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Problems
Contests
National and Regional Contests
Mexico Contests
Regional Olympiad of Mexico West
2024 Regional Olympiad of Mexico West
2024 Regional Olympiad of Mexico West
Part of
Regional Olympiad of Mexico West
Subcontests
(6)
6
1
Hide problems
vitrual triangles, a^{2024}+b^{2024}> c^{2024}, ineq. system
We say that a triangle of sides
a
,
b
,
c
a,b,c
a
,
b
,
c
is virtual if such measures satisfy
{
a
2024
+
b
2024
>
c
2024
,
b
2024
+
c
2024
>
a
2024
,
c
2024
+
a
2024
>
b
2024
\begin{cases} a^{2024}+b^{2024}> c^{2024},\\ b^{2024}+c^{2024}> a^{2024},\\ c^{2024}+a^{2024}> b^{2024} \end{cases}
⎩
⎨
⎧
a
2024
+
b
2024
>
c
2024
,
b
2024
+
c
2024
>
a
2024
,
c
2024
+
a
2024
>
b
2024
Find the number of ordered triples
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
such that
a
,
b
,
c
a,b,c
a
,
b
,
c
are integers between
1
1
1
and
2024
2024
2024
(inclusive) and
a
,
b
,
c
a,b,c
a
,
b
,
c
are the sides of a virtual triangle.
5
1
Hide problems
a_{n+1}=\frac{a_n}{p}+p, where p is the greatest prime factor of a_n
Consider a sequence of positive integers
a
1
,
a
2
,
a
3
,
.
.
.
a_1,a_2,a_3,...
a
1
,
a
2
,
a
3
,
...
such that
a
1
>
1
a_1>1
a
1
>
1
and
a
n
+
1
=
a
n
p
+
p
,
a_{n+1}=\frac{a_n}{p}+p,
a
n
+
1
=
p
a
n
+
p
,
where
p
p
p
is the greatest prime factor of
a
n
a_n
a
n
. Prove that for any choice of
a
1
a_1
a
1
, the sequence
a
1
,
a
2
,
a
3
,
.
.
.
a_1,a_2,a_3,...
a
1
,
a
2
,
a
3
,
...
has an infinite terms that are equal between them.
4
1
Hide problems
BQCR is # iff AC=BC
Let
△
A
B
C
\triangle ABC
△
A
BC
be a triangle and
ω
\omega
ω
its circumcircle. The tangent to
ω
\omega
ω
through
B
B
B
cuts the parallel to
B
C
BC
BC
through
A
A
A
at
P
P
P
. The line
C
P
CP
CP
cuts the circumcircle of
△
A
B
P
\triangle ABP
△
A
BP
again in
Q
Q
Q
and line
A
Q
AQ
A
Q
cuts
ω
\omega
ω
at
R
R
R
. Prove that
B
Q
C
R
BQCR
BQCR
is parallelogram if and only if
A
C
=
B
C
AC=BC
A
C
=
BC
.
3
1
Hide problems
positive integrs in 9x9 grid
In each box of a
9
×
9
9\times 9
9
×
9
grid we write a positive integer such that, between any
2
2
2
boxes on the same row or column that have the same number
n
n
n
written, there's at least
n
n
n
boxes between them. What is the minimum sum possible for the numbers on the grid?
2
1
Hide problems
mexican regional collinearity
Let
△
A
B
C
\triangle ABC
△
A
BC
be a triangle and
H
H
H
its orthocenter. We draw the circumference
C
1
\mathcal{C}_1
C
1
that passes through
H
H
H
and its tangent to
B
C
BC
BC
at
B
B
B
and the circumference
C
2
\mathcal{C}_2
C
2
that passes through
H
H
H
and its tangent to
B
C
BC
BC
at
C
C
C
. If
C
1
\mathcal{C}_1
C
1
cuts line
A
B
AB
A
B
again at
X
X
X
and
C
2
\mathcal{C}_2
C
2
cuts line
A
C
AC
A
C
again at
Y
Y
Y
. Prove that
X
,
Y
X,Y
X
,
Y
and
H
H
H
are collinear.
1
1
Hide problems
Initially, the numbers 1,3,4 are written on a board.
Initially, the numbers
1
,
3
,
4
1,3,4
1
,
3
,
4
are written on a board. We do the following process repeatedly. Consider all of the numbers that can be obtained as the sum of
3
3
3
distinct numbers written on the board and that aren't already written, and we write those numbers on the board. We repeat this process, until at a certain step, all of the numbers in that step are greater than
2024
2024
2024
. Determine all of the integers
1
≤
k
≤
2024
1\leq k\leq 2024
1
≤
k
≤
2024
that were not written on the board.