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Contests
National and Regional Contests
Ecuador Contests
Ecuador Juniors
2016 Ecuador Juniors
2016 Ecuador Juniors
Part of
Ecuador Juniors
Subcontests
(6)
4
1
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1 + 2 + 3 + 4 +...=100 + 98 + 96 + 94 +... - 2016 Ecuador Juniors (OMEC) L2 p4
Two sums, each consisting of
n
n
n
addends , are shown below:
S
=
1
+
2
+
3
+
4
+
.
.
.
S = 1 + 2 + 3 + 4 + ...
S
=
1
+
2
+
3
+
4
+
...
T
=
100
+
98
+
96
+
94
+
.
.
.
T = 100 + 98 + 96 + 94 +...
T
=
100
+
98
+
96
+
94
+
...
. For what value of
n
n
n
is it true that
S
=
T
S = T
S
=
T
?
3
1
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computational geo with cyclic 2016-gon - 2016 Ecuador Juniors (OMEC) L2 p3
Let
P
1
P
2
.
.
.
P
2016
P_1P_2 . . . P_{2016 }
P
1
P
2
...
P
2016
be a cyclic polygon of
2016
2016
2016
sides. Let
K
K
K
be a point inside the polygon and let
M
M
M
be the midpoint of the segment
P
1000
P
2000
P_{1000}P_{2000}
P
1000
P
2000
. Knowing that
K
P
1
=
K
P
2011
=
2016
KP_1 = KP_{2011} = 2016
K
P
1
=
K
P
2011
=
2016
and
K
M
KM
K
M
is perpendicular to
P
1000
P
2000
P_{1000}P_{2000}
P
1000
P
2000
, find the length of segment
K
P
2016
KP_{2016}
K
P
2016
.
1
1
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Ecuadorian 5-digit numbers - 2016 Ecuador Juniors (OMEC) L2 p1
A natural number of five digits is called Ecuadorian if it satisfies the following conditions:
∙
\bullet
∙
All its digits are different.
∙
\bullet
∙
The digit on the far left is equal to the sum of the other four digits. Example:
91350
91350
91350
is an Ecuadorian number since
9
=
1
+
3
+
5
+
0
9 = 1 + 3 + 5 + 0
9
=
1
+
3
+
5
+
0
, but
54210
54210
54210
is not since
5
≠
4
+
2
+
1
+
0
5 \ne 4 + 2 + 1 + 0
5
=
4
+
2
+
1
+
0
. Find how many Ecuadorian numbers exist.
6
1
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digits (2a -1)(2b -1) (2c-1)(2d-1) = 2abcd -1 2016 Ecuador NMO (OMEC) 3.5 2.5
Determine the number of positive integers
N
=
a
b
c
d
‾
N = \overline{abcd}
N
=
ab
c
d
, with
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
nonzero digits, which satisfy
(
2
a
−
1
)
(
2
b
−
1
)
(
2
c
−
1
)
(
2
d
−
1
)
=
2
a
b
c
d
−
1
(2a -1) (2b -1) (2c- 1) (2d - 1) = 2abcd -1
(
2
a
−
1
)
(
2
b
−
1
)
(
2
c
−
1
)
(
2
d
−
1
)
=
2
ab
c
d
−
1
.
5
1
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area chasing in parallelogram 2016 Ecuador NMO (OMEC) 3.4 1.6
In the parallelogram
A
B
C
D
ABCD
A
BC
D
, a line through
C
C
C
intersects the diagonal
B
D
BD
B
D
at
E
E
E
and
A
B
AB
A
B
at
F
F
F
. If
F
F
F
is the midpoint of
A
B
AB
A
B
and the area of
△
B
E
C
\vartriangle BEC
△
BEC
is
100
100
100
, find the area of the quadrilateral
A
F
E
D
AFED
A
FE
D
.
2
1
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(x + 1)^2 + (x + 2)^2 +...+ (x + 9)^2 = y^2 2016 Ecuador NMO (OMEC) 3.1 2.2
Prove that there are no positive integers
x
,
y
x, y
x
,
y
such that:
(
x
+
1
)
2
+
(
x
+
2
)
2
+
.
.
.
+
(
x
+
9
)
2
=
y
2
(x + 1)^2 + (x + 2)^2 +...+ (x + 9)^2 = y^2
(
x
+
1
)
2
+
(
x
+
2
)
2
+
...
+
(
x
+
9
)
2
=
y
2