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Contests
National and Regional Contests
Ecuador Contests
Ecuador Mathematical Olympiad (OMEC)
2018 Ecuador NMO (OMEC)
2018 Ecuador NMO (OMEC)
Part of
Ecuador Mathematical Olympiad (OMEC)
Subcontests
(6)
5
1
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concyclic, AB=AN+PB, BC=BP+MC, CA=CM+AN 2018 Ecuador NMO (OMEC) 3.5
Let
A
B
C
ABC
A
BC
be an acute triangle and let
M
M
M
,
N
N
N
, and
P
P
P
be on
C
B
CB
CB
,
A
C
AC
A
C
, and
A
B
AB
A
B
, respectively, such that
A
B
=
A
N
+
P
B
AB = AN + PB
A
B
=
A
N
+
PB
,
B
C
=
B
P
+
M
C
BC = BP + MC
BC
=
BP
+
MC
,
C
A
=
C
M
+
A
N
CA = CM + AN
C
A
=
CM
+
A
N
. Let
ℓ
\ell
ℓ
be a line in a different half plane than
C
C
C
with respect to to the line
A
B
AB
A
B
such that if
X
,
Y
X, Y
X
,
Y
are the projections of
A
,
B
A, B
A
,
B
on
ℓ
\ell
ℓ
respectively,
A
X
=
A
P
AX = AP
A
X
=
A
P
and
B
Y
=
B
P
BY = BP
B
Y
=
BP
. Prove that
N
X
Y
M
NXYM
NX
Y
M
is a cyclic quadrilateral.
6
1
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2/\sqrt{4-3\sqrt[4]{5}+2\sqrt[4]{25}-\sqrt[4]{125}} 2018 Ecuador NMO (OMEC) 3.6
Reduce
2
4
−
3
5
4
+
2
25
4
−
125
4
\frac{2}{\sqrt{4-3\sqrt[4]{5} + 2\sqrt[4]{25}-\sqrt[4]{125}}}
4
−
3
4
5
+
2
4
25
−
4
125
2
to its lowest form.Then generalize this result and show that it holds for any positive
n
n
n
.
4
1
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p(x)=x^3-24x + k has at most 1 integer root 2018 Ecuador NMO (OMEC) 3.4
Let
k
k
k
be a real number. Show that the polynomial
p
(
x
)
=
x
3
−
24
x
+
k
p (x) = x^3-24x + k
p
(
x
)
=
x
3
−
24
x
+
k
has at most an integer root.
3
1
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trapezoid , 2PQ<=d wanted, AE=BF=CG=DH<AB/2 2018 Ecuador NMO (OMEC) 3.3
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral with
A
B
≤
C
D
AB\le CD
A
B
≤
C
D
. Points
E
,
F
E ,F
E
,
F
are chosen on segment
A
B
AB
A
B
and points
G
,
H
G ,H
G
,
H
are chosen on the segment
C
D
CD
C
D
, are chosen such that
A
E
=
B
F
=
C
G
=
D
H
<
A
B
2
AE = BF = CG = DH <\frac{AB}{2}
A
E
=
BF
=
CG
=
DH
<
2
A
B
. Let
P
,
Q
P, Q
P
,
Q
, and
R
R
R
be the midpoints of
E
G
EG
EG
,
F
H
FH
F
H
, and
C
D
CD
C
D
, respectively. It is known that
P
R
PR
PR
is parallel to
A
D
AD
A
D
and
Q
R
QR
QR
is parallel to
B
C
BC
BC
. a) Show that
A
B
C
D
ABCD
A
BC
D
is a trapezoid. b) Let
d
d
d
be the difference of the lengths of the parallel sides. Show that
2
P
Q
≤
d
2PQ\le d
2
PQ
≤
d
.
2
1
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stones into K groups of equal weight , powers of 2 2018 Ecuador NMO (OMEC) 3.2
During his excursion to the historical park, Pepito set out to collect stones whose weight in kilograms is a power of two. Once the first stone has been collected, Pepito only collects stones strictly heavier than the first. At the end of the excursion, her partner Ana chooses a positive integer
K
≥
2
K \ge 2
K
≥
2
and challenges Pepito to divide the stones into
K
K
K
groups of equal weight. a) Can Pepito meet the challenge if all the stones he collected have different weights? b) Can Pepito meet the challenge if some collected stones are allowed to have equal weight?
1
1
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diophantine a^2+b^2=26a has >=12 solutions 2018 Ecuador NMO (OMEC) 3.1
Let
a
,
b
a, b
a
,
b
be integers. Show that the equation
a
2
+
b
2
=
26
a
a^2 + b^2 = 26a
a
2
+
b
2
=
26
a
has at least
12
12
12
solutions.