MathDB
Problems
Contests
National and Regional Contests
Mexico Contests
Regional Olympiad of Mexico Southeast
2015 Regional Olympiad of Mexico Southeast
2015 Regional Olympiad of Mexico Southeast
Part of
Regional Olympiad of Mexico Southeast
Subcontests
(6)
6
1
Hide problems
set 1,2,3,4,...,100
If we separate the numbers
1
,
2
,
3
,
4
,
…
,
100
1,2,3,4,\dots, 100
1
,
2
,
3
,
4
,
…
,
100
in two lists with
a
1
<
a
2
<
⋯
<
a
50
a_1<a_2<\cdots<a_{50}
a
1
<
a
2
<
⋯
<
a
50
and
b
1
>
b
2
>
⋯
>
b
50
b_1>b_2>\cdots>b_{50}
b
1
>
b
2
>
⋯
>
b
50
Prove that, no matter how we do the separation,
∣
a
1
−
b
1
∣
+
∣
a
2
−
b
2
∣
+
⋯
+
∣
a
50
−
b
50
∣
=
2500
\vert a_1-b_1\vert +\vert a_2-b_2\vert+\cdots +\vert a_{50}-b_{50}\vert=2500
∣
a
1
−
b
1
∣
+
∣
a
2
−
b
2
∣
+
⋯
+
∣
a
50
−
b
50
∣
=
2500
5
1
Hide problems
an angle ratio
In the triangle
A
B
C
ABC
A
BC
, let
A
M
AM
A
M
and
C
N
CN
CN
internal bisectors, with
M
M
M
in
B
C
BC
BC
and
N
N
N
in
A
B
AB
A
B
. Prove that if
∠
B
N
M
∠
M
N
C
=
∠
B
M
N
∠
N
M
A
\frac{\angle BNM}{\angle MNC}=\frac{\angle BMN}{\angle NMA}
∠
MNC
∠
BNM
=
∠
NM
A
∠
BMN
then
A
B
C
ABC
A
BC
is isosceles.
4
1
Hide problems
subject with no multiples of 3
Let
A
=
{
1
,
2
,
4
,
5
,
7
,
8
,
…
}
A=\{1,2,4,5,7,8,\dots\}
A
=
{
1
,
2
,
4
,
5
,
7
,
8
,
…
}
the set with naturals not divisible by three. Find all values of
n
n
n
such that exist
2
n
2n
2
n
consecutive elements of
A
A
A
which sum it´s
300
300
300
.
3
1
Hide problems
geometry inequality
If
T
(
n
)
T(n)
T
(
n
)
is the numbers of triangles with integers sizes(not congruent with each other) with it´s perimeter is equal to
n
n
n
, prove that:
T
(
2012
)
<
T
(
2015
)
T(2012)<T(2015)
T
(
2012
)
<
T
(
2015
)
T
(
2013
)
=
T
(
2016
)
T(2013)=T(2016)
T
(
2013
)
=
T
(
2016
)
2
1
Hide problems
prove two lines are perpendiculars
In a acutangle triangle
A
B
C
,
∠
B
>
∠
C
ABC, \angle B>\angle C
A
BC
,
∠
B
>
∠
C
. Let
D
D
D
the foot of the altitude from
A
A
A
to
B
C
BC
BC
and
E
E
E
the foot of the perpendicular from
D
D
D
to
A
C
AC
A
C
. Let
F
F
F
a point in
D
E
DE
D
E
. Prove that
A
F
AF
A
F
and
B
F
BF
BF
are perpendiculars if and only if
E
F
⋅
D
C
=
B
D
⋅
D
E
EF\cdot DC=BD\cdot DE
EF
⋅
D
C
=
B
D
⋅
D
E
.
1
1
Hide problems
find numbers satiesfiying a condition
Find all integers
n
>
1
n>1
n
>
1
such that every prime that divides
n
6
−
1
n^6-1
n
6
−
1
also divides
n
5
−
n
3
−
n
2
+
1
n^5-n^3-n^2+1
n
5
−
n
3
−
n
2
+
1
.