MathDB
Problems
Contests
National and Regional Contests
Mexico Contests
Mexico National Olympiad
2021 Mexico National Olympiad
2021 Mexico National Olympiad
Part of
Mexico National Olympiad
Subcontests
(5)
5
1
Hide problems
$n=\overline{a_1a_2\cdots a_{k-1}a_k}$
If
n
=
a
1
a
2
⋯
a
k
−
1
a
k
‾
n=\overline{a_1a_2\cdots a_{k-1}a_k}
n
=
a
1
a
2
⋯
a
k
−
1
a
k
, be
s
(
n
)
s(n)
s
(
n
)
such that . If
k
k
k
is even,
s
(
n
)
=
a
1
a
2
‾
+
a
3
a
4
‾
⋯
+
a
k
−
1
a
k
‾
s(n)=\overline{a_1a_2}+\overline{a_3a_4}\cdots+\overline{a_{k-1}a_k}
s
(
n
)
=
a
1
a
2
+
a
3
a
4
⋯
+
a
k
−
1
a
k
. If
k
k
k
is odd,
s
(
n
)
=
a
1
+
a
2
a
3
‾
⋯
+
a
k
−
1
a
k
‾
s(n)=a_1+\overline{a_2a_3}\cdots+\overline{a_{k-1}a_k}
s
(
n
)
=
a
1
+
a
2
a
3
⋯
+
a
k
−
1
a
k
For example
s
(
123
)
=
1
+
23
=
24
s(123)=1+23=24
s
(
123
)
=
1
+
23
=
24
and
s
(
2021
)
=
20
+
21
=
41
s(2021)=20+21=41
s
(
2021
)
=
20
+
21
=
41
Be
n
n
n
is
d
i
g
i
t
a
l
digital
d
i
g
i
t
a
l
if
s
(
n
)
s(n)
s
(
n
)
is a divisor of
n
n
n
. Prove that among any 198 consecutive positive integers, all of them less than 2000021 there is one of them that is
d
i
g
i
t
a
l
digital
d
i
g
i
t
a
l
.
4
1
Hide problems
two circles tangent at the euler line
Let
A
B
C
ABC
A
BC
be an acutangle scalene triangle with
∠
B
A
C
=
6
0
∘
\angle BAC = 60^{\circ}
∠
B
A
C
=
6
0
∘
and orthocenter
H
H
H
. Let
ω
b
\omega_b
ω
b
be the circumference passing through
H
H
H
and tangent to
A
B
AB
A
B
at
B
B
B
, and
ω
c
\omega_c
ω
c
the circumference passing through
H
H
H
and tangent to
A
C
AC
A
C
at
C
C
C
. [*] Prove that
ω
b
\omega_b
ω
b
and
ω
c
\omega_c
ω
c
only have
H
H
H
as common point. [*] Prove that the line passing through
H
H
H
and the circumcenter
O
O
O
of triangle
A
B
C
ABC
A
BC
is a common tangent to
ω
b
\omega_b
ω
b
and
ω
c
\omega_c
ω
c
.Note: The orthocenter of a triangle is the intersection point of the three altitudes, whereas the circumcenter of a triangle is the center of the circumference passing through it's three vertices.
6
1
Hide problems
Find all sets that satisfy sum and cardinality condition
Determine all non empty sets
C
1
,
C
2
,
C
3
,
⋯
C_1, C_2, C_3, \cdots
C
1
,
C
2
,
C
3
,
⋯
such that each one of them has a finite number of elements, all their elements are positive integers, and they satisfy the following property: For any positive integers
n
n
n
and
m
m
m
, the number of elements in the set
C
n
C_n
C
n
plus the number of elements in the set
C
m
C_m
C
m
equals the sum of the elements in the set
C
m
+
n
C_{m + n}
C
m
+
n
.Note: We denote
∣
C
n
∣
\lvert C_n \lvert
∣
C
n
∣
the number of elements in the set
C
n
C_n
C
n
, and
S
k
S_k
S
k
as the sum of the elements in the set
C
n
C_n
C
n
so the problem's condition is that for every
n
n
n
and
m
m
m
:
∣
C
n
∣
+
∣
C
m
∣
=
S
n
+
m
\lvert C_n \lvert + \lvert C_m \lvert = S_{n + m}
∣
C
n
∣
+
∣
C
m
∣
=
S
n
+
m
is satisfied.
2
1
Hide problems
midpoints equivalency and a tangent ?
Let
A
B
C
ABC
A
BC
be a triangle with
∠
A
C
B
>
9
0
∘
\angle ACB > 90^{\circ}
∠
A
CB
>
9
0
∘
, and let
D
D
D
be a point on
B
C
BC
BC
such that
A
D
AD
A
D
is perpendicular to
B
C
BC
BC
. Consider the circumference
Γ
\Gamma
Γ
with with diameter
B
C
BC
BC
. A line
ℓ
\ell
ℓ
passes through
D
D
D
and is tangent to
Γ
\Gamma
Γ
at
P
P
P
, cuts
A
C
AC
A
C
at
M
M
M
(such that
M
M
M
is in between
A
A
A
and
C
C
C
), and cuts the side
A
B
AB
A
B
at
N
N
N
. Prove that
M
M
M
is the midpoint of
D
P
DP
D
P
if and only if
N
N
N
is the midpoint of
A
B
AB
A
B
.
1
1
Hide problems
rectangles of equal area, and sides in arithmetic progression
The real positive numbers
a
1
,
a
2
,
a
3
a_1, a_2,a_3
a
1
,
a
2
,
a
3
are three consecutive terms of an arithmetic progression, and similarly,
b
1
,
b
2
,
b
3
b_1, b_2, b_3
b
1
,
b
2
,
b
3
are distinct real positive numbers and consecutive terms of an arithmetic progression. Is it possible to use three segments of lengths
a
1
,
a
2
,
a
3
a_1, a_2, a_3
a
1
,
a
2
,
a
3
as bases, and other three segments of lengths
b
1
,
b
2
,
b
3
b_1, b_2, b_3
b
1
,
b
2
,
b
3
as altitudes, to construct three rectangles of equal area ?