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Montgomery Blair
MBMT Geometry Rounds
2023
2023
Part of
MBMT Geometry Rounds
Problems
(1)
2023 MBMT Geometry Round - Montgomery Blair Math Tournament
Source:
10/14/2023
[hide=B stands for Bernoulli, G stands for Germain]they had two problem sets under those two names B1. If the values of two angles in a triangle are
60
60
60
and
75
75
75
degrees respectively, what is the measure of the third angle? B2. Square
A
B
C
D
ABCD
A
BC
D
has side length
1
1
1
. What is the area of triangle
A
B
C
ABC
A
BC
? B3 / G1. An equilateral triangle and a square have the same perimeter. If the side length of the equilateral triangle is
8
8
8
, what is the square’s side length? B4 / G2. What is the maximum possible number of sides and diagonals of equal length in a quadrilateral? B5. A square of side length
4
4
4
is put within a circle such that all
4
4
4
corners lie on the circle. What is the diameter of the circle? B6 / G3. Patrick is rafting directly across a river
20
20
20
meters across at a speed of
5
5
5
m/s. The river flows in a direction perpendicular to Patrick’s direction at a rate of
12
12
12
m/s. When Patrick reaches the shore on the other end of the river, what is the total distance he has traveled? B7 / G4. Quadrilateral
A
B
C
D
ABCD
A
BC
D
has side lengths
A
B
=
7
AB = 7
A
B
=
7
,
B
C
=
15
BC = 15
BC
=
15
,
C
D
=
20
CD = 20
C
D
=
20
, and
D
A
=
24
DA = 24
D
A
=
24
. It has a diagonal length of
B
D
=
25
BD = 25
B
D
=
25
. Find the measure, in degrees, of the sum of angles
A
B
C
ABC
A
BC
and
A
D
C
ADC
A
D
C
. B8 / G5. What is the largest
P
P
P
such that any rectangle inscribed in an equilateral triangle of side length
1
1
1
has a perimeter of at least
P
P
P
? G6. A circle is inscribed in an equilateral triangle with side length
s
s
s
. Points
A
A
A
,
B
B
B
,
C
C
C
,
D
D
D
,
E
E
E
,
F
F
F
lie on the triangle such that line segments
A
B
AB
A
B
,
C
D
CD
C
D
, and
E
F
EF
EF
are parallel to a side of the triangle, and tangent to the circle. If the area of hexagon
A
B
C
D
E
F
=
9
3
2
ABCDEF = \frac{9\sqrt3}{2}
A
BC
D
EF
=
2
9
3
, find
s
s
s
. G7. Let
△
A
B
C
\vartriangle ABC
△
A
BC
be such that
∠
A
=
10
5
o
\angle A = 105^o
∠
A
=
10
5
o
,
∠
B
=
4
5
o
\angle B = 45^o
∠
B
=
4
5
o
,
∠
C
=
3
0
o
\angle C = 30^o
∠
C
=
3
0
o
. Let
M
M
M
be the midpoint of
A
C
AC
A
C
. What is
∠
M
B
C
\angle MBC
∠
MBC
? G8. Points
A
A
A
,
B
B
B
, and
C
C
C
lie on a circle centered at
O
O
O
with radius
10
10
10
. Let the circumcenter of
△
A
O
C
\vartriangle AOC
△
A
OC
be
P
P
P
. If
A
B
=
16
AB = 16
A
B
=
16
, find the minimum value of
P
B
PB
PB
. 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