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Problems
Contests
National and Regional Contests
Mexico Contests
OMMock - Mexico National Olympiad Mock Exam
2017 OMMock - Mexico National Olympiad Mock Exam
2017 OMMock - Mexico National Olympiad Mock Exam
Part of
OMMock - Mexico National Olympiad Mock Exam
Subcontests
(6)
6
1
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Path by k+2 cities given there are nk/2 highways
In a certain country there are
n
n
n
cities. Some pairs of cities are connected by highways in such a way that for each two cities there is at most one highway connecting them. Assume that for a certain positive integer
k
k
k
, the total number of highways is greater than
n
k
2
\frac{nk}{2}
2
nk
. Show that there exist
k
+
2
k+2
k
+
2
distinct cities
C
1
,
C
2
,
…
,
C
k
+
2
C_1, C_2, \dots, C_{k+2}
C
1
,
C
2
,
…
,
C
k
+
2
such that
C
i
C_i
C
i
and
C
i
+
1
C_{i+1}
C
i
+
1
are connected by a highway for
i
=
1
,
2
,
…
,
k
+
1
i=1, 2, \dots, k+1
i
=
1
,
2
,
…
,
k
+
1
.Proposed by Oriol Solé
5
1
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Determine functions for fixed k
Let
k
k
k
be a positive real number. Determine all functions
f
:
[
−
k
,
k
]
→
[
0
,
k
]
f:[-k, k]\rightarrow[0, k]
f
:
[
−
k
,
k
]
→
[
0
,
k
]
satisfying the equation
f
(
x
)
2
+
f
(
y
)
2
−
2
x
y
=
k
2
+
f
(
x
+
y
)
2
f(x)^2+f(y)^2-2xy=k^2+f(x+y)^2
f
(
x
)
2
+
f
(
y
)
2
−
2
x
y
=
k
2
+
f
(
x
+
y
)
2
for any
x
,
y
∈
[
−
k
,
k
]
x, y\in[-k, k]
x
,
y
∈
[
−
k
,
k
]
such that
x
+
y
∈
[
−
k
,
k
]
x+y\in[-k, k]
x
+
y
∈
[
−
k
,
k
]
.Proposed by Maximiliano Sánchez
4
1
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Two variable diophantine equation has no solutions
Show that the equation
a
2
b
=
2017
(
a
+
b
)
a^2b=2017(a+b)
a
2
b
=
2017
(
a
+
b
)
has no solutions for positive integers
a
a
a
and
b
b
b
.Proposed by Oriol Solé
3
1
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Proving that system has solutions assuming xy=z^2+2
Let
x
,
y
,
z
x, y, z
x
,
y
,
z
be positive integers such that
x
y
=
z
2
+
2
xy=z^2+2
x
y
=
z
2
+
2
. Prove that there exist integers
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
such that the following equalities are satisfied: \begin{eqnarray*} x=a^2+2b^2\\ y=c^2+d^2\\ z=ac+2bd\\ \end{eqnarray*}Proposed by Isaac Jiménez
2
1
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Game on triangular board
Alice and Bob play on an infinite board formed by equilateral triangles. In each turn, Alice first places a white token on an unoccupied cell, and then Bob places a black token on an unoccupied cell. Alice's goal is to eventually have
k
k
k
white tokens on a line. Determine the maximum value of
k
k
k
for which Alice can achieve this no matter how Bob plays.Proposed by Oriol Solé
1
1
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Segment equality with a rhombus
Let
A
B
C
ABC
A
BC
be a triangle with circumcenter
O
O
O
. Point
D
,
E
,
F
D, E, F
D
,
E
,
F
are chosen on sides
A
B
,
B
C
AB, BC
A
B
,
BC
and
A
C
AC
A
C
, respectively, such that
A
D
E
F
ADEF
A
D
EF
is a rhombus. The circumcircles of
B
D
E
BDE
B
D
E
and
C
F
E
CFE
CFE
intersect
A
E
AE
A
E
at
P
P
P
and
Q
Q
Q
respectively. Show that
O
P
=
O
Q
OP=OQ
OP
=
OQ
.Proposed by Ariel García