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BMT Problems
2021 BMT
2021 BMT
Part of
BMT Problems
Subcontests
(35)
19-21
1
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BMT 2021 Guts Round Set 7 p19-21
Guts Round / Set 7p19. Let
a
a
a
be the answer to Problem 19,
b
b
b
be the answer to Problem 20, and
c
c
c
be the answer to Problem 21.Compute the real value of
a
a
a
such that
a
(
101
b
+
1
)
−
1
=
b
(
c
−
1
)
+
10
(
a
−
c
)
b
.
\sqrt{a(101b + 1)} - 1 = \sqrt{b(c - 1)}+ 10\sqrt{(a - c)b}.
a
(
101
b
+
1
)
−
1
=
b
(
c
−
1
)
+
10
(
a
−
c
)
b
.
p20. Let
a
a
a
be the answer to Problem 19,
b
b
b
be the answer to Problem 20, and
c
c
c
be the answer to Problem 21.For some triangle
△
A
B
C
\vartriangle ABC
△
A
BC
, let
ω
\omega
ω
and
ω
A
\omega_A
ω
A
be the incircle and
A
A
A
-excircle with centers
I
I
I
and
I
A
I_A
I
A
, respectively. Suppose
A
C
AC
A
C
is tangent to
ω
\omega
ω
and
ω
A
\omega_A
ω
A
at
E
E
E
and
E
′
E'
E
′
, respectively, and
A
B
AB
A
B
is tangent to
ω
\omega
ω
and
ω
A
\omega_A
ω
A
at
F
F
F
and
F
′
F'
F
′
respectively. Furthermore, let
P
P
P
and
Q
Q
Q
be the intersections of
B
I
BI
B
I
with
E
F
EF
EF
and
C
I
CI
C
I
with
E
F
EF
EF
, respectively, and let
P
′
P'
P
′
and
Q
′
Q'
Q
′
be the intersections of
B
I
A
BI_A
B
I
A
with
E
′
F
′
E'F'
E
′
F
′
and
C
I
A
CI_A
C
I
A
with
E
′
F
′
E'F'
E
′
F
′
, respectively. Given that the circumradius of
△
A
B
C
\vartriangle ABC
△
A
BC
is a, compute the maximum integer value of
B
C
BC
BC
such that the area
[
P
Q
P
′
Q
′
]
[P QP'Q']
[
PQ
P
′
Q
′
]
is less than or equal to
1
1
1
. p21. Let
a
a
a
be the answer to Problem 19,
b
b
b
be the answer to Problem 20, and
c
c
c
be the answer to Problem 21.Let
c
c
c
be a positive integer such that
g
c
d
(
b
,
c
)
=
1
gcd(b, c) = 1
g
c
d
(
b
,
c
)
=
1
. From each ordered pair
(
x
,
y
)
(x, y)
(
x
,
y
)
such that
x
x
x
and
y
y
y
are both integers, we draw two lines through that point in the
x
−
y
x-y
x
−
y
plane, one with slope
b
c
\frac{b}{c}
c
b
and one with slope
−
c
b
-\frac{c}{b}
−
b
c
. Given that the number of intersections of these lines in
[
0
,
1
)
2
[0, 1)^2
[
0
,
1
)
2
is a square number, what is the smallest possible value of
c
c
c
? Note that
[
0
,
1
)
2
[0, 1)^2
[
0
,
1
)
2
refers to all points
(
x
,
y
)
(x, y)
(
x
,
y
)
such that
0
≤
x
<
1
0 \le x < 1
0
≤
x
<
1
and
0
≤
y
<
1
0 \le y < 1
0
≤
y
<
1
.
T5
1
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BMT 2021 General Tiebreaker p5
Let
r
,
s
,
t
,
u
r, s, t, u
r
,
s
,
t
,
u
be the distinct roots of the polynomial
x
4
+
2
x
3
+
3
x
2
+
3
x
+
5
x^4 + 2x^3 + 3x^2 + 3x + 5
x
4
+
2
x
3
+
3
x
2
+
3
x
+
5
. For
n
≥
1
n \ge 1
n
≥
1
, define
s
n
=
r
n
+
s
n
+
t
n
+
u
n
s_n = r^n + s^n + t^n + u^n
s
n
=
r
n
+
s
n
+
t
n
+
u
n
and
t
n
=
s
1
+
s
2
+
.
.
.
+
s
n
t_n = s_1 + s_2 + ...+ s_n
t
n
=
s
1
+
s
2
+
...
+
s
n
. Compute
t
4
+
2
t
3
+
3
t
2
+
3
t
1
+
5
t_4 + 2t_3 + 3t_2 + 3t_1 + 5
t
4
+
2
t
3
+
3
t
2
+
3
t
1
+
5
.
T4
1
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BMT 2021 General Tiebreaker p4
Let
z
1
z_1
z
1
,
z
2
z_2
z
2
, and
z
3
z_3
z
3
be the complex roots of the equation
(
2
z
−
3
z
‾
)
3
=
54
i
+
54
(2z -3\overline{z})^3 = 54i+54
(
2
z
−
3
z
)
3
=
54
i
+
54
. Compute the area of the triangle formed by
z
1
z_1
z
1
,
z
2
z_2
z
2
, and
z
3
z_3
z
3
when plotted in the complex plane.
T3
2
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BMT 2021 Discrete Tiebreaker p3
Let
N
N
N
be the number of tuples
(
a
1
,
a
2
,
.
.
.
,
a
150
)
(a_1, a_2,..., a_{150})
(
a
1
,
a
2
,
...
,
a
150
)
satisfying:
∙
\bullet
∙
a
i
∈
{
2
,
3
,
5
,
7
,
11
}
a_i \in \{2, 3, 5, 7, 11\}
a
i
∈
{
2
,
3
,
5
,
7
,
11
}
for all
1
≤
i
≤
99
1 \le i \le 99
1
≤
i
≤
99
.
∙
\bullet
∙
a
i
∈
{
2
,
4
,
6
,
8
}
a_i \in \{2, 4, 6, 8\}
a
i
∈
{
2
,
4
,
6
,
8
}
for all
100
≤
i
≤
150
100 \le i \le 150
100
≤
i
≤
150
.
∙
\bullet
∙
∑
i
=
1
150
a
i
\sum^{150}_{i=1}a_i
∑
i
=
1
150
a
i
is divisible by
8
8
8
. Compute the last three digits of
N
N
N
.
BMT 2021 General Tiebreaker p3
Dexter and Raquel are playing a game with
N
N
N
stones. Dexter goes first and takes one stone from the pile. After that, the players alternate turns and can take anywhere from
1
1
1
to
x
+
1
x + 1
x
+
1
stones from the pile, where
x
x
x
is the number of stones the other player took on the turn immediately prior. The winner is the one to take the last stone from the pile. Assuming Dexter and Raquel play optimally, compute the number of positive integers
N
≤
2021
N \le 2021
N
≤
2021
where Dexter wins this game.
T1
2
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BMT 2021 Discrete Tiebreaker p1
How many integers
n
n
n
from
1
1
1
to
2020
2020
2020
, inclusive, are there such that
2020
2020
2020
divides
n
2
+
1
n^2 + 1
n
2
+
1
?
BMT 2021 General Tiebreaker p1
The arithmetic mean of
2
,
6
,
8
2, 6, 8
2
,
6
,
8
, and
x
x
x
is
7
7
7
. The arithmetic mean of
2
,
6
,
8
,
x
2, 6, 8, x
2
,
6
,
8
,
x
, and
y
y
y
is
9
9
9
. What is the value of
y
−
x
y - x
y
−
x
?
T2
2
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BMT 2021 Discrete Tiebreaker p2
A gradian is a unit of measurement of angles much like degrees, except that there are
100
100
100
gradians in a right angle. Suppose that the number of gradians in an interior angle of a regular polygon with
m
m
m
sides equals the number of degrees in an interior angle of a regular polygon with
n
n
n
sides. Compute the number of possible distinct ordered pairs
(
m
,
n
)
(m, n)
(
m
,
n
)
.
largest circle that fits entirely within a unit cube - BMT 2021 General Tie 2
Compute the radius of the largest circle that fits entirely within a unit cube.
21
1
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BMT 2021 General p21
There exist integers
a
a
a
and
b
b
b
such that
(
1
+
2
)
12
=
a
+
b
2
(1 +\sqrt2)^{12}= a + b\sqrt2
(
1
+
2
)
12
=
a
+
b
2
. Compute the remainder when
a
b
ab
ab
is divided by
13
13
13
.
20
1
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BMT 2021 General p20
For some positive integer
n
n
n
,
(
1
+
i
)
+
(
1
+
i
)
2
+
(
1
+
i
)
3
+
.
.
.
+
(
1
+
i
)
n
=
(
n
2
−
1
)
(
1
−
i
)
(1 + i) + (1 + i)^2 + (1 + i)^3 + ... + (1 + i)^n = (n^2 - 1)(1 - i)
(
1
+
i
)
+
(
1
+
i
)
2
+
(
1
+
i
)
3
+
...
+
(
1
+
i
)
n
=
(
n
2
−
1
)
(
1
−
i
)
, where
i
=
−
1
i = \sqrt{-1}
i
=
−
1
. Compute the value of
n
n
n
.
Tie 2
2
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BMT 2021 Geometry Tiebreaker #2
Let
△
A
0
B
0
C
0
\vartriangle A_0B_0C_0
△
A
0
B
0
C
0
be an equilateral triangle with area
1
1
1
, and let
A
1
A_1
A
1
,
B
1
B_1
B
1
,
C
1
C_1
C
1
be the midpoints of
A
0
B
0
‾
\overline{A_0B_0}
A
0
B
0
,
B
0
C
0
‾
\overline{B_0C_0}
B
0
C
0
, and
C
0
A
0
‾
\overline{C_0A_0}
C
0
A
0
, respectively. Furthermore, set
A
2
A_2
A
2
,
B
2
B_2
B
2
,
C
2
C_2
C
2
as the midpoints of segments
A
0
A
1
‾
\overline{A_0A_1}
A
0
A
1
,
B
0
B
1
‾
\overline{B_0B_1}
B
0
B
1
, and
C
0
C
1
‾
\overline{C_0C_1}
C
0
C
1
respectively. For
n
≥
1
n \ge 1
n
≥
1
,
A
2
n
+
1
A_{2n+1}
A
2
n
+
1
is recursively defined as the midpoint of
A
2
n
A
2
n
−
1
A_{2n}A_{2n-1}
A
2
n
A
2
n
−
1
, and
A
2
n
+
2
A_{2n+2}
A
2
n
+
2
is recursively defined as the midpoint of
A
2
n
+
1
A
2
n
−
1
‾
\overline{A_{2n+1}A_{2n-1}}
A
2
n
+
1
A
2
n
−
1
. Recursively define
B
n
B_n
B
n
and
C
n
C_n
C
n
the same way. Compute the value of
lim
n
→
∞
[
A
n
B
n
C
n
]
\lim_{n \to \infty }[A_nB_nC_n]
lim
n
→
∞
[
A
n
B
n
C
n
]
, where
[
A
n
B
n
C
n
]
[A_nB_nC_n]
[
A
n
B
n
C
n
]
denotes the area of triangle
△
A
n
B
n
C
n
\vartriangle A_nB_nC_n
△
A
n
B
n
C
n
.
2021 BMT Algebra Tiebreaker #2
Real numbers
x
x
x
and
y
y
y
satisfy the equations
x
2
−
12
y
=
1
7
2
x^2 - 12y = 17^2
x
2
−
12
y
=
1
7
2
and
38
x
−
y
2
=
2
⋅
7
3
38x - y^2 = 2 \cdot 7^3
38
x
−
y
2
=
2
⋅
7
3
. Compute
x
+
y
x + y
x
+
y
.
Tie 3
2
Hide problems
BMT 2021 Geometry Tiebreaker #3
Right triangle
△
A
B
C
\vartriangle ABC
△
A
BC
with its right angle at
B
B
B
has angle bisector
A
D
‾
\overline{AD}
A
D
with
D
D
D
on
B
C
‾
\overline{BC}
BC
, as well as altitude
B
E
‾
\overline{BE}
BE
with
E
E
E
on
A
C
‾
\overline{AC}
A
C
. If
D
E
‾
⊥
B
C
‾
\overline{DE} \perp \overline{BC}
D
E
⊥
BC
and
A
B
=
1
AB = 1
A
B
=
1
, compute
A
C
AC
A
C
.
2021 BMT Algebra Tiebreaker #3
For integers
a
a
a
and
b
b
b
,
a
+
b
a + b
a
+
b
is a root of
x
2
+
a
x
+
b
=
0
x^2 + ax + b = 0
x
2
+
a
x
+
b
=
0
. Compute the smallest possible value of
a
b
ab
ab
.
Tie 1
2
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BMT 2021 Geometry Tiebreaker #1
Regular hexagon
N
O
S
A
M
E
NOSAME
NOS
A
ME
with side length
1
1
1
and square
U
D
O
N
UDON
U
D
ON
are drawn in the plane such that
U
D
O
N
UDON
U
D
ON
lies outside of
N
O
S
A
M
E
NOSAME
NOS
A
ME
. Compute
[
S
A
N
D
]
+
[
S
E
N
D
]
[SAND] + [SEND]
[
S
A
N
D
]
+
[
SEN
D
]
, the sum of the areas of quadrilaterals
S
A
N
D
SAND
S
A
N
D
and
S
E
N
D
SEND
SEN
D
.
2021 BMT Algebra Tiebreaker #1
Let the sequence
{
a
n
}
\{a_n\}
{
a
n
}
for
n
≥
0
n \ge 0
n
≥
0
be defined as
a
0
=
c
a_0 = c
a
0
=
c
, and for
n
≥
0
n \ge 0
n
≥
0
,
a
n
=
2
a
n
−
1
4
a
n
−
1
2
−
1
.
a_n =\frac{2a_{n-1}}{4a^2_{n-1} -1}.
a
n
=
4
a
n
−
1
2
−
1
2
a
n
−
1
.
Compute the sum of all values of
c
c
c
such that
a
2020
a_{2020}
a
2020
exists but
a
2021
a_{2021}
a
2021
does not exist.
27
1
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BMT 2021 Guts Round p27
Let
S
=
1
,
2
,
2
2
,
2
3
,
.
.
.
,
2
2021
S = {1, 2, 2^2, 2^3, ... , 2^{2021}}
S
=
1
,
2
,
2
2
,
2
3
,
...
,
2
2021
. Compute the difference between the number of even digits and the number of odd digits across all numbers in
S
S
S
(written as integers in base
10
10
10
with no leading zeros). If E is the exact answer to this question and A is your answer, your score is given by
max
(
0
,
⌊
25
−
1
2
⋅
1
0
8
∣
E
−
A
∣
4
⌋
)
\max \, \left(0, \left\lfloor 25 - \frac{1}{2 \cdot 10^8}|E - A|^4\right\rfloor \right)
max
(
0
,
⌊
25
−
2
⋅
1
0
8
1
∣
E
−
A
∣
4
⌋
)
.
26
1
Hide problems
BMT 2021 Guts Round p26
Kailey starts with the number
0
0
0
, and she has a fair coin with sides labeled
1
1
1
and
2
2
2
. She repeatedly flips the coin, and adds the result to her number. She stops when her number is a positive perfect square. What is the expected value of Kailey’s number when she stops? If E is your estimate and A is the correct answer, you will receive
⌊
25
e
−
5
∣
E
−
A
∣
2
⌋
\left\lfloor 25e^{-\frac{5|E-A|}{2} }\right\rfloor
⌊
25
e
−
2
5∣
E
−
A
∣
⌋
points.
25
2
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BMT 2021 Guts Round p25
For any
p
,
q
∈
N
p, q \in N
p
,
q
∈
N
, we can express
p
q
\frac{p}{q}
q
p
as the base
10
10
10
decimal
x
1
x
2
.
.
.
x
ℓ
.
x
ℓ
+
1
.
.
.
x
a
y
1
y
2
.
.
.
y
b
‾
x_1x_2... x_{\ell}.x_{\ell+1}... x_a \overline{y_1y_2... y_b}
x
1
x
2
...
x
ℓ
.
x
ℓ
+
1
...
x
a
y
1
y
2
...
y
b
, with the digits
y
1
,
.
.
.
y
b
y_1, . . . y_b
y
1
,
...
y
b
repeating. In other words,
p
q
\frac{p}{q}
q
p
can be expressed with integer part
x
1
x
2
.
.
.
x
ℓ
x_1x_2... x_{\ell}
x
1
x
2
...
x
ℓ
and decimal part
0.
x
ℓ
+
1
.
.
.
x
a
y
1
y
2
.
.
.
y
b
‾
0.x_{\ell+1}... x_a \overline{y_1y_2... y_b}
0.
x
ℓ
+
1
...
x
a
y
1
y
2
...
y
b
. Given that
p
q
=
(
2021
)
2021
2021
!
\frac{p}{q}= \frac{(2021)^{2021}}{2021!}
q
p
=
2021
!
(
2021
)
2021
, estimate the minimum value of
a
a
a
. If
E
E
E
is the exact answer to this question and
A
A
A
is your answer, your score is given by
max
(
0
,
⌊
25
−
1
10
∣
E
−
A
∣
⌋
)
\max \, \left(0, \left\lfloor 25 - \frac{1}{10}|E - A|\right\rfloor \right)
max
(
0
,
⌊
25
−
10
1
∣
E
−
A
∣
⌋
)
.
BMT 2021 General p25
Let
△
B
M
T
\vartriangle BMT
△
BMT
be a triangle with
B
T
=
1
BT = 1
BT
=
1
and height
1
1
1
. Let
O
0
O_0
O
0
be the centroid of
△
B
M
T
\vartriangle BMT
△
BMT
, and let
B
O
0
‾
\overline{BO_0}
B
O
0
and
T
O
0
‾
\overline{TO_0}
T
O
0
intersect
M
T
‾
\overline{MT}
MT
and
B
M
‾
\overline{BM}
BM
at
B
1
B_1
B
1
and
T
1
T_1
T
1
, respectively. Similarly, let
O
1
O_1
O
1
be the centroid of
△
B
1
M
T
1
\vartriangle B_1MT_1
△
B
1
M
T
1
, and in the same way, denote the centroid of
△
B
n
M
T
n
\vartriangle B_nMT_n
△
B
n
M
T
n
by
O
n
O_n
O
n
, the intersection of
B
O
n
‾
\overline{BO_n}
B
O
n
with
M
T
‾
\overline{MT}
MT
by
B
n
+
1
B_{n+1}
B
n
+
1
, and the intersection of
T
O
n
‾
\overline{TO_n}
T
O
n
with
B
M
‾
\overline{BM}
BM
by
T
n
+
1
T_{n+1}
T
n
+
1
. Compute the area of quadrilateral
M
B
O
2021
T
MBO_{2021}T
MB
O
2021
T
.
24
2
Hide problems
BMT 2021 Guts Round p24
Suppose that
a
,
b
,
c
a, b, c
a
,
b
,
c
, and p are positive integers such that
p
p
p
is a prime number and
a
2
+
b
2
+
c
2
=
a
b
+
b
c
+
c
a
+
2021
p
a^2 + b^2 + c^2 = ab + bc + ca + 2021p
a
2
+
b
2
+
c
2
=
ab
+
b
c
+
c
a
+
2021
p
. Compute the least possible value of
max
(
a
,
b
,
c
)
\max \,(a, b, c)
max
(
a
,
b
,
c
)
.
BMT 2021 General p24
Given that
x
,
y
x, y
x
,
y
, and
z
z
z
are a combination of positive integers such that
x
y
z
=
2
(
x
+
y
+
z
)
xyz = 2(x + y + z)
x
yz
=
2
(
x
+
y
+
z
)
, compute the sum of all possible values of
x
+
y
+
z
x + y + z
x
+
y
+
z
.
23
2
Hide problems
BMT 2021 Guts Round p23
Alireza is currently standing at the point
(
0
,
0
)
(0, 0)
(
0
,
0
)
in the
x
−
y
x-y
x
−
y
plane. At any given time, Alireza can move from the point
(
x
,
y
)
(x, y)
(
x
,
y
)
to the point
(
x
+
1
,
y
)
(x + 1, y)
(
x
+
1
,
y
)
or the point
(
x
,
y
+
1
)
(x, y + 1)
(
x
,
y
+
1
)
. However, he cannot move to any point of the form
(
x
,
y
)
(x, y)
(
x
,
y
)
where
y
≡
2
x
(
m
o
d
5
)
y \equiv 2x\,\, (\mod \,\,5)
y
≡
2
x
(
mod
5
)
. Let
p
k
p_k
p
k
be the number of paths Alireza can take starting from the point
(
0
,
0
)
(0, 0)
(
0
,
0
)
to the point
(
k
+
1
,
2
k
+
1
)
(k + 1, 2k + 1)
(
k
+
1
,
2
k
+
1
)
. Evaluate the sum
∑
k
=
1
∞
p
k
5
k
.
\sum^{\infty}_{k=1} \frac{p_k}{5^k}.
k
=
1
∑
∞
5
k
p
k
.
.
BMT 2021 General p23
Shivani has a single square with vertices labeled
A
B
C
D
ABCD
A
BC
D
. She is able to perform the following transformations:
∙
\bullet
∙
She does nothing to the square.
∙
\bullet
∙
She rotates the square by
90
90
90
,
180
180
180
, or
270
270
270
degrees.
∙
\bullet
∙
She reflects the square over one of its four lines of symmetry. For the first three timesteps, Shivani only performs reflections or does nothing. Then for the next three timesteps, she only performs rotations or does nothing. She ends up back in the square’s original configuration. Compute the number of distinct ways she could have achieved this.
22
2
Hide problems
BMT 2021 Guts Round p22
In
△
A
B
C
\vartriangle ABC
△
A
BC
, let
D
D
D
and
E
E
E
be points on the angle bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
such that
∠
A
B
D
=
∠
A
C
E
=
9
0
o
\angle ABD = \angle ACE =90^o
∠
A
B
D
=
∠
A
CE
=
9
0
o
. Furthermore, let
F
F
F
be the intersection of
A
E
AE
A
E
and
B
C
BC
BC
, and let
O
O
O
be the circumcenter of
△
A
F
C
\vartriangle AF C
△
A
FC
. If
A
B
A
C
=
3
4
\frac{AB}{AC} =\frac{3}{4}
A
C
A
B
=
4
3
,
A
E
=
40
AE = 40
A
E
=
40
, and
B
D
BD
B
D
bisects
E
F
EF
EF
, compute the perpendicular distance from
A
A
A
to
O
F
OF
OF
.
BMT 2021 General p22
Austin is at the Lincoln Airport. He wants to take
5
5
5
successive flights whose destinations are randomly chosen among Indianapolis, Jackson, Kansas City, Lincoln, and Milwaukee. The origin and destination of each flight may not be the same city, but Austin must arrive back at Lincoln on the last of his flights. Compute the probability that the cities Austin arrives at are all distinct.
18
1
Hide problems
BMT 2021 Guts Round p18
The equation
x
−
3
8
3
−
3
8
3
=
x
3
+
3
8
\sqrt[3]{\sqrt[3]{x - \frac38} - \frac38} = x^3+ \frac38
3
3
x
−
8
3
−
8
3
=
x
3
+
8
3
has exactly two real positive solutions
r
r
r
and
s
s
s
. Compute
r
+
s
r + s
r
+
s
.
17
2
Hide problems
BMT 2021 Guts Round p17
Triangle
△
A
B
C
\vartriangle ABC
△
A
BC
has circumcenter
O
O
O
and orthocenter
H
H
H
. Let
D
D
D
be the foot of the altitude from
A
A
A
to
B
C
BC
BC
, and suppose
A
D
=
12
AD = 12
A
D
=
12
. If
B
D
=
1
4
B
C
BD = \frac14 BC
B
D
=
4
1
BC
and
O
H
∥
B
C
OH \parallel BC
O
H
∥
BC
, compute
A
B
2
AB^2
A
B
2
. .
BMT 2021 General p17
Simplify
17
+
12
2
4
−
17
−
12
2
4
\sqrt[4]{17 + 12\sqrt2} - \sqrt[4]{17 - 12\sqrt2}
4
17
+
12
2
−
4
17
−
12
2
.
16
2
Hide problems
BMT 2021 Guts Round p16
Sigfried is singing the ABC’s
100
100
100
times straight, for some reason. It takes him
20
20
20
seconds to sing the ABC’s once, and he takes a
5
5
5
second break in between songs. Normally, he sings the ABC’s without messing up, but he gets fatigued when singing correctly repeatedly. For any song, if he sung the previous three songs without messing up, he has a
1
2
\frac12
2
1
chance of messing up and taking
30
30
30
seconds for the song instead. What is the expected number of minutes it takes for Sigfried to sing the ABC’s
100
100
100
times? Round your answer to the nearest minute.
BMT 2021 General p16
Jason and Valerie agree to meet for game night, which runs from
4
:
00
4:00
4
:
00
PM to
5
:
00
5:00
5
:
00
PM. Jason and Valerie each choose a random time from
4
:
00
4:00
4
:
00
PM to
5
:
00
5:00
5
:
00
PM to show up. If Jason arrives first, he will wait
20
20
20
minutes for Valerie before leaving. If Valerie arrives first, she will wait
10
10
10
minutes for Jason before leaving. What is the probability that Jason and Valerie successfully meet each other for game night?
15
2
Hide problems
BMT 2021 Guts Round p15
Compute
cos
(
π
12
)
cos
(
π
24
)
cos
(
π
48
)
cos
(
π
96
)
.
.
.
cos
(
π
4
)
cos
(
π
8
)
cos
(
π
16
)
cos
(
π
32
)
.
.
.
\frac{\cos \left(\frac{\pi}{12}\right)\cos \left(\frac{\pi}{24}\right)\cos \left(\frac{\pi}{48}\right)\cos \left(\frac{\pi}{96}\right)...}{\cos \left(\frac{\pi}{4}\right)\cos \left(\frac{\pi}{8}\right)\cos \left(\frac{\pi}{16}\right)\cos \left(\frac{\pi}{32}\right)...}
cos
(
4
π
)
cos
(
8
π
)
cos
(
16
π
)
cos
(
32
π
)
...
cos
(
12
π
)
cos
(
24
π
)
cos
(
48
π
)
cos
(
96
π
)
...
BMT 2021 General p15
Benji has a
2
×
2
2\times 2
2
×
2
grid, which he proceeds to place chips on. One by one, he places a chip on one of the unit squares of the grid at random. However, if at any point there is more than one chip on the same square, Benji moves two chips on that square to the two adjacent squares, which he calls a chip-fire. He keeps adding chips until there is an infinite loop of chip-fires. What is the expected number of chips that will be added to the board?
14
2
Hide problems
BMT 2021 Guts Round p14
Given an integer
c
c
c
, the sequence
a
0
,
a
1
,
a
2
,
.
.
.
a_0, a_1, a_2, ...
a
0
,
a
1
,
a
2
,
...
is generated using the recurrence relation
a
0
=
c
a_0 = c
a
0
=
c
and
a
i
=
a
i
−
1
i
+
2021
a
i
−
1
a_i = a^i_{i-1} + 2021a_{i-1}
a
i
=
a
i
−
1
i
+
2021
a
i
−
1
for all
i
≥
1
i \ge 1
i
≥
1
. Given that
a
0
=
c
a_0 = c
a
0
=
c
, let
f
(
c
)
f(c)
f
(
c
)
be the smallest positive integer
n
n
n
such that
a
n
−
1
a_n - 1
a
n
−
1
is a multiple of
47
47
47
. Compute
∑
k
=
1
46
f
(
k
)
.
\sum^{46}_{k=1} f(k).
k
=
1
∑
46
f
(
k
)
.
BMT 2021 General p14
Let
r
1
,
r
2
,
.
.
.
,
r
47
r_1, r_2, ..., r_{47}
r
1
,
r
2
,
...
,
r
47
be the roots of
x
47
−
1
=
0
x^{47} - 1 = 0
x
47
−
1
=
0
. Compute
∑
i
=
1
47
r
i
2020
.
\sum^{47}_{i=1}r^{2020}_i .
i
=
1
∑
47
r
i
2020
.
13
2
Hide problems
BMT 2021 Guts Round p13
How many ways are there to completely fill a
3
×
3
3 \times 3
3
×
3
grid of unit squares with the letters
B
,
M
B, M
B
,
M
, and
T
T
T
, assigning exactly one of the three letters to each of the squares, such that no
2
2
2
adjacent unit squares contain the same letter? Two unit squares are adjacent if they share a side.
BMT 2021 General p13
A six-sided die is rolled four times. What is the probability that the minimum value of the four rolls is
4
4
4
?
12
2
Hide problems
BMT 2021 Guts Round p12
Unit square
A
B
C
D
ABCD
A
BC
D
is drawn on a plane. Point
O
O
O
is drawn outside of
A
B
C
D
ABCD
A
BC
D
such that lines
A
O
AO
A
O
and
B
O
BO
BO
are perpendicular. Square
F
R
O
G
F ROG
FROG
is drawn with
F
F
F
on
A
B
AB
A
B
such that
A
F
=
2
3
AF =\frac23
A
F
=
3
2
,
R
R
R
is on
B
O
‾
\overline{BO}
BO
, and
G
G
G
is on
A
O
‾
\overline{AO}
A
O
. Extend segment
O
F
‾
\overline{OF}
OF
past
A
B
‾
\overline{AB}
A
B
to intersect side
C
D
‾
\overline{CD}
C
D
at
E
E
E
. Compute
D
E
DE
D
E
.
BMT 2021 General p12
Let
a
a
a
,
b
b
b
, and
c
c
c
be the solutions of the equation
x
3
−
3
⋅
202
1
2
x
=
2
⋅
20213.
x^3 - 3 \cdot 2021^2x = 2 \cdot 20213.
x
3
−
3
⋅
202
1
2
x
=
2
⋅
20213.
Compute
1
a
+
1
b
+
1
c
.
\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.
a
1
+
b
1
+
c
1
.
11
2
Hide problems
BMT 2021 Guts Round p11
Compute the sum of all prime numbers
p
p
p
with
p
≥
5
p \ge 5
p
≥
5
such that
p
p
p
divides
(
p
+
3
)
p
−
3
+
(
p
+
5
)
p
−
5
(p + 3)^{p-3} + (p + 5)^{p-5}
(
p
+
3
)
p
−
3
+
(
p
+
5
)
p
−
5
. .
BMT 2021 General p11
Compute the number of sequences of five positive integers
a
1
,
.
.
.
,
a
5
a_1,..., a_5
a
1
,
...
,
a
5
where all
a
i
≤
5
a_i \le 5
a
i
≤
5
and the greatest common divisor of all five integers is
1
1
1
.
10
5
Show problems
9
5
Show problems
8
5
Show problems
7
5
Show problems
6
5
Show problems
5
5
Show problems
4
5
Show problems
3
5
Show problems
2
4
Show problems
1
5
Show problems