MathDB
Problems
Contests
National and Regional Contests
Mexico Contests
Regional Olympiad of Mexico Northeast
2017 Regional Olympiad of Mexico Northeast
2017 Regional Olympiad of Mexico Northeast
Part of
Regional Olympiad of Mexico Northeast
Subcontests
(6)
6
1
Hide problems
b^2 = 4a(\sqrt{c} - 1), c^2 = 4b (\sqrt{a} - 1), a^2 = 4c(\sqrt{b} - 1)
Find all triples of real numbers
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
that satisfy the system of equations
{
b
2
=
4
a
(
c
−
1
)
c
2
=
4
b
(
a
−
1
)
a
2
=
4
c
(
b
−
1
)
\begin{cases} b^2 = 4a(\sqrt{c} - 1) \\ c^2 = 4b (\sqrt{a} - 1) \\ a^2 = 4c(\sqrt{b} - 1) \end{cases}
⎩
⎨
⎧
b
2
=
4
a
(
c
−
1
)
c
2
=
4
b
(
a
−
1
)
a
2
=
4
c
(
b
−
1
)
5
1
Hide problems
no of ways to arrange numbers 1-1000 on 2x500 grid , each 2x2 is ordered
The figure shows a
2
×
2
2\times 2
2
×
2
grid that has been filled with the numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
, and
d
d
d
. We say that this grid is ordered if it is true that
a
>
b
>
c
>
d
a > b > c > d
a
>
b
>
c
>
d
or that
a
>
d
>
c
>
b
a > d > c > b
a
>
d
>
c
>
b
.\begin{tabular}{|l|l|} \hline a & b \\ \hline d & c \\ \hline \end{tabular} In how many ways can the numbers from
1
1
1
to
1000
1000
1000
be arranged in the cells of a
2
×
500
2 \times 500
2
×
500
grid (
2
2
2
rows and
500
500
500
columns) so that each
2
×
2
2 \times 2
2
×
2
sub-grid is ordered?
4
1
Hide problems
APQM cyclic wanted, circumcircle of ABC, arc midpoint, BP = CQ
Let
Γ
\Gamma
Γ
be the circumcircle of the triangle
A
B
C
ABC
A
BC
and let
M
M
M
be the midpoint of the arc
Γ
\Gamma
Γ
containing
A
A
A
and bounded by
B
B
B
and
C
C
C
. Let
P
P
P
and
Q
Q
Q
be points on the segments
A
B
AB
A
B
and
A
C
AC
A
C
, respectively, such that
B
P
=
C
Q
BP = CQ
BP
=
CQ
. Prove that
A
P
Q
M
APQM
A
PQM
is a cyclic quadrilateral.
3
1
Hide problems
a^3 + 2017a = b^3 -2017b diophantine
Prove that there is no pair of relatively prime positive integers
(
a
,
b
)
(a, b)
(
a
,
b
)
that satisfy the equation
a
3
+
2017
a
=
b
3
−
2017
b
.
a^3 + 2017a = b^3 -2017b.
a
3
+
2017
a
=
b
3
−
2017
b
.
2
1
Hide problems
AX bisects < CAB , 2 midpoints, foot of altitude, 2 circumcircles related
Let
A
B
C
ABC
A
BC
be a triangle and let
N
N
N
and
M
M
M
be the midpoints of
A
B
AB
A
B
and
C
A
CA
C
A
, respectively. Let
H
H
H
be the foot of altitude from
A
A
A
. The circumcircle of
A
B
H
ABH
A
B
H
intersects
M
N
MN
MN
at
P
P
P
, with
P
P
P
and
M
M
M
on the same side relative to
N
N
N
, and the circumcircle of
A
C
H
ACH
A
C
H
intersects
M
N
MN
MN
at
Q
Q
Q
, with
Q
Q
Q
and
N
N
N
on the same side relative to
M
M
M
.
B
P
BP
BP
and
C
Q
CQ
CQ
intersect at
X
X
X
. Prove that
A
X
AX
A
X
is the angle bisector of
∠
C
A
B
\angle CAB
∠
C
A
B
.
1
1
Hide problems
sum of remainders divided by 2, 2^2, 2^3, ... , 2^8,2^9 is 137
Let
n
n
n
be a positive integer less than
1000
1000
1000
. The remainders obtained when dividing
n
n
n
by
2
,
2
2
,
2
3
,
.
.
.
,
2
8
2, 2^2, 2^3, ... , 2^8
2
,
2
2
,
2
3
,
...
,
2
8
, and
2
9
2^9
2
9
, are calculated. If the sum of all these remainders is
137
137
137
, what are all the possible values of
n
n
n
?