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Problems
Contests
National and Regional Contests
Mexico Contests
Regional Olympiad of Mexico Northeast
2016 Regional Olympiad of Mexico Northeast
2016 Regional Olympiad of Mexico Northeast
Part of
Regional Olympiad of Mexico Northeast
Subcontests
(6)
5
1
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a -1/b=b - 1/c=c - 1/a
Find all triples of reals
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
such that
a
−
1
b
=
b
−
1
c
=
c
−
1
a
.
a - \frac{1}{b}=b - \frac{1}{c}=c - \frac{1}{a}.
a
−
b
1
=
b
−
c
1
=
c
−
a
1
.
6
1
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for each digit d > 0, there exists a divisor of N whose last digit is d
A positive integer
N
N
N
is called northern if for each digit
d
>
0
d > 0
d
>
0
, there exists a divisor of
N
N
N
whose last digit is
d
d
d
. How many northern numbers less than
2016
2016
2016
are there with the fewest number of divisors as possible?
3
1
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3 rectangles, apart unit squares into nxn board
Consider a grid board of
n
×
n
n \times n
n
×
n
, with
n
≥
5
n \ge 5
n
≥
5
. Two unit squares are said to be far apart if they are neither on the same row nor on consecutive rows and neither in the same column nor in consecutive columns. Take
3
3
3
rectangles with vertices and sides on the points and lines of board so that if two unit squares belong to different rectangles, then they are apart . In how many ways is it possible to do this?
1
1
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a^3 + b^3 + c^3 = 2016 diophantine
Determine if there is any triple of nonnegative integers, not necessarily different,
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
such that:
a
3
+
b
3
+
c
3
=
2016
a^3 + b^3 + c^3 = 2016
a
3
+
b
3
+
c
3
=
2016
4
1
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MN^2 = AM x BN , semicircle with diameter AB side of square ABCD
Let
A
B
C
D
ABCD
A
BC
D
be a square. Let
P
P
P
be a point on the semicircle of diameter
A
B
AB
A
B
outside the square. Let
M
M
M
and
N
N
N
be the intersections of
P
D
PD
P
D
and
P
C
PC
PC
with
A
B
AB
A
B
, respectively. Prove that
M
N
2
=
A
M
⋅
B
N
MN^2 = AM \cdot BN
M
N
2
=
A
M
⋅
BN
.
2
1
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MBOG cyclic wanted, ABC isosceles
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
=
A
C
AB = AC
A
B
=
A
C
with centroid
G
G
G
. Let
M
M
M
and
N
N
N
be the midpoints of
A
B
AB
A
B
and
A
C
AC
A
C
respectively and
O
O
O
be the circumcenter of triangle
B
C
N
BCN
BCN
. Prove that
M
B
O
G
MBOG
MBOG
is a cyclic quadrilateral .