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Problems
Contests
National and Regional Contests
Mexico Contests
Regional Olympiad of Mexico Center Zone
2009 Regional Olympiad of Mexico Center Zone
2009 Regional Olympiad of Mexico Center Zone
Part of
Regional Olympiad of Mexico Center Zone
Subcontests
(6)
1
1
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<ADO = 90 ^o wanted, diameter , 2midpoints, perpendiculars related
Let
Γ
\Gamma
Γ
be a circle with the center
O
O
O
and let
A
A
A
,
A
′
A ^ \prime
A
′
be two diametrically opposite points in
Γ
\Gamma
Γ
. Let
P
P
P
be the midpoint of
O
A
′
OA ^ \prime
O
A
′
and
ℓ
\ell
ℓ
a line that passes through
P
P
P
, different from the line
A
A
′
AA ^ \prime
A
A
′
and different from the line perpendicular on
A
A
′
AA ^ \prime
A
A
′
. Let
B
B
B
and
C
C
C
be the intersection points of
ℓ
\ell
ℓ
with
Γ
\Gamma
Γ
, let
H
H
H
be the foot of the altitude from
A
A
A
on
B
C
BC
BC
, let
M
M
M
be the midpoint of
B
C
BC
BC
, and let
D
D
D
be the intersection of the line
A
′
M
A ^ \prime M
A
′
M
with
A
H
AH
A
H
. Show that the angle
∠
A
D
O
=
9
0
∘
\angle ADO = 90 ^ \circ
∠
A
D
O
=
9
0
∘
.
4
1
Hide problems
299..98200..0029 = sum of squares of 3 consecutive naturals
Let
N
=
2
99
…
9
⏟
n
times
82
00
…
0
⏟
n
times
29
N = 2 \: \: \underbrace {99… 9} _{n \,\,\text {times}} \: \: 82 \: \: \underbrace {00… 0} _{n \,\, \text {times} } \: \: 29
N
=
2
n
times
99
…
9
82
n
times
00
…
0
29
. Prove that
N
N
N
can be written as the sum of the squares of
3
3
3
consecutive natural numbers.
5
1
Hide problems
AN _|_ NM wanted, AE _|_ BE, AF _|_ CF, altitude, midpoints
Let
A
B
C
ABC
A
BC
be a triangle and let
D
D
D
be the foot of the altitude from
A
A
A
. Let points
E
E
E
and
F
F
F
on a line through
D
D
D
such that
A
E
AE
A
E
is perpendicular to
B
E
BE
BE
,
A
F
AF
A
F
is perpendicular to
C
F
CF
CF
, where
E
E
E
and
F
F
F
are points other than the point
D
D
D
. Let
M
M
M
and
N
N
N
be the midpoints of
B
C
BC
BC
and
E
F
EF
EF
, respectively. Prove that
A
N
AN
A
N
is perpendicular to
N
M
NM
NM
.
6
1
Hide problems
sum of differences of largest to smallest of subsets of {1,2,...,n}
For each subset
A
A
A
of
{
1
,
2
,
…
,
n
}
\{1,2, \dots, n \}
{
1
,
2
,
…
,
n
}
, let
M
A
M_A
M
A
be the difference between the largest of the elements of
A
A
A
and the smallest of the elements of
A
A
A
. Finds the sum of all values of
M
A
M_A
M
A
when all possible subsets
A
A
A
of
{
1
,
2
,
…
,
n
}
\{1,2, \dots, n \}
{
1
,
2
,
…
,
n
}
are considered.
3
1
Hide problems
max no of triangles, equilateral triangles inside equilateral
An equilateral triangle
A
B
C
ABC
A
BC
has sides of length
n
n
n
, a positive integer. Divide the triangle into equilateral triangles of length
1
1
1
, drawing parallel lines (at distance
1
1
1
) to all sides of the triangle. A path is a continuous path, starting at the triangle with vertex
A
A
A
and always crossing from one small triangle to another on the side that both triangles share, in such a way that it never passes through a small triangle twice. Find the maximum number of triangles that can be visited.
2
1
Hide problems
a+p and a^2+p^2
Let
p
≥
2
p \ge 2
p
≥
2
be a prime number and
a
≥
1
a \ge 1
a
≥
1
a positive integer with
p
≠
a
p \neq a
p
=
a
. Find all pairs
(
a
,
p
)
(a,p)
(
a
,
p
)
such that:
a
+
p
∣
a
2
+
p
2
a+p \mid a^2+p^2
a
+
p
∣
a
2
+
p
2