MathDB
Problems
Contests
National and Regional Contests
Mexico Contests
Mexico National Olympiad
2015 Mexico National Olympiad
2015 Mexico National Olympiad
Part of
Mexico National Olympiad
Subcontests
(6)
2
1
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Bijections
Let
n
n
n
be a positive integer and let
k
k
k
be an integer between
1
1
1
and
n
n
n
inclusive. There is a white board of
n
×
n
n \times n
n
×
n
. We do the following process. We draw
k
k
k
rectangles with integer sides lenghts and sides parallel to the ones of the
n
×
n
n \times n
n
×
n
board, and such that each rectangle covers the top-right corner of the
n
×
n
n \times n
n
×
n
board. Then, the
k
k
k
rectangles are painted of black. This process leaves a white figure in the board. How many different white figures are possible to do with
k
k
k
rectangles that can't be done with less than
k
k
k
rectangles?Proposed by David Torres Flores
6
1
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Prove n is square-free
Let
n
n
n
be a positive integer and let
d
1
,
d
2
,
…
,
d
k
d_1, d_2, \dots, d_k
d
1
,
d
2
,
…
,
d
k
be its positive divisors. Consider the number
f
(
n
)
=
(
−
1
)
d
1
d
1
+
(
−
1
)
d
2
d
2
+
⋯
+
(
−
1
)
d
k
d
k
f(n) = (-1)^{d_1}d_1 + (-1)^{d_2}d_2 + \dots + (-1)^{d_k}d_k
f
(
n
)
=
(
−
1
)
d
1
d
1
+
(
−
1
)
d
2
d
2
+
⋯
+
(
−
1
)
d
k
d
k
Assume
f
(
n
)
f(n)
f
(
n
)
is a power of 2. Show if
m
m
m
is an integer greater than 1, then
m
2
m^2
m
2
does not divide
n
n
n
.
5
1
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Incenter Geometry
Let
I
I
I
be the incenter of an acute-angled triangle
A
B
C
ABC
A
BC
. Line
A
I
AI
A
I
cuts the circumcircle of
B
I
C
BIC
B
I
C
again at
E
E
E
. Let
D
D
D
be the foot of the altitude from
A
A
A
to
B
C
BC
BC
, and let
J
J
J
be the reflection of
I
I
I
across
B
C
BC
BC
. Show
D
D
D
,
J
J
J
and
E
E
E
are collinear.
4
1
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Combinatorics?
Let
n
n
n
be a positive integer. Mary writes the
n
3
n^3
n
3
triples of not necessarily distinct integers, each between
1
1
1
and
n
n
n
inclusive on a board. Afterwards, she finds the greatest (possibly more than one), and erases the rest. For example, in the triple
(
1
,
3
,
4
)
(1, 3, 4)
(
1
,
3
,
4
)
she erases the numbers 1 and 3, and in the triple
(
1
,
2
,
2
)
(1, 2, 2)
(
1
,
2
,
2
)
she erases only the number 1,Show after finishing this process, the amount of remaining numbers on the board cannot be a perfect square.
1
1
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Geo problem
Let
A
B
C
ABC
A
BC
be an acuted-angle triangle and let
H
H
H
be it's orthocenter. Let
P
Q
PQ
PQ
be a segment through
H
H
H
such that
P
P
P
lies on
A
B
AB
A
B
and
Q
Q
Q
lies on
A
C
AC
A
C
and such that
∠
P
H
B
=
∠
C
H
Q
\angle PHB= \angle CHQ
∠
P
H
B
=
∠
C
H
Q
. Finally, in the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
, consider
M
M
M
such that
M
M
M
is the mid point of the arc
B
C
BC
BC
that doesn't contain
A
A
A
. Prove that
M
P
=
M
Q
MP=MQ
MP
=
MQ
Proposed by Eduardo Velasco/Marco Figueroa
3
1
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functional equation
Let
N
=
{
1
,
2
,
3
,
.
.
.
}
\mathbb{N} =\{1, 2, 3, ...\}
N
=
{
1
,
2
,
3
,
...
}
be the set of positive integers. Let
f
:
N
→
N
f : \mathbb{N} \rightarrow \mathbb{N}
f
:
N
→
N
be a function that gives a positive integer value, to every positive integer. Suppose that
f
f
f
satisfies the following conditions:
f
(
1
)
=
1
f(1)=1
f
(
1
)
=
1
f
(
a
+
b
+
a
b
)
=
a
+
b
+
f
(
a
b
)
f(a+b+ab)=a+b+f(ab)
f
(
a
+
b
+
ab
)
=
a
+
b
+
f
(
ab
)
Find the value of
f
(
2015
)
f(2015)
f
(
2015
)
Proposed by Jose Antonio Gomez Ortega