MathDB
Problems
Contests
National and Regional Contests
Mexico Contests
Regional Olympiad of Mexico Northeast
2018 Regional Olympiad of Mexico Northeast
2018 Regional Olympiad of Mexico Northeast
Part of
Regional Olympiad of Mexico Northeast
Subcontests
(6)
6
1
Hide problems
AB + AT = TC - archimedes broken chord
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
<
A
C
AB < AC
A
B
<
A
C
and
M
M
M
the midpoint of the arc
B
C
BC
BC
containing
A
A
A
, plus
T
T
T
the foot of the perpendicular from
M
M
M
on side
A
C
AC
A
C
. Prove that
A
B
+
A
T
=
T
C
AB + AT = TC
A
B
+
A
T
=
TC
. https://cdn.artofproblemsolving.com/attachments/0/a/5c90d7001f73c2f8ff2b0e69078f9a2a5cd606.png
5
1
Hide problems
3color painting of 2x1 tiles in 300x300 board
A
300
×
300
300\times 300
300
×
300
board is arbitrarily filled with
2
×
1
2\times 1
2
×
1
dominoes with no overflow, underflow, or overlap. (Tokens can be placed vertically or horizontally.) Decide if it is possible to paint the tiles with three different colors, so that the following conditions are met:
∙
\bullet
∙
Each token is painted in one and only one of the colors.
∙
\bullet
∙
The same number of tiles are painted in each color.
∙
\bullet
∙
No piece is a neighbor of more than two pieces of the same color.Note: Two dominoes are neighbors if they share an edge.
4
1
Hide problems
x_r - x_s = 2018 if x_n < x_{n+1} <=2n
We have an infinite sequence of integers
{
x
n
}
\{x_n\}
{
x
n
}
, such that
x
1
=
1
x_1 = 1
x
1
=
1
, and, for all
n
≥
1
n \ge 1
n
≥
1
, it holds that
x
n
<
x
n
+
1
≤
2
n
x_n < x_{n+1} \le 2n
x
n
<
x
n
+
1
≤
2
n
. Prove that there are two terms of the sequence,
x
r
x_r
x
r
and
x
s
x_s
x
s
, such that
x
r
−
x
s
=
2018
x_r - x_s = 2018
x
r
−
x
s
=
2018
.
3
1
Hide problems
(x+1)^3 + (x + 2)^3 + (x + 3)^3 + (x + 4)^3 = (x + n)^3 diophantine
Find the smallest natural number
n
n
n
for which there exists a natural number
x
x
x
such that
(
x
+
1
)
3
+
(
x
+
2
)
3
+
(
x
+
3
)
3
+
(
x
+
4
)
3
=
(
x
+
n
)
3
.
(x+1)^3 + (x + 2)^3 + (x + 3)^3 + (x + 4)^3 = (x + n)^3.
(
x
+
1
)
3
+
(
x
+
2
)
3
+
(
x
+
3
)
3
+
(
x
+
4
)
3
=
(
x
+
n
)
3
.
2
1
Hide problems
IP _|_ BC if EP _|_ MAn and NP // IA
Let
A
B
C
ABC
A
BC
be a triangle such that
M
M
M
and
N
N
N
are the midpoints of
A
C
AC
A
C
and
B
C
BC
BC
, respectively. Let
I
I
I
be the incenter of
A
B
C
ABC
A
BC
and
E
E
E
be the intersection of
M
N
MN
MN
with
B
l
Bl
Bl
. Let
P
P
P
be a point such that
E
P
EP
EP
is perpendicular to
M
N
MN
MN
and
N
P
NP
NP
parallel to
I
A
IA
I
A
. Prove that
I
P
IP
I
P
is perpendicular to
B
C
BC
BC
.
1
1
Hide problems
sum of every 5 consecutive numbers in a circle is multiple of 13
N
N
N
different positive integers are arranged around a circle , in such a way that the sum of every
5
5
5
consecutive numbers in the circle is a multiple of
13
13
13
. Let
A
A
A
be the smallest possible sum of the
n
n
n
numbers. Calculate the value of
A
A
A
for
∙
\bullet
∙
n
=
99
n = 99
n
=
99
,
∙
\bullet
∙
n
=
100
n = 100
n
=
100
.