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Contests
National and Regional Contests
Ecuador Contests
Ecuador Mathematical Olympiad (OMEC)
2022 Ecuador NMO (OMEC)
2022 Ecuador NMO (OMEC)
Part of
Ecuador Mathematical Olympiad (OMEC)
Subcontests
(6)
6
1
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Sweet NT
Prove that for all prime
p
≥
5
p \ge 5
p
≥
5
, there exist an odd prime
q
≠
p
q \not= p
q
=
p
such that
q
q
q
divides
(
p
−
1
)
p
+
1
(p-1)^p + 1
(
p
−
1
)
p
+
1
5
1
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Similar triangles and parallelism
Let
A
B
C
ABC
A
BC
be a 90-degree triangle with hypotenuse
B
C
BC
BC
. Let
D
D
D
and
E
E
E
distinct points on segment
B
C
BC
BC
and
P
,
Q
P, Q
P
,
Q
be the foot of the perpendicular from
D
D
D
to
A
B
AB
A
B
and
E
E
E
to
A
C
AC
A
C
, respectively.
D
P
DP
D
P
and
E
Q
EQ
EQ
intersect at
R
R
R
. Lines
C
R
CR
CR
and
A
B
AB
A
B
intersect at
M
M
M
and lines
B
R
BR
BR
and
A
C
AC
A
C
intersect at
N
N
N
. Prove that
M
N
∥
B
C
MN \parallel BC
MN
∥
BC
if and only if
B
D
=
C
E
BD=CE
B
D
=
CE
.
4
1
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Average problem (Combi or NT?)
Find the number of sets with
10
10
10
distinct positive integers such that the average of its elements is less than 6.
3
1
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90 everywhere
A polygon is gridded if the internal angles of the polygon are either
90
90
90
or
270
270
270
, it has integer side lengths and its sides don't intersect with each other. Prove that for all
n
≥
8
n \ge 8
n
≥
8
, it exist a gridded polygon with area
2
n
2n
2
n
and perimeter
2
n
2n
2
n
.
2
1
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Functional equation :o
Determine all functions
f
:
R
→
R
f: \mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
such that for all real numbers
x
,
y
x, y
x
,
y
f
(
x
+
y
)
=
f
(
f
(
x
)
)
+
y
+
2022
f(x + y)=f(f(x)) + y + 2022
f
(
x
+
y
)
=
f
(
f
(
x
))
+
y
+
2022
1
1
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You look nice with that geo costume!
Prove that it is impossible to divide a square with side length
7
7
7
into exactly
36
36
36
squares with integer side lengths.