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Contests
National and Regional Contests
Ecuador Contests
Ecuador Mathematical Olympiad (OMEC)
2023 Ecuador NMO (OMEC)
2023 Ecuador NMO (OMEC)
Part of
Ecuador Mathematical Olympiad (OMEC)
Subcontests
(6)
6
1
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All possible lengths of sides in that triangle
Let
D
E
DE
D
E
the diameter of a circunference
Γ
\Gamma
Γ
. Let
B
,
C
B, C
B
,
C
on
Γ
\Gamma
Γ
such that
B
C
BC
BC
is perpendicular to
D
E
DE
D
E
, and let
Q
Q
Q
the intersection of
B
C
BC
BC
with
D
E
DE
D
E
. Let
P
P
P
a point on segment
B
C
BC
BC
such that
B
P
=
4
P
Q
BP=4PQ
BP
=
4
PQ
. Let
A
A
A
the second intersection of
P
E
PE
PE
with
Γ
\Gamma
Γ
. If
D
E
=
2
DE=2
D
E
=
2
and
E
Q
=
1
2
EQ=\frac{1}{2}
EQ
=
2
1
, find all possible values of the sides of triangle
A
B
C
ABC
A
BC
.
5
1
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Classic attack to this type of problem
Find all positive integers
n
n
n
such that
4
n
+
4
n
+
1
4^n + 4n + 1
4
n
+
4
n
+
1
is a perfect square.
4
1
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Counting problem with digits
A number is additive if it has three digits, all of them are different and the sum of two of the digits is equal to the remaining one. (For example,
123
(
1
+
2
=
3
)
,
945
(
4
+
5
=
9
)
123 (1+2=3), 945 (4+5=9)
123
(
1
+
2
=
3
)
,
945
(
4
+
5
=
9
)
). Find the sum of all additive numbers.
3
1
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A 'polynomial tree', with 'a' as its leaves
We define a sequence of numbers
a
n
a_n
a
n
such that
a
0
=
1
a_0=1
a
0
=
1
and for all
n
≥
0
n\ge0
n
≥
0
:
2
a
n
+
1
3
+
2
a
n
3
=
3
a
n
+
1
2
a
n
+
3
a
n
+
1
a
n
2
2a_{n+1} ^3 + 2a_n ^3 = 3 a_{n +1} ^2 a_n + 3a_{n+1}a_n^2
2
a
n
+
1
3
+
2
a
n
3
=
3
a
n
+
1
2
a
n
+
3
a
n
+
1
a
n
2
Find the sum of all
a
2023
a_{2023}
a
2023
's possible values.
2
1
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Something seems off
Let
A
B
C
D
ABCD
A
BC
D
a cyclic convex quadrilateral. There is a line
l
l
l
parallel to
D
C
DC
D
C
containing
A
A
A
. Let
P
P
P
a point on
l
l
l
closer to
A
A
A
than to
B
B
B
. Let
B
′
B'
B
′
the reflection of
B
B
B
over the midpoint of
A
D
AD
A
D
. Prove that
∠
B
′
A
P
=
∠
B
A
C
\angle B'AP = \angle BAC
∠
B
′
A
P
=
∠
B
A
C
1
1
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absolute value, squares and roots in the same dish
Find all reals
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
such that
{
a
2
+
b
2
+
c
2
=
1
∣
a
+
b
∣
=
2
\begin{cases}a^2+b^2+c^2=1\\ |a+b|=\sqrt{2}\end{cases}
{
a
2
+
b
2
+
c
2
=
1
∣
a
+
b
∣
=
2