.
k\geq \frac{n^2+2n}{3}.
k
≥
3
n
2
+
2
n
.
2
1
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Proving Collinearity in a Triangle
Let
A
B
C
ABC
A
BC
be an acute-angled triangle,
D
D
D
be the foot of the altitude from
A
A
A
, the circle with diameter
A
D
AD
A
D
intersect
A
B
AB
A
B
at
F
F
F
and
A
C
AC
A
C
at
E
E
E
. Let
P
P
P
be the orthocenter of triangle
A
E
F
AEF
A
EF
and
O
O
O
be the circumcenter of
A
B
C
ABC
A
BC
. Prove that
A
,
P
,
A, P,
A
,
P
,
and
O
O
O
are collinear.
1
1
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Distinct 7-Digit Non-Multiples
Victor writes down all
7
−
7-
7
−
digit numbers using the digits
1
,
2
,
3
,
4
,
5
,
6
,
1, 2, 3, 4, 5, 6,
1
,
2
,
3
,
4
,
5
,
6
,
and
7
7
7
exactly once. Prove that there are no two numbers among them where one is a multiple of the other.