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Problems
Contests
National and Regional Contests
Mexico Contests
Regional Olympiad of Mexico Southeast
2021 Regional Olympiad of Mexico Southeast
2021 Regional Olympiad of Mexico Southeast
Part of
Regional Olympiad of Mexico Southeast
Subcontests
(4)
4
1
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A board painted
Hernan wants to paint a
8
×
8
8\times 8
8
×
8
board such that every square is painted with blue or red. Also wants to every
3
×
3
3\times 3
3
×
3
subsquare have exactly
a
a
a
blue squares and every
2
×
4
2\times 4
2
×
4
or
4
×
2
4\times 2
4
×
2
rectangle have exactly
b
b
b
blue squares. Find all couples
(
a
,
b
)
(a,b)
(
a
,
b
)
such that Hernan can do the required.
2
1
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Primes in arithmetic progression
Let
n
≥
2021
n\geq 2021
n
≥
2021
. Let
a
1
<
a
2
<
⋯
<
a
n
a_1<a_2<\cdots<a_n
a
1
<
a
2
<
⋯
<
a
n
an arithmetic sequence such that
a
1
>
2021
a_1>2021
a
1
>
2021
and
a
i
a_i
a
i
is a prime number for all
1
≤
i
≤
n
1\leq i\leq n
1
≤
i
≤
n
. Prove that for all
p
p
p
prime with
p
<
2021
,
p
p<2021, p
p
<
2021
,
p
divides the diference of the arithmetic sequence.
1
1
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Prove a point does not depend of the choice of other point
Let
A
,
B
A, B
A
,
B
and
C
C
C
three points on a line
l
l
l
, in that order .Let
D
D
D
a point outside
l
l
l
and
Γ
\Gamma
Γ
the circumcircle of
△
B
C
D
\triangle BCD
△
BC
D
, the tangents from
A
A
A
to
Γ
\Gamma
Γ
touch
Γ
\Gamma
Γ
on
S
S
S
and
T
T
T
. Let
P
P
P
the intersection of
S
T
ST
ST
and
A
C
AC
A
C
. Prove that
P
P
P
does not depend of the choice of
D
D
D
.
3
1
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min and max inequality in positive reals
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
positive reals such that
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
. Prove that
min
{
a
(
1
−
b
)
,
b
(
1
−
c
)
,
c
(
1
−
a
)
}
≤
1
4
\min\{a(1-b),b(1-c),c(1-a)\}\leq \frac{1}{4}
min
{
a
(
1
−
b
)
,
b
(
1
−
c
)
,
c
(
1
−
a
)}
≤
4
1
max
{
a
(
1
−
b
)
,
b
(
1
−
c
)
,
c
(
1
−
a
)
}
≥
2
9
\max\{a(1-b),b(1-c),c(1-a)\}\geq \frac{2}{9}
max
{
a
(
1
−
b
)
,
b
(
1
−
c
)
,
c
(
1
−
a
)}
≥
9
2