MathDB
Problems
Contests
National and Regional Contests
Brazil Contests
Girls in Mathematics Tournament
2023 Girls in Mathematics Tournament
2023 Girls in Mathematics Tournament
Part of
Girls in Mathematics Tournament
Subcontests
(4)
4
2
Hide problems
Combinatorics+Number Theory and a bord to solve
Determine all
n
n
n
positive integers such that exists an
n
×
n
n\times n
n
×
n
where we can write
n
n
n
times each of the numbers from
1
1
1
to
n
n
n
(one number in each cell), such that the
n
n
n
sums of numbers in each line leave
n
n
n
distinct remainders in the division by
n
n
n
, and the
n
n
n
sums of numbers in each column leave
n
n
n
distinct remainders in the division by
n
n
n
.
Can you construct the incenter of a triangle ABC?
Given points
P
P
P
and
Q
Q
Q
, Jaqueline has a ruler that allows tracing the line
P
Q
PQ
PQ
. Jaqueline also has a special object that allows the construction of a circle of diameter
P
Q
PQ
PQ
. Also, always when two circles (or a circle and a line, or two lines) intersect, she can mark the points of the intersection with a pencil and trace more lines and circles using these dispositives by the points marked. Initially, she has an acute scalene triangle
A
B
C
ABC
A
BC
. Show that Jaqueline can construct the incenter of
A
B
C
ABC
A
BC
.
3
2
Hide problems
FM bisects the angle AFD?
Let
A
B
C
ABC
A
BC
an acute triangle and
D
D
D
and
E
E
E
the feet of heights by
A
A
A
and
B
B
B
, respectively, and let
M
M
M
be the midpoint of
A
C
AC
A
C
. The circle that passes through
D
D
D
and
B
B
B
and is tangent to
B
E
BE
BE
in
B
B
B
intersects the line
B
M
BM
BM
in
F
,
F
≠
B
F, F\neq B
F
,
F
=
B
. Show that
F
M
FM
FM
is the angle bisector of
∠
A
F
D
\angle AFD
∠
A
F
D
.
Set of positive integers and coloring points in red
Let
S
S
S
be a set not empty of positive integers and
A
B
AB
A
B
a segment with, initially, only points
A
A
A
and
B
B
B
colored by red. An operation consists of choosing two distinct points
X
,
Y
X, Y
X
,
Y
colored already by red and
n
∈
S
n\in S
n
∈
S
an integer, and painting in red the
n
n
n
points
A
1
,
A
2
,
.
.
.
,
A
n
A_1, A_2,..., A_n
A
1
,
A
2
,
...
,
A
n
of segment
X
Y
XY
X
Y
such that
X
A
1
=
A
1
A
2
=
A
2
A
3
=
.
.
.
=
A
n
−
1
A
n
=
A
n
Y
XA_1= A_1A_2= A_2A_3=...= A_{n-1}A_n= A_nY
X
A
1
=
A
1
A
2
=
A
2
A
3
=
...
=
A
n
−
1
A
n
=
A
n
Y
and
X
A
1
<
X
A
2
<
.
.
.
<
X
A
n
XA_1<XA_2<...<XA_n
X
A
1
<
X
A
2
<
...
<
X
A
n
. Find the least positive integer
m
m
m
such exists a subset
S
S
S
of
{
1
,
2
,
.
.
,
m
}
\{1,2,.., m\}
{
1
,
2
,
..
,
m
}
such that, after a finite number of operations, we can paint in red the point
K
K
K
in the segment
A
B
AB
A
B
defined by
A
K
K
B
=
2709
2022
\frac{AK}{KB}= \frac{2709}{2022}
K
B
A
K
=
2022
2709
. Also, find the number of such subsets for such a value of
m
m
m
.
2
2
Hide problems
One of a, b, c is zero?
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
real numbers such that
a
n
+
b
n
=
c
n
a^n+b^n= c^n
a
n
+
b
n
=
c
n
for three positive integers consecutive of
n
n
n
. Prove that
a
b
c
=
0
abc= 0
ab
c
=
0
Quadruples (a,b,x,y) such that n= ax+by, and a>b
Given
n
n
n
a positive integer, define
T
n
T_n
T
n
the number of quadruples of positive integers
(
a
,
b
,
x
,
y
)
(a,b,x,y)
(
a
,
b
,
x
,
y
)
such that
a
>
b
a>b
a
>
b
and
n
=
a
x
+
b
y
n= ax+by
n
=
a
x
+
b
y
. Prove that
T
2023
T_{2023}
T
2023
is odd.
1
1
Hide problems
Is 2023 in the sequence?
Define
(
a
n
)
(a_n)
(
a
n
)
a sequence, where
a
1
=
12
,
a
2
=
24
a_1= 12, a_2= 24
a
1
=
12
,
a
2
=
24
and for
n
≥
3
n\geq 3
n
≥
3
, we have:
a
n
=
a
n
−
2
+
14
a_n= a_{n-2}+14
a
n
=
a
n
−
2
+
14
a) Is
2023
2023
2023
in the sequence? b) Show that there are no perfect squares in the sequence.