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BMT Problems
2015 BMT Spring
2015 BMT Spring
Part of
BMT Problems
Subcontests
(25)
Tie 3
2
Hide problems
BMT 2015 Geometry Tiebreaker #3
The permutohedron of order
3
3
3
is the hexagon determined by points
(
1
,
2
,
3
)
(1, 2, 3)
(
1
,
2
,
3
)
,
(
1
,
3
,
2
)
(1, 3, 2)
(
1
,
3
,
2
)
,
(
2
,
1
,
3
)
(2, 1, 3)
(
2
,
1
,
3
)
,
(
2
,
3
,
1
)
(2, 3, 1)
(
2
,
3
,
1
)
,
(
3
,
1
,
2
)
(3, 1, 2)
(
3
,
1
,
2
)
, and
(
3
,
2
,
1
)
(3, 2, 1)
(
3
,
2
,
1
)
. The pyramid determined by these six points and the origin has a unique inscribed sphere of maximal volume. Determine its radius.
BMT 2015 Spring - Individual Tiebreaker #3
A bag contains
12
12
12
marbles:
3
3
3
red,
4
4
4
green, and
5
5
5
blue. Repeatedly draw marbles with replacement until you draw two marbles of the same color in a row. What is the expected number of times that you will draw a marble?
Tie 2
2
Hide problems
BMT 2015 Geometry Tiebreaker #2
The unit square
A
B
C
D
ABCD
A
BC
D
has
E
E
E
as midpoint of
A
D
AD
A
D
and a circle of radius
r
r
r
tangent to
A
B
AB
A
B
,
B
C
BC
BC
, and
C
E
CE
CE
. Determine
r
r
r
.
BMT 2015 Spring - Individual Tiebreaker #2
Let
S
n
=
1
+
2
+
,
,
,
+
n
S_n = 1 + 2 + ,,, + n
S
n
=
1
+
2
+
,,,
+
n
. Define
T
n
=
S
2
S
2
−
1
⋅
S
3
S
3
−
1
⋅
.
.
.
⋅
S
n
S
n
−
1
.
T_n =\frac{S_2}{S_2- 1}\cdot \frac{S_3}{S_3 - 1}\cdot ... \cdot \frac{S_n}{S_n - 1}.
T
n
=
S
2
−
1
S
2
⋅
S
3
−
1
S
3
⋅
...
⋅
S
n
−
1
S
n
.
Find
T
2015
.
T_{2015}.
T
2015
.
Tie 1
2
Hide problems
BMT 2015 Geometry Tiebreaker #1
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram. Suppose that
E
E
E
is on line
D
C
DC
D
C
such that
C
C
C
lies on segment
E
D
ED
E
D
. Then say lines
A
E
AE
A
E
and
B
D
BD
B
D
intersect at
X
X
X
and lines
C
X
CX
CX
intersects AB at F. If
A
B
=
7
AB = 7
A
B
=
7
,
B
C
=
13
BC = 13
BC
=
13
, and
C
E
=
91
CE = 91
CE
=
91
, then find
A
F
F
B
\frac{AF}{FB}
FB
A
F
.
BMT 2015 Spring - Individual Tiebreaker #1
Compute the surface area of a rectangular prism with side lengths
2
,
3
,
4
2, 3, 4
2
,
3
,
4
.
19
2
Hide problems
2015 BMT Team 19
Two sequences
(
x
n
)
n
∈
N
(x_n)_{n\in N}
(
x
n
)
n
∈
N
and
(
y
n
)
n
∈
N
(y_n)_{n\in N}
(
y
n
)
n
∈
N
are defined recursively as follows:
x
0
=
2015
x_0 = 2015
x
0
=
2015
and
x
n
+
1
=
⌊
x
n
⋅
y
n
+
1
y
n
−
1
⌋
x_{n+1} =\left \lfloor x_n \cdot \frac{y_{n+1}}{y_{n-1}} \right \rfloor
x
n
+
1
=
⌊
x
n
⋅
y
n
−
1
y
n
+
1
⌋
for all
n
≥
0
n \ge 0
n
≥
0
,
y
0
=
307
y_0 = 307
y
0
=
307
and
y
n
+
1
=
y
n
+
1
y_{n+1} = y_n + 1
y
n
+
1
=
y
n
+
1
for all
n
≥
0
n \ge 0
n
≥
0
.Compute
lim
n
→
∞
x
n
(
y
n
)
2
\lim_{n\to \infty} \frac{x_n}{(y_n)^2}
lim
n
→
∞
(
y
n
)
2
x
n
.
BMT 2015 Spring - Individual 19
It is known that
4
4
4
people
A
,
B
,
C
A, B, C
A
,
B
,
C
, and
D
D
D
each have a
1
/
3
1/3
1/3
probability of telling the truth. Suppose that
∙
\bullet
∙
A
A
A
makes a statement.
∙
\bullet
∙
B
B
B
makes a statement about the truthfulness of
A
A
A
’s statement.
∙
\bullet
∙
C
C
C
makes a statement about the truthfulness of
B
B
B
’s statement.
∙
\bullet
∙
D
D
D
says that
C
C
C
says that
B
B
B
says that
A
A
A
was telling the truth. What is the probability that
A
A
A
was actually telling the truth?
15
2
Hide problems
2015 BMT Team 15 integral
Compute
∫
1
/
2
2
x
2
+
1
x
2
(
x
2015
+
1
)
d
x
.
\int_{1/2}^{2} \frac{x^2 + 1}{x^2(x^{2015} + 1)} dx.
∫
1/2
2
x
2
(
x
2015
+
1
)
x
2
+
1
d
x
.
BMT 2015 Spring - Individual 15
Recall that an icosahedron is a
3
3
3
-dimensional solid characterized by its
20
20
20
congruent faces, each of which is an equilateral triangle. Determine the number of rigid rotations that preserve the symmetry of the icosahedron. (Each vertex moves to the location of another vertex.)
12
2
Hide problems
2015 BMT Team 12
Let
f
(
n
)
f(n)
f
(
n
)
be the number of ordered pairs
(
k
,
ℓ
)
(k, \ell)
(
k
,
ℓ
)
of positive integers such that
n
=
(
2
ℓ
−
1
)
⋅
2
k
−
k
n = (2\ell-1)\cdot 2^k - k
n
=
(
2
ℓ
−
1
)
⋅
2
k
−
k
, and let
g
(
n
)
g(n)
g
(
n
)
be the number of ordered pairs
(
k
,
ℓ
)
(k, \ell)
(
k
,
ℓ
)
of positive integers such that
n
=
ℓ
⋅
2
k
+
1
−
k
n = \ell \cdot 2^{k+1}-k
n
=
ℓ
⋅
2
k
+
1
−
k
. Compute
∑
i
=
1
∞
f
(
i
)
−
g
(
i
)
2
i
\sum_{i=1}^{\infty}\frac{f(i) - g(i)}{2^i}
∑
i
=
1
∞
2
i
f
(
i
)
−
g
(
i
)
..
BMT 2015 Spring - Individual 12
How many possible arrangements of bishops are there on a
8
×
8
8 \times 8
8
×
8
chessboard such that no bishop threatens a square on which another lies and the maximum number of bishops are used? (Note that a bishop threatens any square along a diagonal containing its square.)
20
2
Hide problems
2015 BMT Team 20
The Tower of Hanoi is a puzzle with
n
n
n
disks of different sizes and
3
3
3
vertical rods on it. All of the disks are initially placed on the leftmost rod, sorted by size such that the largest disk is on the bottom. On each turn, one may move the topmost disk of any nonempty rod onto any other rod, provided that it is smaller than the current topmost disk of that rod, if it exists. (For instance, if there were two disks on different rods, the smaller disk could move to either of the other two rods, but the larger disk could only move to the empty rod.) The puzzle is solved when all of the disks are moved to the rightmost rod. The specifications normally include an intelligent monk to move the disks, but instead there is a monkey making random moves (with each valid move having an equal probability of being selected). Given
64
64
64
disks, what is the expected number of moves the monkey will have to make to solve the puzzle?
BMT 2015 Spring - Individual 20
Let
a
a
a
and
b
b
b
be real numbers for which the equation
2
x
4
+
2
a
x
3
+
b
x
2
+
2
a
x
+
2
=
0
2x^4 + 2ax^3 + bx^2 + 2ax + 2 = 0
2
x
4
+
2
a
x
3
+
b
x
2
+
2
a
x
+
2
=
0
has at least one real solution. For all such pairs
(
a
,
b
)
(a, b)
(
a
,
b
)
, find the minimum value of
8
a
2
+
b
2
8a^2 + b^2
8
a
2
+
b
2
.
18
2
Hide problems
2015 BMT Team 18
A value
x
∈
[
0
,
1
]
x \in [0, 1]
x
∈
[
0
,
1
]
is selected uniformly at random. A point
(
a
,
b
)
(a, b)
(
a
,
b
)
is called friendly to
x
x
x
if there exists a circle between the lines
y
=
0
y = 0
y
=
0
and
y
=
1
y = 1
y
=
1
that contains both
(
a
,
b
)
(a, b)
(
a
,
b
)
and
(
0
,
x
)
(0, x)
(
0
,
x
)
. Find the area of the region of the plane determined by possible locations of friendly points.
BMT 2015 Spring - Individual 18
Evaluate
∑
n
=
1
∞
1
(
2
n
−
1
)
(
3
n
−
1
)
\sum_{n=1}^{\infty}\frac{1}{(2n - 1)(3n - 1)}
∑
n
=
1
∞
(
2
n
−
1
)
(
3
n
−
1
)
1
.
14
2
Hide problems
2015 BMT Team 14
Alice is at coordinate point
(
0
,
0
)
(0, 0)
(
0
,
0
)
and wants to go to point
(
11
,
6
)
(11, 6)
(
11
,
6
)
. Similarly, Bob is at coordinate point
(
5
,
6
)
(5, 6)
(
5
,
6
)
and wants to go to point
(
16
,
0
)
(16, 0)
(
16
,
0
)
. Both of them choose a lattice path from their current position to their target position at random (such that each lattice path has an equal probability of being chosen), where a lattice path is defined to be a path composed of unit segments with orthogonal direction (parallel to x-axis or y-axis) and of minimal length. (For instance, there are six lattice paths from
(
0
,
0
)
(0, 0)
(
0
,
0
)
to
(
2
,
2
)
(2, 2)
(
2
,
2
)
.) If they walk with the same speed, find the probability that they meet.
BMT 2015 Spring - Individual 14
Determine \left|\prod^{10}_{k=1}(e^{\frac{i \pi}{2^k}}+ 1) \right|
13
2
Hide problems
2015 BMT Team 13
There exist right triangles with integer side lengths such that the legs differ by
1
1
1
. For example,
3
−
4
−
5
3-4-5
3
−
4
−
5
and
20
−
21
−
29
20-21-29
20
−
21
−
29
are two such right triangles. What is the perimeter of the next smallest Pythagorean right triangle with legs differing by
1
1
1
?
BMT 2015 Spring - Individual 13
On a
2
×
40
2\times 40
2
×
40
chessboard colored black and white in the standard alternating pattern,
20
20
20
rooks are placed randomly on the black squares. The expected number of white squares with only rooks as neighbors can be expressed as
a
/
b
a/b
a
/
b
, where
a
a
a
and
b
b
b
are coprime positive integers. What is
a
+
b
a + b
a
+
b
? (Two squares are said to be neighbors if they share an edge.)
11
2
Hide problems
2015 BMT Team 11
Write down
1
,
2
,
3
,
.
.
.
,
2015
1, 2, 3, ... , 2015
1
,
2
,
3
,
...
,
2015
in a row on a whiteboard. Every minute, select a pair of adjacent numbers at random, erase them, and insert their sum where you selected the numbers. (For instance, selecting
3
3
3
and
4
4
4
from
1
,
2
,
3
,
4
,
5
1, 2, 3, 4, 5
1
,
2
,
3
,
4
,
5
would result in
1
,
2
,
7
,
5
1, 2, 7, 5
1
,
2
,
7
,
5
.) Repeat this process until you have two numbers remaining. What is the probability that the smaller number is less than or equal to
2015
2015
2015
?
BMT 2015 Spring - Individual 11
Let
r
,
s
r, s
r
,
s
, and
t
t
t
be the three roots of the equation
8
x
3
+
1001
x
+
2008
=
0
8x^3 + 1001x + 2008 = 0
8
x
3
+
1001
x
+
2008
=
0
. Find
(
r
+
s
)
3
+
(
s
+
t
)
3
+
(
t
+
r
)
3
(r + s)^3 + (s + t)^3 + (t + r)^3
(
r
+
s
)
3
+
(
s
+
t
)
3
+
(
t
+
r
)
3
.
16
2
Hide problems
AB=BC=CD=DE=EA=1, <ABC=<BCD=<CDE=<DEA=90^o 2015 BMT Team 16
Five points
A
,
B
,
C
,
D
A, B, C, D
A
,
B
,
C
,
D
, and
E
E
E
in three-dimensional Euclidean space have the property that
A
B
=
B
C
=
C
D
=
D
E
=
E
A
=
1
AB = BC = CD = DE = EA = 1
A
B
=
BC
=
C
D
=
D
E
=
E
A
=
1
and
∠
A
B
C
=
∠
B
C
D
=
∠
C
D
E
=
∠
D
E
A
=
9
0
o
\angle ABC = \angle BCD =\angle CDE = \angle DEA = 90^o
∠
A
BC
=
∠
BC
D
=
∠
C
D
E
=
∠
D
E
A
=
9
0
o
. Find all possible
cos
(
∠
E
A
B
)
\cos(\angle EAB)
cos
(
∠
E
A
B
)
.
BMT 2015 Spring - Individual 16
A binary decision tree is a list of
n
n
n
yes/no questions, together with instructions for the order in which they should be asked (without repetition). For instance, if
n
=
3
n = 3
n
=
3
, there are
12
12
12
possible binary decision trees, one of which asks question
2
2
2
first, then question
3
3
3
(followed by question
1
1
1
) if the answer was yes or question
1
1
1
(followed by question
3
3
3
) if the answer was no. Determine the greatest possible
k
k
k
such that
2
k
2^k
2
k
divides the number of binary decision trees on
n
=
13
n = 13
n
=
13
questions.
17
2
Hide problems
2015 BMT Team 17
There exist real numbers
x
x
x
and
y
y
y
such that
x
(
a
3
+
b
3
+
c
3
)
+
3
y
a
b
c
≥
(
x
+
y
)
(
a
2
b
+
b
2
c
+
c
2
a
)
x(a^3 + b^3 + c^3) + 3yabc \ge (x + y)(a^2b + b^2c + c^2a)
x
(
a
3
+
b
3
+
c
3
)
+
3
y
ab
c
≥
(
x
+
y
)
(
a
2
b
+
b
2
c
+
c
2
a
)
holds for all positive real numbers
a
,
b
a, b
a
,
b
, and
c
c
c
. Determine the smallest possible value of
x
/
y
x/y
x
/
y
. .
area if union circle and square, AE + AF = 2(BE + DF) 2015 BMT Individual 17
A circle intersects square
A
B
C
D
ABCD
A
BC
D
at points
A
,
E
A, E
A
,
E
, and
F
F
F
, where
E
E
E
lies on
A
B
AB
A
B
and
F
F
F
lies on
A
D
AD
A
D
, such that
A
E
+
A
F
=
2
(
B
E
+
D
F
)
AE + AF = 2(BE + DF)
A
E
+
A
F
=
2
(
BE
+
D
F
)
. If the square and the circle each have area
1
1
1
, determine the area of the union of the circle and square.
P2
3
Hide problems
BMT 2015 Spring - Geometry P2
Suppose that fixed circle
C
1
C_1
C
1
with radius
a
>
0
a > 0
a
>
0
is tangent to the fixed line
ℓ
\ell
ℓ
at
A
A
A
. Variable circle
C
2
C_2
C
2
, with center
X
X
X
, is externally tangent to
C
1
C_1
C
1
at
B
≠
A
B \ne A
B
=
A
and
ℓ
\ell
ℓ
at
C
C
C
. Prove that the set of all
X
X
X
is a parabola minus a point
2015 BMT Spring Analysis P2
Let
f
(
x
)
f(x)
f
(
x
)
be a nonconstant monic polynomial of degree
n
n
n
with rational coefficents that is irreducible, meaning it cannot be factored into two nonconstant rational polynomials. Find and prove a formula for the number of monic complex polynomials that divide
f
f
f
.
2015 BMT Spring Discrete P2
Suppose
k
>
3
k>3
k
>
3
is a divisor of
2
p
+
1
2^p+1
2
p
+
1
, where
p
p
p
is prime. Prove that
k
≥
2
p
+
1
k\ge2p+1
k
≥
2
p
+
1
.
P1
3
Hide problems
BMT 2015 Spring - Geometry P1
Suppose that circles
C
1
C_1
C
1
and
C
2
C_2
C
2
intersect at
X
X
X
and
Y
Y
Y
. Let
A
,
B
A, B
A
,
B
be on
C
1
C_1
C
1
,
C
2
C_2
C
2
, respectively, such that
A
,
X
,
B
A, X, B
A
,
X
,
B
lie on a line in that order. Let
A
,
C
A, C
A
,
C
be on
C
1
C_1
C
1
,
C
2
C_2
C
2
, respectively, such that
A
,
Y
,
C
A, Y, C
A
,
Y
,
C
lie on a line in that order. Let
A
′
,
B
′
,
C
′
A', B', C'
A
′
,
B
′
,
C
′
be another similarly defined triangle with
A
≠
A
′
A \ne A'
A
=
A
′
. Prove that
B
B
′
=
C
C
′
BB' = CC'
B
B
′
=
C
C
′
.
2015 BMT Spring Analysis P1
Suppose
z
0
,
z
1
,
…
,
z
n
−
1
z_0,z_1,\ldots,z_{n-1}
z
0
,
z
1
,
…
,
z
n
−
1
are complex numbers such that
z
k
=
e
2
k
π
i
/
n
z_k=e^{2k\pi i/n}
z
k
=
e
2
kπi
/
n
for
k
=
0
,
1
,
2
,
…
,
n
−
1
k=0,1,2,\ldots,n-1
k
=
0
,
1
,
2
,
…
,
n
−
1
. Prove that for any complex number
z
z
z
,
∑
k
=
0
n
−
1
∣
z
−
z
k
∣
≥
n
\sum_{k=0}^{n-1}|z-z_k|\ge n
∑
k
=
0
n
−
1
∣
z
−
z
k
∣
≥
n
.
2015 BMT Spring Discrete P1
Find two disjoint sets
N
1
N_1
N
1
and
N
2
N_2
N
2
with
N
1
∪
N
2
=
N
N_1\cup N_2=\mathbb N
N
1
∪
N
2
=
N
, so that neither set contains an infinite arithmetic progression.
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5
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