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National and Regional Contests
Mexico Contests
OMMock - Mexico National Olympiad Mock Exam
2019 OMMock - Mexico National Olympiad Mock Exam
2019 OMMock - Mexico National Olympiad Mock Exam
Part of
OMMock - Mexico National Olympiad Mock Exam
Subcontests
(6)
6
1
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Intersection point of perpendicular bisectors satisfies property
Let
A
B
C
ABC
A
BC
be a scalene triangle with circumcenter
O
O
O
, and let
D
D
D
and
E
E
E
be points inside angle
∡
B
A
C
\measuredangle BAC
∡
B
A
C
such that
A
A
A
lies on line
D
E
DE
D
E
, and
∠
A
D
B
=
∠
C
B
A
\angle ADB=\angle CBA
∠
A
D
B
=
∠
CB
A
and
∠
A
E
C
=
∠
B
C
A
\angle AEC=\angle BCA
∠
A
EC
=
∠
BC
A
. Let
M
M
M
be the midpoint of
B
C
BC
BC
and
K
K
K
be a point such that
O
K
OK
O
K
is perpendicular to
A
O
AO
A
O
and
∠
B
A
K
=
∠
M
A
C
\angle BAK=\angle MAC
∠
B
A
K
=
∠
M
A
C
. Finally, let
P
P
P
be the intersection of the perpendicular bisectors of
B
D
BD
B
D
and
C
E
CE
CE
. Show that
K
O
=
K
P
KO=KP
K
O
=
K
P
.Proposed by Victor Domínguez
5
1
Hide problems
n people at a party and friendship
There are
n
≥
2
n\geq 2
n
≥
2
people at a party. Each person has at least one friend inside the party. Show that it is possible to choose a group of no more than
n
2
\frac{n}{2}
2
n
people at the party, such that any other person outside the group has a friend inside it.
4
1
Hide problems
Split numbers from 1 to 2n into two groups
Find all positive integers
n
n
n
such that it is possible to split the numbers from
1
1
1
to
2
n
2n
2
n
in two groups
(
a
1
,
a
2
,
.
.
,
a
n
)
(a_1,a_2,..,a_n)
(
a
1
,
a
2
,
..
,
a
n
)
,
(
b
1
,
b
2
,
.
.
.
,
b
n
)
(b_1,b_2,...,b_n)
(
b
1
,
b
2
,
...
,
b
n
)
in such a way that
2
n
∣
a
1
a
2
⋯
a
n
+
b
1
b
2
⋯
b
n
−
1
2n\mid a_1a_2\cdots a_n+b_1b_2\cdots b_n-1
2
n
∣
a
1
a
2
⋯
a
n
+
b
1
b
2
⋯
b
n
−
1
.Proposed by Alef Pineda
3
1
Hide problems
f(m^2)+f(mf(n))=f(m+n)f(m)
Let
Z
\mathbb{Z}
Z
be the set of integers. Find all functions
f
:
Z
→
Z
f: \mathbb{Z}\rightarrow \mathbb{Z}
f
:
Z
→
Z
such that, for any two integers
m
,
n
m, n
m
,
n
,
f
(
m
2
)
+
f
(
m
f
(
n
)
)
=
f
(
m
+
n
)
f
(
m
)
.
f(m^2)+f(mf(n))=f(m+n)f(m).
f
(
m
2
)
+
f
(
m
f
(
n
))
=
f
(
m
+
n
)
f
(
m
)
.
Proposed by Victor Domínguez and Pablo Valeriano
2
1
Hide problems
Expression in variables m and n divides other two
Find all pairs of positive integers
(
m
,
n
)
(m, n)
(
m
,
n
)
such that
m
2
−
m
n
+
n
2
+
1
m^2-mn+n^2+1
m
2
−
mn
+
n
2
+
1
divides both numbers
3
m
+
n
+
(
m
+
n
)
!
3^{m+n}+(m+n)!
3
m
+
n
+
(
m
+
n
)!
and
3
m
3
+
n
3
+
m
+
n
3^{m^3+n^3}+m+n
3
m
3
+
n
3
+
m
+
n
.Proposed by Dorlir Ahmeti
1
1
Hide problems
Similarity with centers of two circles
Let
C
1
C_1
C
1
and
C
2
C_2
C
2
be two circles with centers
O
1
O_1
O
1
and
O
2
O_2
O
2
, respectively, intersecting at
A
A
A
and
B
B
B
. Let
l
1
l_1
l
1
be the line tangent to
C
1
C_1
C
1
passing trough
A
A
A
, and
l
2
l_2
l
2
the line tangent to
C
2
C_2
C
2
passing through
B
B
B
. Suppose that
l
1
l_1
l
1
and
l
2
l_2
l
2
intersect at
P
P
P
and
l
1
l_1
l
1
intersects
C
2
C_2
C
2
again at
Q
Q
Q
. Show that
P
O
1
B
PO_1B
P
O
1
B
and
P
O
2
Q
PO_2Q
P
O
2
Q
are similar triangles.Proposed by Pablo Valeriano