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2019 BMT Spring
2019 BMT Spring
Part of
BMT Problems
Subcontests
(26)
Tie 5
1
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2019 BMT Individual Tiebreaker 5
Ankit, Box, and Clark are taking the tiebreakers for the geometry round, consisting of three problems. Problem
k
k
k
takes each
k
k
k
minutes to solve. If for any given problem there is a
1
3
\frac13
3
1
chance for each contestant to solve that problem first, what is the probability that Ankit solves a problem first?
20
1
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2019 BMT Individual 20
Define a sequence
F
n
F_n
F
n
such that
F
1
=
1
F_1 = 1
F
1
=
1
,
F
2
=
x
F_2 = x
F
2
=
x
,
F
n
+
1
=
x
F
n
+
y
F
n
−
1
F_{n+1} = xF_n + yF_{n-1}
F
n
+
1
=
x
F
n
+
y
F
n
−
1
where and
x
x
x
and
y
y
y
are positive integers. Suppose
1
F
k
=
∑
n
=
1
∞
F
n
d
n
\frac{1}{F_k}= \sum_{n=1}^{\infty}\frac{F_n}{d^n}
F
k
1
=
∑
n
=
1
∞
d
n
F
n
has exactly two solutions
(
d
,
k
)
(d, k)
(
d
,
k
)
with
d
>
0
d > 0
d
>
0
is a positive integer. Find the least possible positive value of
d
d
d
.
19
1
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2019 BMT Individual 19
Let
a
a
a
and
b
b
b
be real numbers such that
max
0
≤
x
≤
1
∣
x
3
−
a
x
−
b
∣
\max_{0\le x\le 1} |x^3 - ax - b|
max
0
≤
x
≤
1
∣
x
3
−
a
x
−
b
∣
is as small as possible. Find
a
+
b
a + b
a
+
b
in simplest radical form. (Hint: If
f
(
x
)
=
x
3
−
c
x
−
d
f(x) = x^3 - cx - d
f
(
x
)
=
x
3
−
c
x
−
d
, then the maximum (or minimum) of
f
(
x
)
f(x)
f
(
x
)
either occurs when
x
=
0
x = 0
x
=
0
and/or
x
=
1
x = 1
x
=
1
and/or when x satisfies
3
x
2
−
c
=
0
3x^2 - c = 0
3
x
2
−
c
=
0
).
18
1
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2019 BMT Individual 18
Define
f
(
x
,
y
)
f(x, y)
f
(
x
,
y
)
to be
∣
x
∣
∣
y
∣
\frac{|x|}{|y|}
∣
y
∣
∣
x
∣
if that value is a positive integer,
∣
y
∣
∣
x
∣
\frac{|y|}{|x|}
∣
x
∣
∣
y
∣
if that value is a positive integer, and zero otherwise. We say that a sequence of integers
ℓ
1
\ell_1
ℓ
1
through
ℓ
n
\ell_n
ℓ
n
is good if
f
(
ℓ
i
,
ℓ
i
+
1
)
f(\ell_i, \ell_{i+1})
f
(
ℓ
i
,
ℓ
i
+
1
)
is nonzero for all
i
i
i
where
1
≤
i
≤
n
−
1
1 \le i \le n - 1
1
≤
i
≤
n
−
1
, and the score of the sequence is
∑
i
=
1
n
−
1
f
(
ℓ
i
,
ℓ
i
+
1
)
\sum^{n-1}_{i=1} f(\ell_i, \ell_{i+1})
∑
i
=
1
n
−
1
f
(
ℓ
i
,
ℓ
i
+
1
)
15
2
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2019 BMT Team 15
A group of aliens from Gliese
667
667
667
Cc come to Earth to test the hypothesis that mathematics is indeed a universal language. To do this, they give you the following information about their mathematical system:
∙
\bullet
∙
For the purposes of this experiment, the Gliesians have decided to write their equations in the same syntactic format as in Western math. For example, in Western math, the expression “
5
+
4
5+4
5
+
4
” is interpreted as running the “
+
+
+
” operation on numbers
5
5
5
and
4
4
4
. Similarly, in Gliesian math, the expression
α
γ
β
\alpha \gamma \beta
α
γ
β
is interpreted as running the “
γ
\gamma
γ
” operation on numbers
α
\alpha
α
and
β
\beta
β
.
∙
\bullet
∙
You know that
γ
\gamma
γ
and
η
\eta
η
are the symbols for addition and multiplication (which works the same in Gliesian math as in Western math), but you don’t know which is which. By some bizarre coincidence, the symbol for equality is the same in Gliesian math as it is in Western math; equality is denoted with an “
=
=
=
” symbol between the two equal values.
∙
\bullet
∙
Two symbols that look exactly the same have the same meaning. Two symbols that are different have different meanings and, therefore, are not equal.They then provide you with the following equations, written in Gliesian, which are known to be true: https://cdn.artofproblemsolving.com/attachments/b/e/e2e44c257830ce8eee7c05535046c17ae3b7e6.png
2019 BMT Individual 15
How many distinct positive integers can be formed by choosing their digits from the string
04072019
04072019
04072019
?
12
2
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2019 BMT Team 12
2019
2019
2019
people (all of whom are perfect logicians), labeled from
1
1
1
to
2019
2019
2019
, partake in a paintball duel. First, they decide to stand in a circle, in order, so that Person
1
1
1
has Person
2
2
2
to his left and person
2019
2019
2019
to his right. Then, starting with Person
1
1
1
and moving to the left, every person who has not been eliminated takes a turn shooting. On their turn, each person can choose to either shoot one non-eliminated person of his or her choice (which eliminates that person from the game), or deliberately miss. The last person standing wins. If, at any point, play goes around the circle once with no one getting eliminated (that is, if all the people playing decide to miss), then automatic paint sprayers will turn on, and end the game with everyone losing. Each person will, on his or her turn, always pick a move that leads to a win if possible, and, if there is still a choice in what move to make, will prefer shooting over missing, and shooting a person closer to his or her left over shooting someone farther from their left. What is the number of the person who wins this game? Put “
0
0
0
” if no one wins.
2019 BMT Individual 12
Box is thinking of a number, whose digits are all “
1
1
1
”. When he squares the number, the sum of its digit is
85
85
85
. How many digits is Box’s number?
Tie 1
3
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2019 BMT Algebra Tiebreakers 1
Compute the maximum real value of
a
a
a
for which there is an integer
b
b
b
such that
a
b
2
a
+
2
b
=
2019
\frac{ab^2}{a+2b} = 2019
a
+
2
b
a
b
2
=
2019
. Compute the maximum possible value of
a
a
a
.
2019 BMT Discrete Tiebreakers 1
Compute the probability that a random permutation of the letters in BERKELEY does not have the three E’s all on the same side of the Y.
2019 BMT Individual Tiebreaker 1
Let
p
p
p
be a prime and
n
n
n
a positive integer below
100
100
100
. What’s the probability that
p
p
p
divides
n
n
n
?
Tie 4
1
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surface area of the circumsphere of the pyramid 2019 BMT Individual Tiebreaker 4
Consider a regular triangular pyramid with base
△
A
B
C
\vartriangle ABC
△
A
BC
and apex
D
D
D
. If we have
A
B
=
B
C
=
A
C
=
6
AB = BC =AC = 6
A
B
=
BC
=
A
C
=
6
and
A
D
=
B
D
=
C
D
=
4
AD = BD = CD = 4
A
D
=
B
D
=
C
D
=
4
, calculate the surface area of the circumsphere of the pyramid.
14
2
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laser beam inside a regular hexagon, min path a>2019 2019 BMT Team 14
A regular hexagon has positive integer side length. A laser is emitted from one of the hexagon’s corners, and is reflected off the edges of the hexagon until it hits another corner. Let
a
a
a
be the distance that the laser travels. What is the smallest possible value of
a
2
a^2
a
2
such that
a
>
2019
a > 2019
a
>
2019
?You need not simplify/compute exponents.
2019 BMT Individual 14
On a
24
24
24
hour clock, there are two times after
01
:
00
01:00
01
:
00
for which the time expressed in the form
h
h
:
m
m
hh:mm
hh
:
mm
and in minutes are both perfect squares. One of these times is
01
:
21
01:21
01
:
21
, since
121
121
121
and
60
+
21
=
81
60+21 = 81
60
+
21
=
81
are both perfect squares. Find the other time, expressed in the form
h
h
:
m
m
hh:mm
hh
:
mm
.
17
1
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area among 2 segments and an arc 2019 BMT Individual 17
Let
C
C
C
be a circle of radius
1
1
1
and
O
O
O
its center. Let
A
B
‾
\overline{AB}
A
B
be a chord of the circle and
D
D
D
a point on
A
B
‾
\overline{AB}
A
B
such that
O
D
=
2
2
OD =\frac{\sqrt2}{2}
O
D
=
2
2
such that
D
D
D
is closer to
A
A
A
than it is to
B
B
B
, and if the perpendicular line at
D
D
D
with respect to
A
B
‾
\overline{AB}
A
B
intersects the circle at
E
E
E
and
F
F
F
,
A
D
=
D
E
AD = DE
A
D
=
D
E
. The area of the region of the circle enclosed by
A
D
‾
\overline{AD}
A
D
,
D
E
‾
\overline{DE}
D
E
, and the minor arc
A
E
AE
A
E
may be expressed as
a
+
b
c
+
d
π
e
\frac{a + b\sqrt{c} + d\pi}{e}
e
a
+
b
c
+
d
π
where
a
,
b
,
c
,
d
,
e
a, b, c, d, e
a
,
b
,
c
,
d
,
e
are integers, gcd
(
a
,
b
,
d
,
e
)
=
1
(a, b, d, e) = 1
(
a
,
b
,
d
,
e
)
=
1
, and
c
c
c
is squarefree. Find
a
+
b
+
c
+
d
+
e
a + b + c + d + e
a
+
b
+
c
+
d
+
e
13
2
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area of quadr. wanted, 13-14-15 triangle, incenter excenter 2019 BMT Team 13
Triangle
△
A
B
C
\vartriangle ABC
△
A
BC
has
A
B
=
13
AB = 13
A
B
=
13
,
B
C
=
14
BC = 14
BC
=
14
, and
C
A
=
15
CA = 15
C
A
=
15
.
△
A
B
C
\vartriangle ABC
△
A
BC
has incircle
γ
\gamma
γ
and circumcircle
ω
\omega
ω
.
γ
\gamma
γ
has center at
I
I
I
. Line
A
I
AI
A
I
is extended to hit
ω
\omega
ω
at
P
P
P
. What is the area of quadrilateral
A
B
P
C
ABPC
A
BPC
?
(CADB)=, 2 intersecting circles, AC_|_AB, BD_|_AB 2019 BMT Individual 13
Two circles
O
1
O_1
O
1
and
O
2
O_2
O
2
intersect at points
A
A
A
and
B
B
B
. Lines
A
C
‾
\overline{AC}
A
C
and
B
D
‾
\overline{BD}
B
D
are drawn such that
C
C
C
is on
O
1
O_1
O
1
and
D
D
D
is on
O
2
O_2
O
2
and
A
C
‾
⊥
A
B
‾
\overline{AC} \perp \overline{AB}
A
C
⊥
A
B
and
B
D
‾
⊥
A
B
‾
\overline{BD} \perp \overline{AB}
B
D
⊥
A
B
. If minor arc
A
B
=
45
AB= 45
A
B
=
45
degrees relative to
O
1
O_1
O
1
and minor arc
A
B
=
60
AB= 60
A
B
=
60
degrees relative to
O
2
O_2
O
2
and the radius of
O
2
=
10
O_2 = 10
O
2
=
10
, the area of quadrilateral
C
A
D
B
CADB
C
A
D
B
can be expressed in simplest form as
a
+
b
k
+
c
ℓ
a + b\sqrt{k} + c\sqrt{\ell}
a
+
b
k
+
c
ℓ
. Compute
a
+
b
+
c
+
k
+
ℓ
a + b + c + k +\ell
a
+
b
+
c
+
k
+
ℓ
.
16
1
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min area of an hexagon, lines through O // sides ABC 2019 BMT Individual 16
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
=
26
AB = 26
A
B
=
26
,
B
C
=
51
BC = 51
BC
=
51
, and
C
A
=
73
CA = 73
C
A
=
73
, and let
O
O
O
be an arbitrary point in the interior of
△
A
B
C
\vartriangle ABC
△
A
BC
. Lines
ℓ
1
\ell_1
ℓ
1
,
ℓ
2
\ell_2
ℓ
2
, and
ℓ
3
\ell_3
ℓ
3
pass through
O
O
O
and are parallel to
A
B
‾
\overline{AB}
A
B
,
B
C
‾
\overline{BC}
BC
, and
C
A
‾
\overline{CA}
C
A
, respectively. The intersections of
ℓ
1
\ell_1
ℓ
1
,
ℓ
2
\ell_2
ℓ
2
, and
ℓ
3
\ell_3
ℓ
3
and the sides of
△
A
B
C
\vartriangle ABC
△
A
BC
form a hexagon whose area is
A
A
A
. Compute the minimum value of
A
A
A
.
11
2
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2019 BMT Team 11
A baseball league has
64
64
64
people, each with a different
6
6
6
-digit binary number whose base-
10
10
10
value ranges from
0
0
0
to
63
63
63
. When any player bats, they do the following: for each pitch, they swing if their corresponding bit number is a
1
1
1
, otherwise, they decide to wait and let the ball pass. For example, the player with the number
11
11
11
has binary number
001011
001011
001011
. For the first and second pitch, they wait; for the third, they swing, and so on. Pitchers follow a similar rule to decide whether to throw a splitter or a fastball, if the bit is
0
0
0
, they will throw a splitter, and if the bit is
1
1
1
, they will throw a fastball. If a batter swings at a fastball, then they will score a hit; if they swing on a splitter, they will miss and get a “strike.” If a batter waits on a fastball, then they will also get a strike. If a batter waits on a splitter, then they get a “ball.” If a batter gets
3
3
3
strikes, then they are out; if a batter gets
4
4
4
balls, then they automatically get a hit. For example, if player
11
11
11
pitched against player
6
6
6
(binary is
000110
000110
000110
), the batter would get a ball for the first pitch, a ball for the second pitch, a strike for the third pitch, a strike for the fourth pitch, and a hit for the fifth pitch; as a result, they will count that as a “hit.” If player
11
11
11
pitched against player
5
5
5
(binary is
000101
000101
000101
), however, then the fifth pitch would be the batter’s third strike, so the batter would be “out.” Each player in the league plays against every other player exactly twice; once as batter, and once as pitcher. They are then given a score equal to the number of outs they throw as a pitcher plus the number of hits they get as a batter. What is the highest score received?
chord rotates inside a 17-gon 2019 BMT Individual 11
A regular
17
17
17
-gon with vertices
V
1
,
V
2
,
.
.
.
,
V
17
V_1, V_2, . . . , V_{17}
V
1
,
V
2
,
...
,
V
17
and sides of length
3
3
3
has a point
P
P
P
on
V
1
V
2
V_1V_2
V
1
V
2
such that
V
1
P
=
1
V_1P = 1
V
1
P
=
1
. A chord that stretches from
V
1
V_1
V
1
to
V
2
V_2
V
2
containing
P
P
P
is rotated within the interior of the heptadecagon around
V
2
V_2
V
2
such that the chord now stretches from
V
2
V_2
V
2
to
V
3
V_3
V
3
. The chord then hinges around
V
3
V_3
V
3
, then
V
4
V_4
V
4
, and so on, continuing until
P
P
P
is back at its original position. Find the total length traced by
P
P
P
.
Tie 3
4
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Tie 2
4
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Tie1
1
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BMT 2019 Spring - Geometry Tiebreaker 1
We inscribe a circle
ω
\omega
ω
in equilateral triangle
A
B
C
ABC
A
BC
with radius
1
1
1
. What is the area of the region inside the triangle but outside the circle?
2
5
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1
5
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10
5
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9
5
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8
5
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7
5
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6
5
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5
5
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4
5
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3
5
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