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Problems
Contests
National and Regional Contests
Mexico Contests
Regional Olympiad of Mexico Northeast
2020 Regional Olympiad of Mexico Northeast
2020 Regional Olympiad of Mexico Northeast
Part of
Regional Olympiad of Mexico Northeast
Subcontests
(4)
4
1
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4p-3 is perfect square if n |(p-1) and p|(n^3-1)
Let
n
>
1
n > 1
n
>
1
be an integer and
p
p
p
be a prime. Prove that if
n
∣
p
−
1
n|p-1
n
∣
p
−
1
and
p
∣
n
3
−
1
p|n^3-1
p
∣
n
3
−
1
, then
4
p
−
3
4p-3
4
p
−
3
is a perfect square.
3
1
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no of permutations satisfy that none of s_1,...,s_{100} is divisiblle by 3
A permutation of the integers
2020
,
2021
,
.
.
.
,
2118
,
2119
2020, 2021,...,2118, 2119
2020
,
2021
,
...
,
2118
,
2119
is a list
a
1
,
a
2
,
a
3
,
.
.
.
,
a
100
a_1,a_2,a_3,...,a_{100}
a
1
,
a
2
,
a
3
,
...
,
a
100
where each one of the numbers appears exactly once. For each permutation we define the partial sums.
s
1
=
a
1
s_1=a_1
s
1
=
a
1
s
2
=
a
1
+
a
2
s_2=a_1+a_2
s
2
=
a
1
+
a
2
s
3
=
a
1
+
a
2
+
a
3
s_3=a_1+a_2+a_3
s
3
=
a
1
+
a
2
+
a
3
.
.
.
...
...
s
100
=
a
1
+
a
2
+
.
.
.
+
a
100
s_{100}=a_1+a_2+...+a_{100}
s
100
=
a
1
+
a
2
+
...
+
a
100
How many of these permutations satisfy that none of the numbers
s
1
,
.
.
.
,
s
100
s_1,...,s_{100}
s
1
,
...
,
s
100
is divisible by
3
3
3
?
2
1
Hide problems
PQ bisects AC if <BCD=90^o, A, B,C,D concyclic
Let
A
A
A
,
B
B
B
,
C
C
C
and
D
D
D
be points on the same circumference with
∠
B
C
D
=
9
0
∘
\angle BCD=90^\circ
∠
BC
D
=
9
0
∘
. Let
P
P
P
and
Q
Q
Q
be the projections of
A
A
A
onto
B
D
BD
B
D
and
C
D
CD
C
D
, respectively. Prove that
P
Q
PQ
PQ
cuts the segment
A
C
AC
A
C
into equal parts.
1
1
Hide problems
[a_{2020}] =? a_{n+1}=\sqrt{2020+a_n}
Let
a
1
=
2020
a_1=2020
a
1
=
2020
and let
a
n
+
1
=
2020
+
a
n
a_{n+1}=\sqrt{2020+a_n}
a
n
+
1
=
2020
+
a
n
for
n
≥
1
n\ge 1
n
≥
1
. How much is
⌊
a
2020
⌋
\left\lfloor a_{2020}\right\rfloor
⌊
a
2020
⌋
?Note:
⌊
x
⌋
\lfloor x\rfloor
⌊
x
⌋
denotes the integer part of a number, that is that is, the immediate integer less than
x
x
x
. For example,
⌊
2.71
⌋
=
2
\lfloor 2.71\rfloor=2
⌊
2.71
⌋
=
2
and
⌊
π
⌋
=
3
\lfloor \pi\rfloor=3
⌊
π
⌋
=
3
.