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Problems
Contests
National and Regional Contests
Brazil Contests
Girls in Mathematics Tournament
2022 Girls in Mathematics Tournament
2022 Girls in Mathematics Tournament
Part of
Girls in Mathematics Tournament
Subcontests
(4)
4
1
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Coprime with 1/2 terms
The sequence of positive integers
a
1
,
a
2
,
a
3
,
…
a_1,a_2,a_3,\dots
a
1
,
a
2
,
a
3
,
…
is brazilian if
a
1
=
1
a_1=1
a
1
=
1
and
a
n
a_n
a
n
is the least integer greater than
a
n
−
1
a_{n-1}
a
n
−
1
and
a
n
a_n
a
n
is coprime with at least half elements of the set
{
a
1
,
a
2
,
…
,
a
n
−
1
}
\{a_1,a_2,\dots, a_{n-1}\}
{
a
1
,
a
2
,
…
,
a
n
−
1
}
. Is there any odd number which does not belong to the brazilian sequence?
3
1
Hide problems
Max + Lewis = Fight?
There are
n
n
n
cards. Max and Lewis play, alternately, the following game Max starts the game, he removes exactly
1
1
1
card, in each round the current player can remove any quantity of cards, from
1
1
1
card to
t
+
1
t+1
t
+
1
cards, which
t
t
t
is the number of removed cards by the previous player, and the winner is the player who remove the last card. Determine all the possible values of
n
n
n
such that Max has the winning strategy.
2
1
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A symmetric pair of functions?
Determine all the integers solutions
(
x
,
y
)
(x,y)
(
x
,
y
)
of the following equation
x
2
−
4
2
x
−
1
+
y
2
−
4
2
y
−
1
=
x
+
y
\frac{x^2-4}{2x-1}+\frac{y^2-4}{2y-1}=x+y
2
x
−
1
x
2
−
4
+
2
y
−
1
y
2
−
4
=
x
+
y
1
1
Hide problems
Many Equal Sides
Let
A
B
C
ABC
A
BC
be a triangle with
B
A
=
B
C
BA=BC
B
A
=
BC
and
∠
A
B
C
=
9
0
∘
\angle ABC=90^{\circ}
∠
A
BC
=
9
0
∘
. Let
D
D
D
and
E
E
E
be the midpoints of
C
A
CA
C
A
and
B
A
BA
B
A
respectively. The point
F
F
F
is inside of
△
A
B
C
\triangle ABC
△
A
BC
such that
△
D
E
F
\triangle DEF
△
D
EF
is equilateral. Let
X
=
B
F
∩
A
C
X=BF\cap AC
X
=
BF
∩
A
C
and
Y
=
A
F
∩
D
B
Y=AF\cap DB
Y
=
A
F
∩
D
B
. Prove that
D
X
=
Y
D
DX=YD
D
X
=
Y
D
.