MathDB
Problems
Contests
National and Regional Contests
Mexico Contests
Mexico National Olympiad
2013 Mexico National Olympiad
2013 Mexico National Olympiad
Part of
Mexico National Olympiad
Subcontests
(6)
5
1
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2013 Mexican MO, Problem 5
A pair of integers is special if it is of the form
(
n
,
n
−
1
)
(n, n-1)
(
n
,
n
−
1
)
or
(
n
−
1
,
n
)
(n-1, n)
(
n
−
1
,
n
)
for some positive integer
n
n
n
. Let
n
n
n
and
m
m
m
be positive integers such that pair
(
n
,
m
)
(n, m)
(
n
,
m
)
is not special. Show
(
n
,
m
)
(n, m)
(
n
,
m
)
can be expressed as a sum of two or more different special pairs if and only if
n
n
n
and
m
m
m
satisfy the inequality
n
+
m
≥
(
n
−
m
)
2
n+m\geq (n-m)^2
n
+
m
≥
(
n
−
m
)
2
. Note: The sum of two pairs is defined as
(
a
,
b
)
+
(
c
,
d
)
=
(
a
+
c
,
b
+
d
)
(a, b)+(c, d) = (a+c, b+d)
(
a
,
b
)
+
(
c
,
d
)
=
(
a
+
c
,
b
+
d
)
.
4
1
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Colored cube
A
n
×
n
×
n
n \times n \times n
n
×
n
×
n
cube is constructed using
1
×
1
×
1
1 \times 1 \times 1
1
×
1
×
1
cubes, some of them black and others white, such that in each
n
×
1
×
1
n \times 1 \times 1
n
×
1
×
1
,
1
×
n
×
1
1 \times n \times 1
1
×
n
×
1
, and
1
×
1
×
n
1 \times 1 \times n
1
×
1
×
n
subprism there are exactly two black cubes, and they are separated by an even number of white cubes (possibly 0). Show it is possible to replace half of the black cubes with white cubes such that each
n
×
1
×
1
n \times 1 \times 1
n
×
1
×
1
,
1
×
n
×
1
1 \times n \times 1
1
×
n
×
1
and
1
×
1
×
n
1 \times 1 \times n
1
×
1
×
n
subprism contains exactly one black cube.
3
1
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Largest Subset
What is the largest amount of elements that can be taken from the set
{
1
,
2
,
.
.
.
,
2012
,
2013
}
\{1, 2, ... , 2012, 2013\}
{
1
,
2
,
...
,
2012
,
2013
}
, such that within them there are no distinct three, say
a
a
a
,
b
b
b
,and
c
c
c
, such that
a
a
a
is a divisor or multiple of
b
−
c
b-c
b
−
c
?
6
1
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Octagon Inequality
Let
A
1
A
2
.
.
.
A
8
A_1A_2 ... A_8
A
1
A
2
...
A
8
be a convex octagon such that all of its sides are equal and its opposite sides are parallel. For each
i
=
1
,
.
.
.
,
8
i = 1, ... , 8
i
=
1
,
...
,
8
, define
B
i
B_i
B
i
as the intersection between segments
A
i
A
i
+
4
A_iA_{i+4}
A
i
A
i
+
4
and
A
i
−
1
A
i
+
1
A_{i-1}A_{i+1}
A
i
−
1
A
i
+
1
, where
A
j
+
8
=
A
j
A_{j+8} = A_j
A
j
+
8
=
A
j
and
B
j
+
8
=
B
j
B_{j+8} = B_j
B
j
+
8
=
B
j
for all
j
j
j
. Show some number
i
i
i
, amongst 1, 2, 3, and 4 satisfies
A
i
A
i
+
4
B
i
B
i
+
4
≤
3
2
\frac{A_iA_{i+4}}{B_iB_{i+4}} \leq \frac{3}{2}
B
i
B
i
+
4
A
i
A
i
+
4
≤
2
3
2
1
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Angles in parallelogram
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram with the angle at
A
A
A
obtuse. Let
P
P
P
be a point on segment
B
D
BD
B
D
. The circle with center
P
P
P
passing through
A
A
A
cuts line
A
D
AD
A
D
at
A
A
A
and
Y
Y
Y
and cuts line
A
B
AB
A
B
at
A
A
A
and
X
X
X
. Line
A
P
AP
A
P
intersects
B
C
BC
BC
at
Q
Q
Q
and
C
D
CD
C
D
at
R
R
R
. Prove
∠
X
P
Y
=
∠
X
Q
Y
+
∠
X
R
Y
\angle XPY = \angle XQY + \angle XRY
∠
XP
Y
=
∠
XQ
Y
+
∠
XR
Y
.
1
1
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Primes dividing their indices
All the prime numbers are written in order,
p
1
=
2
,
p
2
=
3
,
p
3
=
5
,
.
.
.
p_1 = 2, p_2 = 3, p_3 = 5, ...
p
1
=
2
,
p
2
=
3
,
p
3
=
5
,
...
Find all pairs of positive integers
a
a
a
and
b
b
b
with
a
−
b
≥
2
a - b \geq 2
a
−
b
≥
2
, such that
p
a
−
p
b
p_a - p_b
p
a
−
p
b
divides
2
(
a
−
b
)
2(a-b)
2
(
a
−
b
)
.