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BMT Problems
2022 BMT
2022 BMT
Part of
BMT Problems
Subcontests
(30)
19-21
1
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BMT 2022 Guts Round Set 7 p19-21
Guts Round / Set 7p19. Let
N
≥
3
N \ge 3
N
≥
3
be the answer to Problem 21.A regular
N
N
N
-gon is inscribed in a circle of radius
1
1
1
. Let
D
D
D
be the set of diagonals, where we include all sides as diagonals. Then, let
D
′
D'
D
′
be the set of all unordered pairs of distinct diagonals in
D
D
D
. Compute the sum
∑
{
d
,
d
′
}
∈
D
′
ℓ
(
d
)
2
ℓ
(
d
′
)
2
,
\sum_{\{d,d'\}\in D'} \ell (d)^2 \ell (d')^2,
{
d
,
d
′
}
∈
D
′
∑
ℓ
(
d
)
2
ℓ
(
d
′
)
2
,
where
ℓ
(
d
)
\ell (d)
ℓ
(
d
)
denotes the length of diagonal
d
d
d
. p20. Let
N
N
N
be the answer to Problem
19
19
19
, and let
M
M
M
be the last digit of
N
N
N
.Let
ω
\omega
ω
be a primitive
M
M
M
th root of unity, and define
P
(
x
)
P(x)
P
(
x
)
such that
P
(
x
)
=
∏
k
=
1
M
(
x
−
ω
i
k
)
,
P(x) = \prod^M_{k=1} (x - \omega^{i_k}),
P
(
x
)
=
k
=
1
∏
M
(
x
−
ω
i
k
)
,
where the
i
k
i_k
i
k
are chosen independently and uniformly at random from the range
{
0
,
1
,
.
.
.
,
M
−
1
}
\{0, 1, . . . ,M-1\}
{
0
,
1
,
...
,
M
−
1
}
. Compute
E
[
P
(
⌋
1250
N
⌋
)
]
.
E \left[P\left(\sqrt{\rfloor \frac{1250}{N} \rfloor } \right)\right].
E
[
P
(
⌋
N
1250
⌋
)
]
.
p21. Let
N
N
N
be the answer to Problem
20
20
20
.Define the polynomial
f
(
x
)
=
x
34
+
x
33
+
x
32
+
.
.
.
+
x
+
1
f(x) = x^{34} +x^{33} +x^{32} +...+x+1
f
(
x
)
=
x
34
+
x
33
+
x
32
+
...
+
x
+
1
. Compute the number of primes
p
<
N
p < N
p
<
N
such that there exists an integer
k
k
k
with
f
(
k
)
f(k)
f
(
k
)
divisible by
p
p
p
.
Tie 4
1
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BMT 2022 General Tiebreaker #4
How many positive integers less than
2022
2022
2022
contain at least one digit less than
5
5
5
and also at least one digit greater than
4
4
4
?
20
1
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BMT 2022 General p20
The game Boddle uses eight cards numbered
6
,
11
,
12
,
14
,
24
,
47
,
54
6, 11, 12, 14, 24, 47, 54
6
,
11
,
12
,
14
,
24
,
47
,
54
, and
n
n
n
, where
0
≤
n
≤
56
0 \le n \le 56
0
≤
n
≤
56
. An integer D is announced, and players try to obtain two cards, which are not necessarily distinct, such that one of their differences (positive or negative) is congruent to
D
D
D
modulo
57
57
57
. For example, if
D
=
27
D = 27
D
=
27
, then the pair
24
24
24
and
54
54
54
would work because
24
−
54
≡
27
24 - 54 \equiv 27
24
−
54
≡
27
mod
57
57
57
. Compute
n
n
n
such that this task is always possible for all
D
D
D
.
27
1
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BMT 2022 Guts #27
Submit a positive integer
n
n
n
less than
1
0
5
10^5
1
0
5
. Let the sum of the valid submissions from all teams to this question be
S
S
S
. If you submit an invalid answer, you will receive
0
0
0
points. Otherwise, your score will be
max
(
0
,
⌊
25
−
∣
S
′
−
n
∣
10
⌋
)
\max \left(0,\lfloor 25 - \frac{|S'-n|}{10} \rfloor \right)
max
(
0
,
⌊
25
−
10
∣
S
′
−
n
∣
⌋
)
, where
S
′
S'
S
′
is the sum of the squares of the digits of
S
S
S
.
26
1
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BMT 2022 Guts #26
Compute the number of positive integers
n
n
n
less than
1
0
8
10^8
1
0
8
such that at least two of the last five digits of
⌊
1000
25
n
2
+
50
9
n
+
2022
⌋
\lfloor 1000\sqrt{25n^2 + \frac{50}{9}n + 2022}\rfloor
⌊
1000
25
n
2
+
9
50
n
+
2022
⌋
are
6
6
6
. If your submitted estimate is a positive number
E
E
E
and the true value is
A
A
A
, then your score is given by
max
(
0
,
⌊
25
min
(
E
A
,
A
E
)
7
⌋
)
\max \left(0, \left\lfloor 25 \min \left( \frac{E}{A}, \frac{A}{E}\right)^7\right\rfloor \right)
max
(
0
,
⌊
25
min
(
A
E
,
E
A
)
7
⌋
)
.
25
1
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BMT 2022 Guts #25
For triangle
△
A
B
C
\vartriangle ABC
△
A
BC
, define its
A
A
A
-excircle to be the circle that is externally tangent to line segment
B
C
BC
BC
and extensions of
A
B
↔
\overleftrightarrow{AB}
A
B
and
A
C
↔
\overleftrightarrow{AC}
A
C
, and define the
B
B
B
-excircle and
C
C
C
-excircle likewise. Then, define the
A
A
A
-veryexcircle to be the unique circle externally tangent to both the
A
A
A
-excircle as well as the extensions of
A
B
↔
\overleftrightarrow{AB}
A
B
and
A
C
↔
\overleftrightarrow{AC}
A
C
, but that shares no points with line
B
C
↔
\overleftrightarrow{BC}
BC
, and define the
B
B
B
-veryexcircle and
C
C
C
-veryexcircle likewise. Compute the smallest integer
N
≥
337
N \ge 337
N
≥
337
such that for all
N
1
≥
N
N_1 \ge N
N
1
≥
N
, the area of a triangle with lengths
3
N
1
2
3N^2_1
3
N
1
2
,
3
N
1
2
+
1
3N^2_1 + 1
3
N
1
2
+
1
, and
2022
N
1
2022N_1
2022
N
1
is at most
1
22022
\frac{1}{22022}
22022
1
times the area of the triangle formed by connecting the centers of its three veryexcircles. If your submitted estimate is a positive number
E
E
E
and the true value is
A
A
A
, then your score is given by
max
(
0
,
⌊
25
min
(
E
A
,
A
E
)
3
⌋
)
\max \left(0, \left\lfloor 25 \min \left( \frac{E}{A}, \frac{A}{E}\right)^3\right\rfloor \right)
max
(
0
,
⌊
25
min
(
A
E
,
E
A
)
3
⌋
)
.
24
1
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BMT 2022 Guts #24
Let
△
B
C
D
\vartriangle BCD
△
BC
D
be an equilateral triangle and
A
A
A
be a point on the circumcircle of
△
B
C
D
\vartriangle BCD
△
BC
D
such that
A
A
A
is on the minor arc
B
D
BD
B
D
. Then, let
P
P
P
be the intersection of
A
B
‾
\overline{AB}
A
B
with
C
D
‾
\overline{CD}
C
D
,
Q
Q
Q
be the intersection of
A
C
‾
\overline{AC}
A
C
with
D
B
‾
\overline{DB}
D
B
, and
R
R
R
be the intersection of
A
D
‾
\overline{AD}
A
D
with
B
C
‾
\overline{BC}
BC
. Finally, let
X
X
X
,
Y
Y
Y
, and
Z
Z
Z
be the feet of the altitudes from
P
P
P
,
Q
Q
Q
, and
R
R
R
, respectively, in triangle
△
P
Q
R
\vartriangle PQR
△
PQR
. Given
B
Q
=
3
−
5
BQ = 3 -\sqrt5
BQ
=
3
−
5
and
B
C
=
2
BC = 2
BC
=
2
, compute the product of the areas
[
△
X
C
D
]
⋅
[
△
Y
D
B
]
⋅
[
△
Z
B
C
]
[\vartriangle XCD] \cdot [\vartriangle Y DB] \cdot [\vartriangle ZBC]
[
△
XC
D
]
⋅
[
△
Y
D
B
]
⋅
[
△
ZBC
]
.
23
1
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BMT 2022 Guts #23
Carson the farmer has a plot of land full of crops in the shape of a
6
×
6
6 \times 6
6
×
6
grid of squares. Each day, he uniformly at random chooses a row or a column of the plot that he hasn’t chosen before and harvests all of the remaining crops in the row or column. Compute the expected number of connected components that the remaining crops form after
6
6
6
days. If all crops have been harvested, we say there are
0
0
0
connected components.
22
1
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BMT 2022 Guts #22
Set
n
=
425425
n = 425425
n
=
425425
. Let
S
S
S
be the set of proper divisors of
n
n
n
. Compute the remainder when
∑
k
∈
S
ϕ
(
k
)
(
2
n
/
k
n
/
k
)
\sum_{k\in S} \phi (k) {2n/k \choose n/k}
k
∈
S
∑
ϕ
(
k
)
(
n
/
k
2
n
/
k
)
is divided by
2
n
2n
2
n
, where
ϕ
(
x
)
\phi (x)
ϕ
(
x
)
is the number of positive integers at most
x
x
x
that are relatively prime to it.
18
1
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BMT 2022 Guts #18
Nir finds integers
a
0
,
a
1
,
.
.
.
,
a
208
a_0, a_1, ... , a_{208}
a
0
,
a
1
,
...
,
a
208
such that
(
x
+
2
)
208
=
a
0
x
0
+
a
1
x
1
+
a
2
x
2
+
.
.
.
+
a
208
x
208
.
(x + 2)^{208} = a_0x^0 + a_1x^1 + a_2x^2 +... + a_{208}x^{208}.
(
x
+
2
)
208
=
a
0
x
0
+
a
1
x
1
+
a
2
x
2
+
...
+
a
208
x
208
.
Let
S
S
S
be the sum of all an such that
n
−
3
n -3
n
−
3
is divisible by
5
5
5
. Compute the remainder when
S
S
S
is divided by
103
103
103
.
17
2
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BMT 2022 Guts #17
Compute the number of ordered triples
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
of integers between
−
100
-100
−
100
and
100
100
100
inclusive satisfying the simultaneous equations
a
3
−
2
a
=
a
b
c
−
b
−
c
a^3 - 2a = abc - b - c
a
3
−
2
a
=
ab
c
−
b
−
c
b
3
−
2
b
=
b
c
a
−
c
−
a
b^3 - 2b = bca - c - a
b
3
−
2
b
=
b
c
a
−
c
−
a
c
3
−
2
c
=
c
a
b
−
a
−
b
.
c^3 - 2c = cab - a - b.
c
3
−
2
c
=
c
ab
−
a
−
b
.
BMT 2022 General p17
Midori and Momoi are arguing over chores. Each of
5
5
5
chores may either be done by Midori, done by Momoi, or put off for tomorrow. Today, each of them must complete at least one chore, and more than half of the chores must be completed. How many ways can they assign chores for today? (The order in which chores are completed does not matter.)
16
2
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BMT 2022 Guts #16
Let triangle
△
A
B
C
\vartriangle ABC
△
A
BC
be a triangle with
A
B
=
5
AB = 5
A
B
=
5
,
B
C
=
7
BC = 7
BC
=
7
, and
C
A
=
8
CA = 8
C
A
=
8
, and let
I
I
I
be the incenter of
△
A
B
C
\vartriangle ABC
△
A
BC
. Let circle
C
A
C_A
C
A
denote the circle with center
A
A
A
and radius
A
I
‾
\overline{AI}
A
I
, denote
C
B
C_B
C
B
and circle
C
C
C_C
C
C
similarly. Besides all intersecting at
I
I
I
, the circles
C
A
C_A
C
A
,
C
B
C_B
C
B
,
C
C
C_C
C
C
also intersect pairwise at
F
F
F
,
G
G
G
, and
H
H
H
. Compute the area of triangle
△
F
G
H
\vartriangle FGH
△
FG
H
.
BMT 2022 General p16
A street on Stanford can be modeled by a number line. Four Stanford students, located at positions
1
1
1
,
9
9
9
,
25
25
25
and
49
49
49
along the line, want to take an UberXL to Berkeley, but are not sure where to meet the driver. Find the smallest possible total distance walked by the students to a single position on the street. (For example, if they were to meet at position
46
46
46
, then the total distance walked by the students would be
45
+
37
+
21
+
3
=
106
45 + 37 + 21 + 3 = 106
45
+
37
+
21
+
3
=
106
, where the distances walked by the students at positions
1
1
1
,
9
9
9
,
25
25
25
and
49
49
49
are summed in that order.)
15
1
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BMT 2022 Guts #15
Let
f
(
x
)
f(x)
f
(
x
)
be a function acting on a string of
0
0
0
s and
1
1
1
s, defined to be the number of substrings of
x
x
x
that have at least one
1
1
1
, where a substring is a contiguous sequence of characters in
x
x
x
. Let
S
S
S
be the set of binary strings with
24
24
24
ones and
100
100
100
total digits. Compute the maximum possible value of
f
(
s
)
f(s)
f
(
s
)
over all
s
∈
S
s\in S
s
∈
S
. For example,
f
(
110
)
=
5
f(110) = 5
f
(
110
)
=
5
as
1
‾
10
\underline{1}10
1
10
,
1
1
‾
0
1\underline{1}0
1
1
0
,
11
‾
0
\underline{11}0
11
0
,
1
10
‾
1\underline{10}
1
10
, and
110
‾
\underline{110}
110
are all substrings including a
1
1
1
. Note that
11
0
‾
11\underline{0}
11
0
is not such a substring.
14
1
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BMT 2022 Guts #14
Isaac writes each fraction
1
2
300
\frac{1^2}{300}
300
1
2
,
2
2
300
\frac{2^2}{300}
300
2
2
,
.
.
.
...
...
,
30
0
2
300
\frac{300^2}{300}
300
30
0
2
in reduced form. Compute the sum of all denominators over all the reduced fractions that Isaac writes down.
13
2
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BMT 2022 Guts #13
Real numbers
x
x
x
and
y
y
y
satisfy the system of equations
x
3
+
3
x
2
=
−
3
y
−
1
x^3 + 3x^2 = -3y - 1
x
3
+
3
x
2
=
−
3
y
−
1
y
3
+
3
y
2
=
−
3
x
−
1.
y^3 + 3y^2 = -3x - 1.
y
3
+
3
y
2
=
−
3
x
−
1.
What is the greatest possible value of
x
x
x
?
BMT 2022 General p13
Three standard six-sided dice are rolled. What is the probability that the product of the values on the top faces of the three dice is a perfect cube?
12
2
Hide problems
BMT 2022 Guts #12
Let circles
C
1
C_1
C
1
and
C
2
C_2
C
2
be internally tangent at point
P
P
P
, with
C
1
C_1
C
1
being the smaller circle. Consider a line passing through
P
P
P
which intersects
C
1
C_1
C
1
at
Q
Q
Q
and
C
2
C_2
C
2
at
R
R
R
. Let the line tangent to
C
2
C_2
C
2
at
R
R
R
and the line perpendicular to
P
R
‾
\overline{PR}
PR
passing through
Q
Q
Q
intersect at a point
S
S
S
outside both circles. Given that
S
R
=
5
SR = 5
SR
=
5
,
R
Q
=
3
RQ = 3
RQ
=
3
, and
Q
P
=
2
QP = 2
QP
=
2
, compute the radius of
C
2
C_2
C
2
.
BMT 2022 General p12
Parallelograms
A
B
G
F
ABGF
A
BGF
,
C
D
G
B
CDGB
C
D
GB
and
E
F
G
D
EFGD
EFG
D
are drawn so that
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
is a convex hexagon, as shown. If
∠
A
B
G
=
5
3
o
\angle ABG = 53^o
∠
A
BG
=
5
3
o
and
∠
C
D
G
=
5
6
o
\angle CDG = 56^o
∠
C
D
G
=
5
6
o
, what is the measure of
∠
E
F
G
\angle EFG
∠
EFG
, in degrees? https://cdn.artofproblemsolving.com/attachments/9/f/79d163662e02bc40d2636a76b73f632e59d584.png
11
1
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BMT 2022 Guts #11
Kylie is trying to count to
202250
202250
202250
. However, this would take way too long, so she decides to only write down positive integers from
1
1
1
to
202250
202250
202250
, inclusive, that are divisible by
125
125
125
. How many times does she write down the digit
2
2
2
?
Tie 3
4
Show problems
Tie 2
4
Show problems
Tie 1
3
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BMT 2022 Algebra Tiebreaker p1
For all
a
a
a
and
b
b
b
, let
a
♣
b
=
3
a
+
2
b
+
1
a\clubsuit b = 3a + 2b + 1
a
♣
b
=
3
a
+
2
b
+
1
. Compute
c
c
c
such that
(
2
c
)
♣
(
5
♣
(
c
+
3
)
)
=
60
(2c)\clubsuit (5\clubsuit (c + 3)) = 60
(
2
c
)
♣
(
5♣
(
c
+
3
))
=
60
.
BMT 2022 Discrete Tiebreaker p1
How many three-digit positive integers have digits which sum to a multiple of
10
10
10
?
BMT 2022 Geometry Tiebreaker #1
Let
A
B
C
D
E
F
G
H
ABCDEF GH
A
BC
D
EFG
H
be a unit cube such that
A
B
C
D
ABCD
A
BC
D
is one face of the cube and
A
E
‾
\overline{AE}
A
E
,
B
F
‾
\overline{BF}
BF
,
C
G
‾
\overline{CG}
CG
, and
D
H
‾
\overline{DH}
DH
are all edges of the cube. Points
I
,
J
,
K
I, J, K
I
,
J
,
K
, and
L
L
L
are the respective midpoints of
A
F
‾
\overline{AF}
A
F
,
B
G
‾
\overline{BG}
BG
,
C
H
‾
\overline{CH}
C
H
, and
D
E
‾
\overline{DE}
D
E
. The inscribed circle of
I
J
K
L
IJKL
I
J
K
L
is the largest cross-section of some sphere. Compute the volume of this sphere.
10
5
Show problems
9
5
Show problems
8
5
Show problems
7
5
Show problems
6
4
Show problems
5
5
Show problems
4
5
Show problems
3
4
Show problems
2
4
Show problems
1
5
Show problems