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USA - Middle School Tournaments
Montgomery Blair
MBMT Geometry Rounds
2022
2022
Part of
MBMT Geometry Rounds
Problems
(1)
2022 MBMT Geometry Round - Montgomery Blair Math Tournament
Source:
10/14/2023
[hide=D stands for Dedekind, Z stands for Zermelo]they had two problem sets under those two names D1. A Giant Hopper is
200
200
200
meters away from you. It can hop
50
50
50
meters. How many hops would it take for it to reach you? D2. A rope of length
6
6
6
is used to form the edges of an equilateral triangle (a triangle with equal side lengths). What is the length of one of these edges? D3 / Z1. Point
E
E
E
is on side
A
B
AB
A
B
of rectangle
A
B
C
D
ABCD
A
BC
D
. Find the area of triangle
E
C
D
ECD
EC
D
divided by the area of rectangle
A
B
C
D
ABCD
A
BC
D
. D4 / Z2. Garb and Grunt have two rectangular pastures of area
30
30
30
. Garb notices that his has a side length of
3
3
3
, while Grunt’s has a side length of
5
5
5
. What’s the positive difference between the perimeters of their pastures? D5. Let points
A
A
A
and
B
B
B
be on a circle with radius
6
6
6
and center
O
O
O
. If
∠
A
O
B
=
9
0
o
\angle AOB = 90^o
∠
A
OB
=
9
0
o
, find the area of triangle
A
O
B
AOB
A
OB
. D6 / Z3. A scalene triangle (the
3
3
3
side lengths are all different) has integer angle measures (in degrees). What is the largest possible difference between two angles in the triangle? D7. Square
A
B
C
D
ABCD
A
BC
D
has side length
6
6
6
. If triangle
A
B
E
ABE
A
BE
has area
9
9
9
, find the sum of all possible values of the distance from
E
E
E
to line
C
D
CD
C
D
. D8 / Z4. Let point
E
E
E
be on side
A
B
‾
\overline{AB}
A
B
of square
A
B
C
D
ABCD
A
BC
D
with side length
2
2
2
. Given
D
E
=
B
C
+
B
E
DE = BC+BE
D
E
=
BC
+
BE
, find
B
E
BE
BE
. Z5. The two diagonals of rectangle
A
B
C
D
ABCD
A
BC
D
meet at point
E
E
E
. If
∠
A
E
B
=
2
∠
B
E
C
\angle AEB = 2\angle BEC
∠
A
EB
=
2∠
BEC
, and
B
C
=
1
BC = 1
BC
=
1
, find the area of rectangle
A
B
C
D
ABCD
A
BC
D
. Z6. In
△
A
B
C
\vartriangle ABC
△
A
BC
, let
D
D
D
be the foot of the altitude from
A
A
A
to
B
C
BC
BC
. Additionally, let
X
X
X
be the intersection of the angle bisector of
∠
A
C
B
\angle ACB
∠
A
CB
and
A
D
AD
A
D
. If
B
D
=
A
C
=
2
A
X
=
6
BD = AC = 2AX = 6
B
D
=
A
C
=
2
A
X
=
6
, find the area of
A
B
C
ABC
A
BC
. Z7. Let
△
A
B
C
\vartriangle ABC
△
A
BC
have
∠
A
B
C
=
4
0
o
\angle ABC = 40^o
∠
A
BC
=
4
0
o
. Let
D
D
D
and
E
E
E
be on
A
B
‾
\overline{AB}
A
B
and
A
C
‾
\overline{AC}
A
C
respectively such that DE is parallel to
B
C
‾
\overline{BC}
BC
, and the circle passing through points
D
D
D
,
E
E
E
, and
C
C
C
is tangent to
A
B
‾
\overline{AB}
A
B
. If the center of the circle is
O
O
O
, find
∠
D
O
E
\angle DOE
∠
D
OE
. Z8. Consider
△
A
B
C
\vartriangle ABC
△
A
BC
with
A
B
=
3
AB = 3
A
B
=
3
,
B
C
=
4
BC = 4
BC
=
4
, and
A
C
=
5
AC = 5
A
C
=
5
. Let
D
D
D
be a point of
A
C
AC
A
C
other than
A
A
A
for which
B
D
=
3
BD = 3
B
D
=
3
, and
E
E
E
be a point on
B
C
BC
BC
such that
∠
B
D
E
=
9
0
o
\angle BDE = 90^o
∠
B
D
E
=
9
0
o
. Find
E
C
EC
EC
. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
MBMT
geometry