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National and Regional Contests
Mexico Contests
OMMock - Mexico National Olympiad Mock Exam
2018 OMMock - Mexico National Olympiad Mock Exam
2018 OMMock - Mexico National Olympiad Mock Exam
Part of
OMMock - Mexico National Olympiad Mock Exam
Subcontests
(6)
6
1
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Difference of set cardinalities eventually constant
Let
A
A
A
be a finite set of positive integers, and for each positive integer
n
n
n
we define
S
n
=
{
x
1
+
x
2
+
⋯
+
x
n
∣
x
i
∈
A
for
i
=
1
,
2
,
…
,
n
}
S_n = \{x_1 + x_2 + \cdots + x_n \;\vert\; x_i \in A \text{ for } i = 1, 2, \dots, n\}
S
n
=
{
x
1
+
x
2
+
⋯
+
x
n
∣
x
i
∈
A
for
i
=
1
,
2
,
…
,
n
}
That is,
S
n
S_n
S
n
is the set of all positive integers which can be expressed as sum of exactly
n
n
n
elements of
A
A
A
, not necessarily different. Prove that there exist positive integers
N
N
N
and
k
k
k
such that
∣
S
n
+
1
∣
=
∣
S
n
∣
+
k
for all
n
≥
N
.
\left\vert S_{n + 1} \right\vert = \left\vert S_n \right\vert + k \text{ for all } n\geq N.
∣
S
n
+
1
∣
=
∣
S
n
∣
+
k
for all
n
≥
N
.
Proposed by Ariel García
5
1
Hide problems
Arc midpoints and lines intersect on circumcircle
Let
A
B
C
ABC
A
BC
be a triangle with circumcirle
Γ
\Gamma
Γ
, and let
M
M
M
and
N
N
N
be the respective midpoints of the minor arcs
A
B
AB
A
B
and
A
C
AC
A
C
of
Γ
\Gamma
Γ
. Let
P
P
P
and
Q
Q
Q
be points such that
A
B
=
B
P
AB=BP
A
B
=
BP
,
A
C
=
C
Q
AC=CQ
A
C
=
CQ
, and
P
P
P
,
B
B
B
,
C
C
C
,
Q
Q
Q
lie on
B
C
BC
BC
in that order. Prove that
P
M
PM
PM
and
Q
N
QN
QN
meet at a point on
Γ
\Gamma
Γ
.Proposed by Victor Domínguez
4
1
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Divisibilities on sum of digits
For each positive integer
n
n
n
let
s
(
n
)
s(n)
s
(
n
)
denote the sum of the decimal digits of
n
n
n
. Find all pairs of positive integers
(
a
,
b
)
(a, b)
(
a
,
b
)
with
a
>
b
a > b
a
>
b
which simultaneously satisfy the following two conditions
a
∣
b
+
s
(
a
)
a \mid b + s(a)
a
∣
b
+
s
(
a
)
b
∣
a
+
s
(
b
)
b \mid a + s(b)
b
∣
a
+
s
(
b
)
Proposed by Victor Domínguez
3
1
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Solving System of Symmetric Equations on Reals
Find all
n
n
n
-tuples of real numbers
(
x
1
,
x
2
,
…
,
x
n
)
(x_1, x_2, \dots, x_n)
(
x
1
,
x
2
,
…
,
x
n
)
such that, for every index
k
k
k
with
1
≤
k
≤
n
1\leq k\leq n
1
≤
k
≤
n
, the following holds:
x
k
2
=
∑
i
<
j
i
,
j
≠
k
x
i
x
j
x_k^2=\sum\limits_{\substack{i < j \\ i, j\neq k}} x_ix_j
x
k
2
=
i
<
j
i
,
j
=
k
∑
x
i
x
j
Proposed by Oriol Solé
2
1
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Minimum number of changes of direction
An equilateral triangle of side
n
n
n
has been divided into little equilateral triangles of side
1
1
1
in the usual way. We draw a path over the segments of this triangulation, in such a way that it visits exactly once each one of the
(
n
+
1
)
(
n
+
2
)
2
\frac{(n+1)(n+2)}{2}
2
(
n
+
1
)
(
n
+
2
)
vertices. What is the minimum number of times the path can change its direction?The figure below shows a valid path on a triangular board of side
4
4
4
, with exactly
9
9
9
changes of direction.[asy] unitsize(30); pair h = (1, 0); pair v = dir(60); pair d = dir(120); for(int i = 0; i < 4; ++i) { draw(i*v -- i*v + (4 - i)*h); draw(i*h -- i*h + (4 - i)*v); draw((i + 1)*h -- (i + 1)*h + (i + 1)*d); }draw(h + v -- v -- (0, 0) -- 2*h -- 2*h + v -- h + 2*v -- 2*v -- 4*v -- 3*h + v -- 3*h -- 4*h, linewidth(2)); draw(3*h -- 4*h, EndArrow); fill(circle(h + v, 0.1));[/asy]Proposed by Oriol Solé
1
1
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Trapezium, circles, and proving parallel lines
Let
A
B
C
D
ABCD
A
BC
D
be a trapezoid with bases
A
D
AD
A
D
and
B
C
BC
BC
, and let
M
M
M
be the midpoint of
C
D
CD
C
D
. The circumcircle of triangle
B
C
M
BCM
BCM
meets
A
C
AC
A
C
and
B
D
BD
B
D
again at
E
E
E
and
F
F
F
, with
E
E
E
and
F
F
F
distinct, and line
E
F
EF
EF
meets the circumcircle of triangle
A
E
M
AEM
A
EM
again at
P
P
P
. Prove that
C
P
CP
CP
is parallel to
B
D
BD
B
D
.Proposed by Ariel García