MathDB
Problems
Contests
National and Regional Contests
Mexico Contests
Mexican Girls' Contest
2022 Mexican Girls' Contest
2022 Mexican Girls' Contest
Part of
Mexican Girls' Contest
Subcontests
(8)
3
2
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color the board and place k tiles without any 2 of them attacking each other
All the squares of a
2022
×
2022
2022 \times 2022
2022
×
2022
board will be colored white or black. Chips will be placed in several of these boxes, at most one per box. We say that two tokens attack each other, when the following two conditions are met: a) There is a path of squares that joins the squares where the pieces were placed. This path can have a horizontal, vertical, or diagonal direction. b) All the squares in this path, including the squares where the pieces are, are of the same color. For example, the following figure shows a small example of a possible coloring of a
6
×
6
6 \times 6
6
×
6
board with
A
,
B
,
C
,
D
A, B, C, D
A
,
B
,
C
,
D
, and
E
E
E
tiles placed. The pairs of checkers that attack each other are
(
D
,
E
)
(D, E)
(
D
,
E
)
,
(
C
,
D
)
(C, D)
(
C
,
D
)
, and
(
B
,
E
)
(B, E)
(
B
,
E
)
. https://cdn.artofproblemsolving.com/attachments/2/0/52ec7b7d1c02e266b666e4f8b25e87c58f0c89.png What is the maximum value of
k
k
k
such that it is possible to color the board and place
k
k
k
tiles without any two of them attacking each other?
lattice points in the plane
Consider a set
S
S
S
of
16
16
16
lattice points. The
16
16
16
points of
S
S
S
are divided into
8
8
8
pairs in such a way thatfor every point
A
A
A
and any of the
7
7
7
pairs of points
(
B
,
C
)
(B,C)
(
B
,
C
)
where
A
A
A
is not included,
A
A
A
is at a distance of at most
5
\sqrt{5}
5
from either
B
B
B
or
C
C
C
Prove that any two points in the set
S
S
S
are at a distance of at most
3
5
3\sqrt5
3
5
.
2
2
Hide problems
4 numbers game
In the training of a state, the coach proposes a game. The coach writes four real numbers on the board in order from least to greatest:
a
<
b
<
c
<
d
a < b < c < d
a
<
b
<
c
<
d
.Each Olympian draws the figure on the right in her notebook and arranges the numbers inside the corner shapes, however she wants, putting a number on each one. Once arranged, on each segment write the square of the difference of the numbers at its ends. Then, add the
4
4
4
numbers obtained. https://cdn.artofproblemsolving.com/attachments/9/a/ea348c637ae266c908e0b97e64605808b3b1d2.png For example, if Vania arranges them as in the figure on the right, then the result would be
(
c
−
b
)
2
+
(
b
−
a
)
2
+
(
a
−
d
)
2
+
(
d
−
c
)
2
.
(c - b)^2 + (b- a)^2 + (a - d)^2 + (d - c)^2.
(
c
−
b
)
2
+
(
b
−
a
)
2
+
(
a
−
d
)
2
+
(
d
−
c
)
2
.
https://cdn.artofproblemsolving.com/attachments/8/b/9c5375d66a4a6344b2bce333534fa7fac2ad6c.png The Olympians with the lowest result win. In what ways can you arrange the numbers to win? Give all the possible solutions.
find the value of an angles
Consider
△
A
B
C
\triangle ABC
△
A
BC
an isosceles triangle such that
A
B
=
B
C
AB = BC
A
B
=
BC
. Let
P
P
P
be a point satisfying
∠
A
B
P
=
8
0
∘
,
∠
C
B
P
=
2
0
∘
,
and
A
C
=
B
P
\angle ABP = 80^\circ, \angle CBP = 20^\circ, \textrm{and} \hspace{0.17cm} AC = BP
∠
A
BP
=
8
0
∘
,
∠
CBP
=
2
0
∘
,
and
A
C
=
BP
Find all possible values of
∠
B
C
P
\angle BCP
∠
BCP
.
5
1
Hide problems
50 lightest frogs represented 30 % of total mass of all the frogs in the pond
A biologist found a pond with frogs. When classifying them by their mass, he noticed the following:The
50
50
50
lightest frogs represented
30
%
30\%
30%
of the total mass of all the frogs in the pond, while the
44
44
44
heaviest frogs represented
27
%
27\%
27%
of the total mass.As fate would have it, the frogs escaped and the biologist only has the above information. How many frogs were in the pond?
6
1
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a not prime if (5a^4+a^2)/(b^4+3b^2+4) is an integer
Let
a
a
a
and
b
b
b
be positive integers such that
5
a
4
+
a
2
b
4
+
3
b
2
+
4
\frac{5a^4+a^2}{b^4+3b^2+4}
b
4
+
3
b
2
+
4
5
a
4
+
a
2
is an integer. Prove that
a
a
a
is not a prime number.
1
1
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ABCD is # if MENF is #
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral,
E
E
E
the midpoint of side
B
C
BC
BC
, and
F
F
F
the midpoint of side
A
D
AD
A
D
. Segment
A
C
AC
A
C
intersects segment
B
F
BF
BF
at
M
M
M
and segment
D
E
DE
D
E
at
N
N
N
. If quadrilateral
M
E
N
F
MENF
MENF
is also known to be a parallelogram, prove that
A
B
C
D
ABCD
A
BC
D
is also a parallelogram.
8
1
Hide problems
Marking vertices in splitted triangle
Let
n
n
n
be a positive integer. Consider a figure of a equilateral triangle of side
n
n
n
and splitted in
n
2
n^2
n
2
small equilateral triangles of side
1
1
1
. One will mark some of the
1
+
2
+
⋯
+
(
n
+
1
)
1+2+\dots+(n+1)
1
+
2
+
⋯
+
(
n
+
1
)
vertices of the small triangles, such that for every integer
k
≥
1
k\geq 1
k
≥
1
, there is not any trapezoid(trapezium), whose the sides are
(
1
,
k
,
1
,
k
+
1
)
(1,k,1,k+1)
(
1
,
k
,
1
,
k
+
1
)
, with all the vertices marked. Furthermore, there are no small triangle(side
1
1
1
) have your three vertices marked. Determine the greatest quantity of marked vertices.
7
1
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Many Many Intersections
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram (non-rectangle) and
Γ
\Gamma
Γ
is the circumcircle of
△
A
B
D
\triangle ABD
△
A
B
D
. The points
E
E
E
and
F
F
F
are the intersections of the lines
B
C
BC
BC
and
D
C
DC
D
C
with
Γ
\Gamma
Γ
respectively. Define
P
=
E
D
∩
B
A
P=ED\cap BA
P
=
E
D
∩
B
A
,
Q
=
F
B
∩
D
A
Q=FB\cap DA
Q
=
FB
∩
D
A
and
R
=
P
Q
∩
C
A
R=PQ\cap CA
R
=
PQ
∩
C
A
. Prove that
P
R
R
Q
=
(
B
C
C
D
)
2
\frac{PR}{RQ}=(\frac{BC}{CD})^2
RQ
PR
=
(
C
D
BC
)
2
4
1
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Power of 2 dividing Power of...
Let
k
k
k
be a positive integer and
m
m
m
be an odd integer. Prove that there exists a positive integer
n
n
n
such that
n
n
−
m
n^n-m
n
n
−
m
is divisible by
2
k
2^k
2
k
.