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National and Regional Contests
Mexico Contests
OMMock - Mexico National Olympiad Mock Exam
2021 OMMock - Mexico National Olympiad Mock Exam
2021 OMMock - Mexico National Olympiad Mock Exam
Part of
OMMock - Mexico National Olympiad Mock Exam
Subcontests
(6)
6
1
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Fun primes
Let
a
a
a
and
b
b
b
be fixed positive integers. We say that a prime
p
p
p
is fun if there exists a positive integer
n
n
n
satisfying the following conditions:[*]
p
p
p
divides
a
n
!
+
b
a^{n!} + b
a
n
!
+
b
. [*]
p
p
p
divides
a
(
n
+
1
)
!
+
b
a^{(n + 1)!} + b
a
(
n
+
1
)!
+
b
. [*]
p
<
2
n
2
+
1
p < 2n^2 + 1
p
<
2
n
2
+
1
.Show that there are finitely many fun primes.
5
1
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Numbers on an infinite chessboard
Consider a chessboard that is infinite in all directions. Alex the T-rex wishes to place a positive integer in each square in such a way that:[*] No two numbers are equal. [*] If a number
m
m
m
is placed on square
C
C
C
, then at least
k
k
k
of the squares orthogonally adjacent to
C
C
C
have a multiple of
m
m
m
written on them.What is the greatest value of
k
k
k
for which this is possible?
4
1
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Lines meet on perpendicular bisector
Let
A
B
C
ABC
A
BC
be an obtuse triangle with
A
B
=
A
C
AB = AC
A
B
=
A
C
, and let
Γ
\Gamma
Γ
be the circle that is tangent to
A
B
AB
A
B
at
B
B
B
and to
A
C
AC
A
C
at
C
C
C
. Let
D
D
D
be the point on
Γ
\Gamma
Γ
furthest from
A
A
A
such that
A
D
AD
A
D
is perpendicular to
B
C
BC
BC
. Point
E
E
E
is the intersection of
A
B
AB
A
B
and
D
C
DC
D
C
, and point
F
F
F
lies on line
A
B
AB
A
B
such that
B
C
=
B
F
BC = BF
BC
=
BF
and
B
B
B
lies on segment
A
F
AF
A
F
. Finally, let
P
P
P
be the intersection of lines
A
C
AC
A
C
and
D
B
DB
D
B
. Show that
P
E
=
P
F
PE = PF
PE
=
PF
.
3
1
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Prove perpendicularity
Let
P
P
P
and
Q
Q
Q
be points in the interior of a triangle
A
B
C
ABC
A
BC
such that
∠
A
P
C
=
∠
A
Q
B
=
9
0
∘
\angle APC = \angle AQB = 90^{\circ}
∠
A
PC
=
∠
A
QB
=
9
0
∘
,
∠
A
C
P
=
∠
P
B
C
\angle ACP = \angle PBC
∠
A
CP
=
∠
PBC
, and
∠
A
B
Q
=
∠
Q
C
B
\angle ABQ = \angle QCB
∠
A
BQ
=
∠
QCB
. Suppose that lines
B
P
BP
BP
and
C
Q
CQ
CQ
meet at a point
R
R
R
. Show that
A
R
AR
A
R
is perpendicular to
P
Q
PQ
PQ
.
2
1
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All pairs are coprime
For which positive integers
n
n
n
does there exist a positive integer
m
m
m
such that among the numbers
m
+
n
,
2
m
+
(
n
−
1
)
,
…
,
n
m
+
1
m + n, 2m + (n - 1), \dots, nm + 1
m
+
n
,
2
m
+
(
n
−
1
)
,
…
,
nm
+
1
, there are no two that share a common factor greater than
1
1
1
?
1
1
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Functional polynomial equation
Find all functions
f
:
R
→
R
f \colon \mathbb{R} \to \mathbb{R}
f
:
R
→
R
that satisfy the following property for all real numbers
x
x
x
and all polynomials
P
P
P
with real coefficients:If
P
(
f
(
x
)
)
=
0
P(f(x)) = 0
P
(
f
(
x
))
=
0
, then
f
(
P
(
x
)
)
=
0
f(P(x)) = 0
f
(
P
(
x
))
=
0
.