MathDB
Problems
Contests
National and Regional Contests
Mexico Contests
Mexico National Olympiad
2019 Mexico National Olympiad
2019 Mexico National Olympiad
Part of
Mexico National Olympiad
Subcontests
(6)
6
1
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Another orthocenter problem <3
Let
A
B
C
ABC
A
BC
be a triangle such that
∠
B
A
C
=
4
5
∘
\angle BAC = 45^{\circ}
∠
B
A
C
=
4
5
∘
. Let
H
,
O
H,O
H
,
O
be the orthocenter and circumcenter of
A
B
C
ABC
A
BC
, respectively. Let
ω
\omega
ω
be the circumcircle of
A
B
C
ABC
A
BC
and
P
P
P
the point on
ω
\omega
ω
such that the circumcircle of
P
B
H
PBH
PB
H
is tangent to
B
C
BC
BC
. Let
X
X
X
and
Y
Y
Y
be the circumcenters of
P
H
B
PHB
P
H
B
and
P
H
C
PHC
P
H
C
respectively. Let
O
1
,
O
2
O_1,O_2
O
1
,
O
2
be the circumcenters of
P
X
O
PXO
PXO
and
P
Y
O
PYO
P
Y
O
respectively. Prove that
O
1
O_1
O
1
and
O
2
O_2
O
2
lie on
A
B
AB
A
B
and
A
C
AC
A
C
, respectively.
5
1
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Another grasshopper problem
Let
a
>
b
a > b
a
>
b
be relatively prime positive integers. A grashopper stands at point
0
0
0
in a number line. Each minute, the grashopper jumps according to the following rules: [*] If the current minute is a multiple of
a
a
a
and not a multiple of
b
b
b
, it jumps
a
a
a
units forward. [*] If the current minute is a multiple of
b
b
b
and not a multiple of
a
a
a
, it jumps
b
b
b
units backward. [*] If the current minute is both a multiple of
b
b
b
and a multiple of
a
a
a
, it jumps
a
−
b
a - b
a
−
b
units forward. [*] If the current minute is neither a multiple of
a
a
a
nor a multiple of
b
b
b
, it doesn't move.Find all positions on the number line that the grasshopper will eventually reach.
4
1
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Very good lists
A list of positive integers is called good if the maximum element of the list appears exactly once. A sublist is a list formed by one or more consecutive elements of a list. For example, the list
10
,
34
,
34
,
22
,
30
,
22
10,34,34,22,30,22
10
,
34
,
34
,
22
,
30
,
22
the sublist
22
,
30
,
22
22,30,22
22
,
30
,
22
is good and
10
,
34
,
34
,
22
10,34,34,22
10
,
34
,
34
,
22
is not. A list is very good if all its sublists are good. Find the minimum value of
k
k
k
such that there exists a very good list of length
2019
2019
2019
with
k
k
k
different values on it.
3
1
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Isabel pairs into segments which do not intersect.
Let
n
≥
2
n\geq 2
n
≥
2
be an integer. Consider
2
n
2n
2
n
points around a circle. Each vertex has been tagged with one integer from
1
1
1
to
n
n
n
, inclusive, and each one of these integers has been used exactly two times. Isabel divides the points into
n
n
n
pairs, and draws the segments joining them, with the condition that the segments do not intersect. Then, she assigns to each segment the greatest integer between its endpoints.a) Show that, no matter how the points have been tagged, Isabel can always choose the pairs in such a way that she uses exactly
⌈
n
/
2
⌉
\lceil n/2\rceil
⌈
n
/2
⌉
numbers to tag the segments.b) Can the points be tagged in such a way that, no matter how Isabel divides the points into pairs, she always uses exactly
⌈
n
/
2
⌉
\lceil n/2\rceil
⌈
n
/2
⌉
numbers to tag the segments?Note. For each real number
x
x
x
,
⌈
x
⌉
\lceil x\rceil
⌈
x
⌉
denotes the least integer greater than or equal to
x
x
x
. For example,
⌈
3.6
⌉
=
4
\lceil 3.6\rceil=4
⌈
3.6
⌉
=
4
and
⌈
2
⌉
=
2
\lceil 2\rceil=2
⌈
2
⌉
=
2
.Proposed by Victor Domínguez
2
1
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Perpendicular lines CM and BF in orthocenter configuration
Let
H
H
H
be the orthocenter of acute-angled triangle
A
B
C
ABC
A
BC
and
M
M
M
be the midpoint of
A
H
AH
A
H
. Line
B
H
BH
B
H
cuts
A
C
AC
A
C
at
D
D
D
. Consider point
E
E
E
such that
B
C
BC
BC
is the perpendicular bisector of
D
E
DE
D
E
. Segments
C
M
CM
CM
and
A
E
AE
A
E
intersect at
F
F
F
. Show that
B
F
BF
BF
is perpendicular to
C
M
CM
CM
.Proposed by Germán Puga
1
1
Hide problems
Perfect powers with divisor-function-like exponent
An integer number
m
≥
1
m\geq 1
m
≥
1
is mexica if it's of the form
n
d
(
n
)
n^{d(n)}
n
d
(
n
)
, where
n
n
n
is a positive integer and
d
(
n
)
d(n)
d
(
n
)
is the number of positive integers which divide
n
n
n
. Find all mexica numbers less than
2019
2019
2019
.Note. The divisors of
n
n
n
include
1
1
1
and
n
n
n
; for example,
d
(
12
)
=
6
d(12)=6
d
(
12
)
=
6
, since
1
,
2
,
3
,
4
,
6
,
12
1, 2, 3, 4, 6, 12
1
,
2
,
3
,
4
,
6
,
12
are all the positive divisors of
12
12
12
.Proposed by Cuauhtémoc Gómez