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Problems
Contests
National and Regional Contests
Mexico Contests
Regional Olympiad of Mexico West
2021 Regional Olympiad of Mexico West
2021 Regional Olympiad of Mexico West
Part of
Regional Olympiad of Mexico West
Subcontests
(6)
6
1
Hide problems
partition a square into n^2+1 disjointed rectangles with // sides to square
Let
n
n
n
be an integer greater than
3
3
3
. Show that it is possible to divide a square into
n
2
+
1
n^2 + 1
n
2
+
1
or more disjointed rectangles and with sides parallel to those of the square so that any line parallel to one of the sides intersects at most the interior of
n
n
n
rectangles.Note: We say that two rectangles are disjointed if they do not intersect or only intersect at their perimeters.
5
1
Hide problems
bisector of <BEA _|_ MR
Let
A
B
C
ABC
A
BC
be a triangle such that
A
C
AC
A
C
is its shortest side. A point
P
P
P
is inside it and satisfies that
B
P
=
A
C
BP = AC
BP
=
A
C
. Let
R
R
R
be the midpoint of
B
C
BC
BC
and let
M
M
M
be the midpoint of
A
P
AP
A
P
. Let
E
E
E
be the intersection of
B
P
BP
BP
and
A
C
AC
A
C
. Prove that the bisector of angle
∠
B
E
A
\angle BE A
∠
BE
A
is perpendicular to segment
M
R
MR
MR
.
4
1
Hide problems
painting red numbers from 1-100, no number between a,b divides ab
Some numbers from
1
1
1
to
100
100
100
are painted red so that the following two conditions are met:
∙
\bullet
∙
The number
1
1
1
is painted red.
∙
\bullet
∙
If the numbers other than
a
a
a
and
b
b
b
are painted red then no number between
a
a
a
and
b
b
b
divides the number
a
b
ab
ab
. What is the maximum number of numbers that can be painted red?
3
1
Hide problems
no 6-th power in terms of a_{n+3} = 5a^6_{n+2} + 3a^3_{n+1} + a^2_n
The sequence of real numbers
a
1
,
a
2
,
a
3
,
.
.
.
a_1, a_2, a_3, ...
a
1
,
a
2
,
a
3
,
...
is defined as follows:
a
1
=
2019
a_1 = 2019
a
1
=
2019
,
a
2
=
2020
a_2 = 2020
a
2
=
2020
,
a
3
=
2021
a_3 = 2021
a
3
=
2021
and for all
n
≥
1
n \ge 1
n
≥
1
a
n
+
3
=
5
a
n
+
2
6
+
3
a
n
+
1
3
+
a
n
2
.
a_{n+3} = 5a^6_{n+2} + 3a^3_{n+1} + a^2_n.
a
n
+
3
=
5
a
n
+
2
6
+
3
a
n
+
1
3
+
a
n
2
.
Show that this sequence does not contain numbers of the form
m
6
m^6
m
6
where
m
m
m
is a positive integer.
2
1
Hide problems
chain of one or more consecutive digits with product perfect square
Prove that in every
16
16
16
-digit number there is a chain of one or more consecutive digits such that the product of those digits is a perfect square.For example, if the original number is
7862328578632785
7862328578632785
7862328578632785
we can take the digits
6
6
6
,
2
2
2
and
3
3
3
whose product is
6
2
6^2
6
2
(note that these appear consecutively in the number).
1
1
Hide problems
1/2 <= (a^3+b^3)/(a^2+b^2) <= 1 for positive with a+b = 1
Let
a
a
a
and
b
b
b
be positive real numbers such that
a
+
b
=
1
a+b = 1
a
+
b
=
1
. Prove that
1
2
≤
a
3
+
b
3
a
2
+
b
2
≤
1
\frac12 \le \frac{a^3+b^3}{a^2+b^2} \le 1
2
1
≤
a
2
+
b
2
a
3
+
b
3
≤
1