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Problems
Contests
National and Regional Contests
Brazil Contests
Girls in Mathematics Tournament
2019 Girls in Mathematics Tournament
2019 Girls in Mathematics Tournament
Part of
Girls in Mathematics Tournament
Subcontests
(5)
5
1
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XZ passes through the midpoint of BK, isosceles, KX = CX, angle bisector
Let
A
B
C
ABC
A
BC
be an isosceles triangle with
A
B
=
A
C
AB = AC
A
B
=
A
C
. Let
X
X
X
and
K
K
K
points over
A
C
AC
A
C
and
A
B
AB
A
B
, respectively, such that
K
X
=
C
X
KX = CX
K
X
=
CX
. Bisector of
∠
A
K
X
\angle AKX
∠
A
K
X
intersects line
B
C
BC
BC
at
Z
Z
Z
. Show that
X
Z
XZ
XZ
passes through the midpoint of
B
K
BK
B
K
.
4
1
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cute numbers less than 2019, n is cute when m! ends in n zeros
A positive integer
n
n
n
is called cute when there is a positive integer
m
m
m
such that
m
!
m!
m
!
ends in exactly
n
n
n
zeros. a) Determine if
2019
2019
2019
is cute. b) How many positive integers less than
2019
2019
2019
are cute?
3
1
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nice number with digits 1 or 2 only
We say that a positive integer N is nice if it satisfies the following conditions:
∙
\bullet
∙
All of its digits are
1
1
1
or
2
2
2
∙
\bullet
∙
All numbers formed by
3
3
3
consecutive digits of
N
N
N
are distinct. For example,
121222
121222
121222
is nice, because the
4
4
4
numbers formed by
3
3
3
consecutive digits of
121222
121222
121222
, which are
121
,
212
,
122
121,212,122
121
,
212
,
122
and
222
222
222
, are distinct. However,
12121
12121
12121
is not nice. What is the largest quantity possible number of numbers that a nice number can have? What is the greatest nice number there is?
2
1
Hide problems
incenter of ABC is circumcenter of APQ, related to a right triangle
Let
A
B
C
ABC
A
BC
be a right triangle with hypotenuse
B
C
BC
BC
and center
I
I
I
. Let bisectors of the angles
∠
B
\angle B
∠
B
and
∠
C
\angle C
∠
C
intersect the sides
A
C
AC
A
C
and
A
B
AB
A
B
in
D
D
D
and
E
E
E
, respectively. Let
P
P
P
and
Q
Q
Q
be the feet of the perpendiculars of the points
D
D
D
and
E
E
E
on the side
B
C
BC
BC
. Prove that
I
I
I
is the circumcenter of
A
P
Q
APQ
A
PQ
.
1
1
Hide problems
ways to write 2019 as difference of two perfect squares
During the factoring class, Esmeralda observed that
1
1
1
,
3
3
3
and
5
5
5
can be written as the difference of two perfect squares, as can be seen:
1
=
1
2
−
0
2
1 = 1^2 - 0^2
1
=
1
2
−
0
2
3
=
2
2
−
1
2
3 = 2^2 - 1^2
3
=
2
2
−
1
2
5
=
3
2
−
2
2
5 = 3^2 - 2^2
5
=
3
2
−
2
2
a) Show that all numbers written in the form
2
∗
m
+
1
2 * m + 1
2
∗
m
+
1
can be written as a difference of two perfect squares. b) Show how to calculate the value of the expression
E
=
1
+
3
+
5
+
.
.
.
+
(
2
m
+
1
)
E = 1 + 3 + 5 + ... + (2m + 1)
E
=
1
+
3
+
5
+
...
+
(
2
m
+
1
)
. c) Esmeralda, happy with what she discovered, decided to look for other ways to write
2019
2019
2019
as the difference of two perfect squares of positive integers. Determine how many ways it can do what you want.