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National and Regional Contests
Ecuador Contests
Ecuador Mathematical Olympiad (OMEC)
2017 Ecuador NMO (OMEC)
2017 Ecuador NMO (OMEC)
Part of
Ecuador Mathematical Olympiad (OMEC)
Subcontests
(6)
6
1
Hide problems
concurrent wanted, hexagon related 2017 Ecuador NMO (OMEC) 3.6
Let
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
be a convex hexagon with sides not parallel and tangent to a circle
Γ
\Gamma
Γ
at the midpoints
P
P
P
,
Q
Q
Q
,
R
R
R
of the sides AB,
C
D
CD
C
D
,
E
F
EF
EF
respectively.
Γ
\Gamma
Γ
is tangent to
B
C
BC
BC
,
D
E
DE
D
E
and
F
A
FA
F
A
at the points
X
,
Y
,
Z
X, Y, Z
X
,
Y
,
Z
respectively. Line
A
B
AB
A
B
intersects lines
E
F
EF
EF
and
C
D
CD
C
D
at points
M
M
M
and
N
N
N
respectively. Lines
M
Z
MZ
MZ
and
N
X
NX
NX
intersect at point
K
K
K
. Let
r
r
r
be the line joining the center of
Γ
\Gamma
Γ
and point
K
K
K
. Prove that the intersection of
P
Y
PY
P
Y
and
Q
Z
QZ
QZ
lies on the line
r
r
r
.
5
1
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x_{n + 2} = 3x_{n + 1}-2x_n, y_n = x^2_n+2^{n + 2} 2017 Ecuador NMO (OMEC) 3.5
Let the sequences
(
x
n
)
(x_n)
(
x
n
)
and
(
y
n
)
(y_n)
(
y
n
)
be defined by
x
0
=
0
x_0 = 0
x
0
=
0
,
x
1
=
1
x_1 = 1
x
1
=
1
,
x
n
+
2
=
3
x
n
+
1
−
2
x
n
x_{n + 2} = 3x_{n + 1}-2x_n
x
n
+
2
=
3
x
n
+
1
−
2
x
n
for
n
=
0
,
1
,
.
.
.
n = 0, 1, ...
n
=
0
,
1
,
...
and
y
n
=
x
n
2
+
2
n
+
2
y_n = x^2_n+2^{n + 2}
y
n
=
x
n
2
+
2
n
+
2
for
n
=
0
,
1
,
.
.
.
,
n = 0, 1, ...,
n
=
0
,
1
,
...
,
respectively. Show that for all n> 0, and n is the square of a odd integer.
4
1
Hide problems
traveling ant on a 7x7 board, through all squares 2017 Ecuador NMO (OMEC) 3.4
Sebastian, the traveling ant, walks on top of some square boards. He just walks horizontally or vertically through the squares of the boards and does not pass through the same square twice. On a board of
7
×
7
7\times 7
7
×
7
, in which squares can Sebastian start his journey so that he can pass through all the squares on the board?
3
1
Hide problems
sum of consecutive cards is 1615 - 2017 Ecuador NMO (OMEC) 3.3
Adrian has
2
n
2n
2
n
cards numbered from
1
1
1
to
2
n
2n
2
n
. He gets rid of
n
n
n
cards that are consecutively numbered. The sum of the numbers of the remaining papers is
1615
1615
1615
. Find all the possible values of
n
n
n
.
2
1
Hide problems
computational geo with perpendiculars 2017 Ecuador NMO (OMEC) 3.2
Let
A
B
C
ABC
A
BC
be a triangle with
A
C
=
18
AC = 18
A
C
=
18
and
D
D
D
is the point on the segment
A
C
AC
A
C
such that
A
D
=
5
AD = 5
A
D
=
5
. Draw perpendiculars from
D
D
D
on
A
B
AB
A
B
and
B
C
BC
BC
which have lengths
4
4
4
and
5
5
5
respectively. Find the area of the triangle
A
B
C
ABC
A
BC
.
1
1
Hide problems
what day of week day was June 6, 1944 ? 2017 Ecuador NMO (OMEC) 3.1
Determine what day of the week day was: June
6
6
6
,
1944
1944
1944
.Note: Leap years are those that are multiples of
4
4
4
and do not end in
00
00
00
or that are multiples of
400
400
400
, for example
1812
1812
1812
,
1816
1816
1816
,
1820
1820
1820
,
1600
1600
1600
,
2000
2000
2000
, but
1800
1800
1800
,
1810
1810
1810
,
2100
2100
2100
are not leaps.Giving the answer without any mathematical justification will not award points.