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Montgomery Blair
MBMT Geometry Rounds
2018
2018
Part of
MBMT Geometry Rounds
Problems
(1)
2018 MBMT Geometry Round - Montgomery Blair Math Tournament
Source:
10/14/2023
[hide=C stands for Cantor, G stands for Gauss]they had two problem sets under those two names C1. A circle has circumference
6
π
6\pi
6
π
. Find the area of this circle. C2 / G2. Points
A
A
A
,
B
B
B
, and
C
C
C
are on a line such that
A
B
=
6
AB = 6
A
B
=
6
and
B
C
=
11
BC = 11
BC
=
11
. Find all possible values of
A
C
AC
A
C
. C3. A trapezoid has area
84
84
84
and one base of length
5
5
5
. If the height is
12
12
12
, what is the length of the other base? C4 / G1.
27
27
27
cubes of side length 1 are arranged to form a
3
×
3
×
3
3 \times 3 \times 3
3
×
3
×
3
cube. If the corner
1
×
1
×
1
1 \times 1 \times 1
1
×
1
×
1
cubes are removed, what fraction of the volume of the big cube is left? C5. There is a
50
50
50
-foot tall wall and a
300
300
300
-foot tall guard tower
50
50
50
feet from the wall. What is the minimum
a
a
a
such that a flat “
X
X
X
” drawn on the ground
a
a
a
feet from the side of the wall opposite the guard tower is visible from the top of the guard tower? C6. Steven’s pizzeria makes pizzas in the shape of equilateral triangles. If a pizza with side length 8 inches will feed 2 people, how many people will a pizza of side length of 16 inches feed? C7 / G3. Consider rectangle
A
B
C
D
ABCD
A
BC
D
, with
1
=
A
B
<
B
C
1 = AB < BC
1
=
A
B
<
BC
. The angle bisector of
∠
D
A
B
\angle DAB
∠
D
A
B
intersects
B
C
‾
\overline{BC}
BC
at
E
E
E
and
D
C
‾
\overline{DC}
D
C
at
F
F
F
. If
F
E
=
F
D
FE = FD
FE
=
F
D
, find
B
C
BC
BC
. C8 / G6.
△
A
B
C
\vartriangle ABC
△
A
BC
. is a right triangle with
∠
A
=
9
0
o
\angle A = 90^o
∠
A
=
9
0
o
. Square
A
D
E
F
ADEF
A
D
EF
is drawn, with
D
D
D
on
A
B
‾
\overline{AB}
A
B
,
F
F
F
on
A
C
‾
\overline{AC}
A
C
, and
E
E
E
inside
△
A
B
C
\vartriangle ABC
△
A
BC
. Point
G
G
G
is chosen on
B
C
‾
\overline{BC}
BC
such that
E
G
EG
EG
is perpendicular to
B
C
BC
BC
. Additionally,
D
E
=
E
G
DE = EG
D
E
=
EG
. Given that
∠
C
=
2
0
o
\angle C = 20^o
∠
C
=
2
0
o
, find the measure of
∠
B
E
G
\angle BEG
∠
BEG
. G4. Consider a lamp in the shape of a hollow cylinder with the circular faces removed with height
48
48
48
cm and radius
7
7
7
cm. A point source of light is situated at the center of the lamp. The lamp is held so that the bottom of the lamp is at a height
48
48
48
cm above an infinite flat sheet of paper. What is the area of the illuminated region on the flat sheet of paper, in
c
m
2
cm^2
c
m
2
? https://cdn.artofproblemsolving.com/attachments/c/6/6e5497a67ae5ff5a7bff7007834a4271ce3ca7.pngG5. There exist two triangles
A
B
C
ABC
A
BC
such that
A
B
=
13
AB = 13
A
B
=
13
,
B
C
=
12
2
BC = 12\sqrt2
BC
=
12
2
, and
∠
C
=
4
5
o
\angle C = 45^o
∠
C
=
4
5
o
. Find the positive difference between their areas. G7. Let
A
B
C
ABC
A
BC
be an equilateral triangle with side length
2
2
2
. Let the circle with diameter
A
B
AB
A
B
be
Γ
\Gamma
Γ
. Consider the two tangents from
C
C
C
to
Γ
\Gamma
Γ
, and let the tangency point closer to
A
A
A
be
D
D
D
. Find the area of
∠
C
A
D
\angle CAD
∠
C
A
D
. G8. Let
A
B
C
ABC
A
BC
be a triangle with
∠
A
=
6
0
o
\angle A = 60^o
∠
A
=
6
0
o
,
A
B
=
37
AB = 37
A
B
=
37
,
A
C
=
41
AC = 41
A
C
=
41
. Let
H
H
H
and
O
O
O
be the orthocenter and circumcenter of
A
B
C
ABC
A
BC
, respectively. Find
O
H
OH
O
H
. The orthocenter of a triangle is the intersection point of the three altitudes. The circumcenter of a triangle is the intersection point of the three perpendicular bisectors of the sides. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
MBMT
geometry