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USA - Middle School Tournaments
Montgomery Blair
MBMT Geometry Rounds
2019
2019
Part of
MBMT Geometry Rounds
Problems
(1)
2019 MBMT Geometry Round - Montgomery Blair Math Tournament
Source:
10/14/2023
[hide=D stands for Descartes, L stands for Leibniz]they had two problem sets under those two names D1. Triangle
A
B
C
ABC
A
BC
has
A
B
=
3
AB = 3
A
B
=
3
,
B
C
=
4
BC = 4
BC
=
4
, and
∠
B
=
9
0
o
\angle B = 90^o
∠
B
=
9
0
o
. Find the area of triangle
A
B
C
ABC
A
BC
. D2 / L1. Let
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
be a regular hexagon. Given that
A
D
=
5
AD = 5
A
D
=
5
, find
A
B
AB
A
B
. D3. Caroline glues two pentagonal pyramids to the top and bottom of a pentagonal prism so that the pentagonal faces coincide. How many edges does Caroline’s figure have? D4 / L3. The hour hand of a clock is
6
6
6
inches long, and the minute hand is
10
10
10
inches long. Find the area of the region swept out by the hands from
8
:
45
8:45
8
:
45
AM to
9
:
15
9:15
9
:
15
AM of a single day, in square inches. D5 / L2. Circles
A
A
A
,
B
B
B
, and
C
C
C
are all externally tangent, with radii
1
1
1
,
10
10
10
, and
100
100
100
, respectively. What is the radius of the smallest circle entirely containing all three circles? D6. Four parallel lines are drawn such that they are equally spaced and pass through the four vertices of a unit square. Find the distance between any two consecutive lines. D7 / L4. In rectangle
A
B
C
D
ABCD
A
BC
D
,
A
B
=
2
AB = 2
A
B
=
2
and
A
D
>
A
B
AD > AB
A
D
>
A
B
. Two quarter circles are drawn inside of
A
B
C
D
ABCD
A
BC
D
with centers at
A
A
A
and
C
C
C
that pass through
B
B
B
and
D
D
D
, respectively. If these two quarter circles are tangent, find the area inside of
A
B
C
D
ABCD
A
BC
D
that is outside both of the quarter circles. D8 / L6. Triangle
A
B
C
ABC
A
BC
is equilateral. A circle passes through
A
A
A
and is tangent to side
B
C
BC
BC
. It intersects sides AB and
A
C
AC
A
C
again at
E
E
E
and
F
F
F
, respectively. If
A
E
=
10
AE = 10
A
E
=
10
and
A
F
=
11
AF = 11
A
F
=
11
, find
A
B
AB
A
B
. L5. Find the area of a triangle with side lengths
2
\sqrt{2}
2
,
58
\sqrt{58}
58
, and
2
17
2\sqrt{17}
2
17
. L7. Triangle
A
B
C
ABC
A
BC
has area
80
80
80
. Point
D
D
D
is in the interior of
△
A
B
C
\vartriangle ABC
△
A
BC
such that
A
D
=
6
AD =6
A
D
=
6
,
B
D
=
4
BD = 4
B
D
=
4
,
C
D
=
16
CD = 16
C
D
=
16
, and the area of
△
A
D
C
=
48
\vartriangle ADC = 48
△
A
D
C
=
48
. Determine the area of
△
A
D
B
\vartriangle ADB
△
A
D
B
. L8. Given two points
A
A
A
and
B
B
B
in the plane with
A
B
=
1
AB = 1
A
B
=
1
, define
f
(
C
)
f(C)
f
(
C
)
to be the circumcenter of triangle
A
B
C
ABC
A
BC
, if it exists. Find the number of points
X
X
X
so that
f
2019
(
X
)
=
X
f^{2019}(X) = X
f
2019
(
X
)
=
X
. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
MBMT
geometry