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BMT Problems
2012 BMT Spring
2012 BMT Spring
Part of
BMT Problems
Subcontests
(17)
Consolation
1
Hide problems
2012 BMT Tournament Round, Consolation Round - Berkley Math Tournament
p1. How many ways can we arrange the elements
{
1
,
2
,
.
.
.
,
n
}
\{1, 2, ..., n\}
{
1
,
2
,
...
,
n
}
to a sequence
a
1
,
a
2
,
.
.
.
,
a
n
a_1, a_2, ..., a_n
a
1
,
a
2
,
...
,
a
n
such that there is only exactly one
a
i
a_i
a
i
,
a
i
+
1
a_{i+1}
a
i
+
1
such that
a
i
>
a
i
+
1
a_i > a_{i+1}
a
i
>
a
i
+
1
? p2. How many distinct (non-congruent) triangles are there with integer side-lengths and perimeter
2012
2012
2012
? p3. Let
ϕ
\phi
ϕ
be the Euler totient function, and let
S
=
{
x
∣
x
ϕ
(
x
)
=
3
}
S = \{x| \frac{x}{\phi (x)} = 3\}
S
=
{
x
∣
ϕ
(
x
)
x
=
3
}
. What is
∑
x
∈
S
1
x
\sum_{x\in S} \frac{1}{x}
∑
x
∈
S
x
1
? p4. Denote
f
(
N
)
f(N)
f
(
N
)
as the largest odd divisor of
N
N
N
. Compute
f
(
1
)
+
f
(
2
)
+
f
(
3
)
+
.
.
.
+
f
(
29
)
+
f
(
30
)
f(1) + f(2) + f(3) +... + f(29) + f(30)
f
(
1
)
+
f
(
2
)
+
f
(
3
)
+
...
+
f
(
29
)
+
f
(
30
)
. p5. Triangle
A
B
C
ABC
A
BC
has base
A
C
AC
A
C
equal to
218
218
218
and altitude
100
100
100
. Squares
s
1
,
s
2
,
s
3
,
.
.
.
s_1, s_2, s_3, ...
s
1
,
s
2
,
s
3
,
...
are drawn such that
s
1
s_1
s
1
has a side on
A
C
AC
A
C
and has one point each touching
A
B
AB
A
B
and
B
C
BC
BC
, and square
s
k
s_k
s
k
has a side on square
s
k
−
1
s_{k-}1
s
k
−
1
and also touches
A
B
AB
A
B
and
B
C
BC
BC
exactly once each. What is the sum of the area of these squares? p6. Let
P
P
P
be a parabola
6
x
2
−
28
x
+
10
6x^2 - 28x + 10
6
x
2
−
28
x
+
10
, and
F
F
F
be the focus. A line
ℓ
\ell
ℓ
passes through
F
F
F
and intersects the parabola twice at points
P
1
=
(
2
,
−
22
)
P_1 = (2,-22)
P
1
=
(
2
,
−
22
)
,
P
2
P_2
P
2
. Tangents to the parabola with points at
P
1
,
P
2
P_1, P_2
P
1
,
P
2
are then drawn, and intersect at a point
Q
Q
Q
. What is
m
∠
P
1
Q
P
2
m\angle P_1QP_2
m
∠
P
1
Q
P
2
? PS. You had better use hide for answers.
Championship
1
Hide problems
2012 BMT Tournament Round, Championship Round - Berkley Math Tournament
p1. If
n
n
n
is a positive integer such that
2
n
+
1
=
14416
9
2
2n+1 = 144169^2
2
n
+
1
=
14416
9
2
, find two consecutive numbers whose squares add up to
n
+
1
n + 1
n
+
1
. p2. Katniss has an
n
n
n
-sided fair die which she rolls. If
n
>
2
n > 2
n
>
2
, she can either choose to let the value rolled be her score, or she can choose to roll a
n
−
1
n - 1
n
−
1
sided fair die, continuing the process. What is the expected value of her score assuming Katniss starts with a
6
6
6
sided die and plays to maximize this expected value? p3. Suppose that
f
(
x
)
=
x
6
+
a
x
5
+
b
x
4
+
c
x
3
+
d
x
2
+
e
x
+
f
f(x) = x^6 + ax^5 + bx^4 + cx^3 + dx^2 + ex + f
f
(
x
)
=
x
6
+
a
x
5
+
b
x
4
+
c
x
3
+
d
x
2
+
e
x
+
f
, and that
f
(
1
)
=
f
(
2
)
=
f
(
3
)
=
f
(
4
)
=
f
(
5
)
=
f
(
6
)
=
7
f(1) = f(2) = f(3) = f(4) = f(5) = f(6) = 7
f
(
1
)
=
f
(
2
)
=
f
(
3
)
=
f
(
4
)
=
f
(
5
)
=
f
(
6
)
=
7
. What is
a
a
a
? p4.
a
a
a
and
b
b
b
are positive integers so that
20
a
+
12
b
20a+12b
20
a
+
12
b
and
20
b
−
12
a
20b-12a
20
b
−
12
a
are both powers of
2
2
2
, but
a
+
b
a+b
a
+
b
is not. Find the minimum possible value of
a
+
b
a + b
a
+
b
. p5. Square
A
B
C
D
ABCD
A
BC
D
and rhombus
C
D
E
F
CDEF
C
D
EF
share a side. If
m
∠
D
C
F
=
3
6
o
m\angle DCF = 36^o
m
∠
D
CF
=
3
6
o
, find the measure of
∠
A
E
C
\angle AEC
∠
A
EC
. p6. Tom challenges Harry to a game. Tom first blindfolds Harry and begins to set up the game. Tom places
4
4
4
quarters on an index card, one on each corner of the card. It is Harry’s job to flip all the coins either face-up or face-down using the following rules: (a) Harry is allowed to flip as many coins as he wants during his turn. (b) A turn consists of Harry flipping as many coins as he wants (while blindfolded). When he is happy with what he has flipped, Harry will ask Tom whether or not he was successful in flipping all the coins face-up or face-down. If yes, then Harry wins. If no, then Tom will take the index card back, rotate the card however he wants, and return it back to Harry, thus starting Harry’s next turn. Note that Tom cannot touch the coins after he initially places them before starting the game. Assuming that Tom’s initial configuration of the coins weren’t all face-up or face-down, and assuming that Harry uses the most efficient algorithm, how many moves maximum will Harry need in order to win? Or will he never win? PS. You had better use hide for answers.
round 5
1
Hide problems
2012 BMT Tournament Round 5 - Berkley Math Tournament
p1. Let
n
n
n
be the number so that
1
−
2
+
3
−
4
+
.
.
.
−
(
n
−
1
)
+
n
=
2012
1 - 2 + 3 - 4 + ... - (n - 1) + n = 2012
1
−
2
+
3
−
4
+
...
−
(
n
−
1
)
+
n
=
2012
. What is
4
2012
4^{2012}
4
2012
(mod
n
n
n
)? p2. Consider three unit squares placed side by side. Label the top left vertex
P
P
P
and the bottom four vertices
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
respectively. Find
∠
P
B
A
+
∠
P
C
A
+
∠
P
D
A
\angle PBA + \angle PCA + \angle PDA
∠
PB
A
+
∠
PC
A
+
∠
P
D
A
. p3. Given
f
(
x
)
=
3
x
−
1
f(x) = \frac{3}{x-1}
f
(
x
)
=
x
−
1
3
, then express
9
(
x
2
−
2
x
+
1
)
x
2
−
8
x
+
16
\frac{9(x^2-2x+1)}{x^2-8x+16}
x
2
−
8
x
+
16
9
(
x
2
−
2
x
+
1
)
entirely in terms of
f
(
x
)
f(x)
f
(
x
)
. In other words,
x
x
x
should not be in your answer, only
f
(
x
)
f(x)
f
(
x
)
. p4. Right triangle with right angle
B
B
B
and integer side lengths has
B
D
BD
B
D
as the altitude.
E
E
E
and
F
F
F
are the incenters of triangles
A
D
B
ADB
A
D
B
and
B
D
C
BDC
B
D
C
respectively. Line
E
F
EF
EF
is extended and intersects
B
C
BC
BC
at
G
G
G
, and
A
B
AB
A
B
at
H
H
H
. If
A
B
=
15
AB = 15
A
B
=
15
and
B
C
=
8
BC = 8
BC
=
8
, find the area of triangle
B
G
H
BGH
BG
H
. p5. Let
a
1
,
a
2
,
.
.
.
,
a
n
a_1, a_2, ..., a_n
a
1
,
a
2
,
...
,
a
n
be a sequence of real numbers. Call a
k
k
k
-inversion
(
0
<
k
≤
n
)
(0 < k\le n)
(
0
<
k
≤
n
)
of a sequence to be indices
i
1
,
i
2
,
.
.
,
i
k
i_1, i_2, .. , i_k
i
1
,
i
2
,
..
,
i
k
such that
i
1
<
i
2
<
.
.
<
i
k
i_1 < i_2 < .. < i_k
i
1
<
i
2
<
..
<
i
k
but
a
i
1
>
a
i
2
>
.
.
.
>
a
i
k
a_{i1} > a_{i2} > ...> a_{ik}
a
i
1
>
a
i
2
>
...
>
a
ik
. Calculate the expected number of
6
6
6
-inversions in a random permutation of the set
{
1
,
2
,
.
.
.
,
10
}
\{1, 2, ... , 10\}
{
1
,
2
,
...
,
10
}
. p6. Chell is given a strip of squares labeled
1
,
.
.
,
6
1, .. , 6
1
,
..
,
6
all placed side to side. For each
k
∈
1
,
.
.
.
,
6
k \in {1, ..., 6}
k
∈
1
,
...
,
6
, she then chooses one square at random in
{
1
,
.
.
.
,
k
}
\{1, ..., k\}
{
1
,
...
,
k
}
and places a Weighted Storage Cube there. After she has placed all
6
6
6
cubes, she computes her score as follows: For each square, she takes the number of cubes in the pile and then takes the square (i.e. if there were 3 cubes in a square, her score for that square would be
9
9
9
). Her overall score is the sum of the scores of each square. What is the expected value of her score? PS. You had better use hide for answers.
round 4
1
Hide problems
2012 BMT Tournament Round 4 - Berkley Math Tournament
p1. Denote
S
n
=
1
+
1
2
+
1
3
+
.
.
.
+
1
n
S_n = 1 + \frac12 + \frac13 + ...+ \frac{1}{n}
S
n
=
1
+
2
1
+
3
1
+
...
+
n
1
. What is
144169
⋅
S
144169
−
(
S
1
+
S
2
+
.
.
.
+
S
144168
)
144169\cdot S_{144169} - (S_1 + S_2 + ... + S_{144168})
144169
⋅
S
144169
−
(
S
1
+
S
2
+
...
+
S
144168
)
? p2. Let
A
,
B
,
C
A,B,C
A
,
B
,
C
be three collinear points, with
A
B
=
4
AB = 4
A
B
=
4
,
B
C
=
8
BC = 8
BC
=
8
, and
A
C
=
12
AC = 12
A
C
=
12
. Draw circles with diameters
A
B
AB
A
B
,
B
C
BC
BC
, and
A
C
AC
A
C
. Find the radius of the two identical circles that will lie tangent to all three circles. p3. Let
s
(
i
)
s(i)
s
(
i
)
denote the number of
1
1
1
’s in the binary representation of
i
i
i
. What is
∑
x
=
1
314
(
∑
i
=
0
2
576
−
2
x
s
(
i
)
)
m
o
d
629
?
\sum_{x=1}{314}\left( \sum_{i=0}^{2^{576}-2} x^{s(i)} \right) \,\, mod \,\,629 ?
x
=
1
∑
314
i
=
0
∑
2
576
−
2
x
s
(
i
)
m
o
d
629
?
p4. Parallelogram
A
B
C
D
ABCD
A
BC
D
has an area of
S
S
S
. Let
k
=
42
k = 42
k
=
42
.
E
E
E
is drawn on AB such that
A
E
=
A
B
k
AE =\frac{AB}{k}
A
E
=
k
A
B
.
F
F
F
is drawn on
C
D
CD
C
D
such that
C
F
=
C
D
k
CF = \frac{CD}{k}
CF
=
k
C
D
.
G
G
G
is drawn on
B
C
BC
BC
such that
B
G
=
B
C
k
BG = \frac{BC}{k}
BG
=
k
BC
.
H
H
H
is drawn on
A
D
AD
A
D
such that
D
H
=
A
D
k
DH = \frac{AD}{k}
DH
=
k
A
D
. Line
C
E
CE
CE
intersects
B
H
BH
B
H
at
M
M
M
, and
D
G
DG
D
G
at
N
N
N
. Line
A
F
AF
A
F
intersects
D
G
DG
D
G
at
P
P
P
, and
B
H
BH
B
H
at
Q
Q
Q
. If
S
1
S_1
S
1
is the area of quadrilateral
M
N
P
Q
MNPQ
MNPQ
, find
S
1
S
\frac{S_1}{S}
S
S
1
. p5. Let
ϕ
\phi
ϕ
be the Euler totient function. What is the sum of all
n
n
n
for which
n
ϕ
(
n
)
\frac{n}{\phi(n)}
ϕ
(
n
)
n
is maximal for
1
≤
n
≤
500
1 \le n \le 500
1
≤
n
≤
500
? p6. Link starts at the top left corner of an
12
×
12
12 \times 12
12
×
12
grid and wants to reach the bottom right corner. He can only move down or right. A turn is defined a down move immediately followed by a right move, or a right move immediately followed by a down move. Given that he makes exactly
6
6
6
turns, in how many ways can he reach his destination? PS. You had better use hide for answers.
round 3
1
Hide problems
2012 BMT Tournament Round 3 - Berkley Math Tournament
p1. Let
A
(
S
)
A(S)
A
(
S
)
denote the average value of a set
S
S
S
. Let
T
T
T
be the set of all subsets of the set
{
1
,
2
,
3
,
4
,
.
.
.
,
2012
}
\{1, 2, 3, 4, ... , 2012\}
{
1
,
2
,
3
,
4
,
...
,
2012
}
, and let
R
R
R
be
{
A
(
K
)
∣
K
∈
T
}
\{A(K)|K \in T \}
{
A
(
K
)
∣
K
∈
T
}
. Compute
A
(
R
)
A(R)
A
(
R
)
. p2. Consider the minute and hour hands of the Campanile, our clock tower. During one single day (
12
:
00
12:00
12
:
00
AM -
12
:
00
12:00
12
:
00
AM), how many times will the minute and hour hands form a right-angle at the center of the clock face? p3. In a regular deck of
52
52
52
face-down cards, Billy flips
18
18
18
face-up and shuffles them back into the deck. Before giving the deck to Penny, Billy tells her how many cards he has flipped over, and blindfolds her so she can’t see the deck and determine where the face-up cards are. Once Penny is given the deck, it is her job to split the deck into two piles so that both piles contain the same number of face-up cards. Assuming that she knows how to do this, how many cards should be in each pile when he is done? p4. The roots of the equation
x
3
+
a
x
2
+
b
x
+
c
=
0
x^3 + ax^2 + bx + c = 0
x
3
+
a
x
2
+
b
x
+
c
=
0
are three consecutive integers. Find the maximum value of
a
2
b
+
1
\frac{a^2}{b+1}
b
+
1
a
2
. p5. Oski has a bag initially filled with one blue ball and one gold ball. He plays the following game: first, he removes a ball from the bag. If the ball is blue, he will put another blue ball in the bag with probability
1
437
\frac{1}{437}
437
1
and a gold ball in the bag the rest of the time. If the ball is gold, he will put another gold ball in the bag with probability
1
437
\frac{1}{437}
437
1
and a blue ball in the bag the rest of the time. In both cases, he will put the ball he drew back into the bag. Calculate the expected number of blue balls after
525600
525600
525600
iterations of this game. p6. Circles
A
A
A
and
B
B
B
intersect at points
C
C
C
and
D
D
D
. Line
A
C
AC
A
C
and circle
B
B
B
meet at
E
E
E
, line
B
D
BD
B
D
and circle
A
A
A
meet at
F
F
F
, and lines
E
F
EF
EF
and
A
B
AB
A
B
meet at
G
G
G
. If
A
B
=
10
AB = 10
A
B
=
10
,
E
F
=
4
EF = 4
EF
=
4
,
F
G
=
8
FG = 8
FG
=
8
, find
B
G
BG
BG
. PS. You had better use hide for answers.
round 2
1
Hide problems
2012 BMT Tournament Round 2 - Berkley Math Tournament
p1.
4
4
4
balls are distributed uniformly at random among
6
6
6
bins. What is the expected number of empty bins? p2. Compute
(
150
20
)
{150 \choose 20 }
(
20
150
)
(mod
221
221
221
). p3. On the right triangle
A
B
C
ABC
A
BC
, with right angle at
B
B
B
, the altitude
B
D
BD
B
D
is drawn.
E
E
E
is drawn on
B
C
BC
BC
such that AE bisects angle
B
A
C
BAC
B
A
C
and F is drawn on
A
C
AC
A
C
such that
B
F
BF
BF
bisects angle
C
B
D
CBD
CB
D
. Let the intersection of
A
E
AE
A
E
and
B
F
BF
BF
be
G
G
G
. Given that
A
B
=
15
AB = 15
A
B
=
15
,
B
C
=
20
BC = 20
BC
=
20
,
A
C
=
25
AC = 25
A
C
=
25
, find
B
G
G
F
\frac{BG}{GF}
GF
BG
. p4. What is the largest integer
n
n
n
so that
n
2
−
2012
n
+
7
\frac{n^2-2012}{n+7}
n
+
7
n
2
−
2012
is also an integer? p5. What is the side length of the largest equilateral triangle that can be inscribed in a regular pentagon with side length
1
1
1
? p6. Inside a LilacBall, you can find one of
7
7
7
different notes, each equally likely. Delcatty must collect all
7
7
7
notes in order to restore harmony and save Kanto from eternal darkness. What is the expected number of LilacBalls she must open in order to do so? PS. You had better use hide for answers.
round 1
1
Hide problems
2012 BMT Tournament Round 1 - Berkley Math Tournament
p1. Find all prime factors of
8051
8051
8051
. p2. Simplify
[
log
x
y
z
(
x
z
)
]
[
1
+
log
x
y
+
log
x
z
]
,
[\log_{xyz}(x^z)][1 + \log_x y + \log_x z],
[
lo
g
x
yz
(
x
z
)]
[
1
+
lo
g
x
y
+
lo
g
x
z
]
,
where
x
=
628
x = 628
x
=
628
,
y
=
233
y = 233
y
=
233
,
z
=
340
z = 340
z
=
340
. p3. In prokaryotes, translation of mRNA messages into proteins is most often initiated at start codons on the mRNA having the sequence AUG. Assume that the mRNA is single-stranded and consists of a sequence of bases, each described by a single letter A,C,U, or G. Consider the set of all pieces of bacterial mRNA six bases in length. How many such mRNA sequences have either no A’s or no U’s? p4. What is the smallest positive
n
n
n
so that
1
7
n
+
n
17^n + n
1
7
n
+
n
is divisible by
29
29
29
? p5. The legs of the right triangle shown below have length
a
=
255
a = 255
a
=
255
and
b
=
32
b = 32
b
=
32
. Find the area of the smaller rectangle (the one labeled
R
R
R
). https://cdn.artofproblemsolving.com/attachments/c/d/566f2ce631187684622dfb43f36c7e759e2f34.png p6. A
3
3
3
dimensional cube contains ”cubes” of smaller dimensions, ie: faces (
2
2
2
-cubes),edges (
1
1
1
-cubes), and vertices (
0
0
0
-cubes). How many 3-cubes are in a
5
5
5
-cube? PS. You had better use hide for answers.
10
2
Hide problems
Spring Round (2012) #10
You are at one vertex of a equilateral triangle with side length
1
1
1
. All of the edges of the equilateral triangle will reflect the laser beam perfectly (angle of incidence is equal to angle of reflection). Given that the laser beam bounces off exactly
137
137
137
edges and returns to the original vertex without touching any other vertices, let
M
M
M
be the maximum possible distance the beam could have traveled, and
m
m
m
be the minimum possible distance the beam could have traveled. Find
M
2
−
m
2
M^2 - m^2
M
2
−
m
2
.
2012 BMT Team 10
Suppose that
728
728
728
coins are set on a table, all facing heads up at first. For each iteration, we randomly choose
314
314
314
coins and flip them (from heads to tails or vice versa). Let
a
/
b
a/b
a
/
b
be the expected number of heads after we finish
4001
4001
4001
iterations, where
a
a
a
and
b
b
b
are relatively prime. Find
a
+
b
a + b
a
+
b
mod
10000
10000
10000
.
9
2
Hide problems
Spring Round (2012) #9
Bowling Pins is a game played between two players in the following way: We start with
14
14
14
bowling pins in a line: X X X X X X X X X X X X X X Players alternate turns. On each turn, the player can knock down one, two or three consecutive pins at a time. For example:Jing Jing bowls: X X \:\:\:\: X X X X X X X X X X Soumya bowls: X X \:\:\:\: X X X X X X X X X Jing Jing bowls again: X X \:\:\:\: X X X X X \:\: X The player who knocks down the last pin wins. In the above game, it is Soumya’s turn. If he plays perfectly from here, he has a winning strategy (In fact, he has four different winning moves.) Imagine it’s Jing Jing’s turn to play and the game looks as follows X \:\: X\dots with 1 X on the left and a string of
k
k
k
consecutive X’s on the right. For what values of
k
k
k
from
1
1
1
to
10
10
10
does she have a winning strategy?
2012 BMT Team 9
A permutation of a set is a bijection from the set to itself. For example, if
σ
\sigma
σ
is the permutation
17
↦
3
1 7\mapsto 3
17
↦
3
,
2
↦
1
2 \mapsto 1
2
↦
1
, and
3
↦
2
3 \mapsto 2
3
↦
2
, and we apply it to the ordered triplet
(
1
,
2
,
3
)
(1, 2, 3)
(
1
,
2
,
3
)
, we get the reordered triplet
(
3
,
1
,
2
)
(3, 1, 2)
(
3
,
1
,
2
)
. Let
σ
\sigma
σ
be a permutation of the set
{
1
,
.
.
.
,
n
}
\{1, ... , n\}
{
1
,
...
,
n
}
. Let
θ
k
(
m
)
=
{
m
+
1
for
m
<
k
1
for
m
=
k
m
for
m
>
k
\theta_k(m) = \begin{cases} m + 1 & \text{for} \,\, m < k\\ 1 & \text{for} \,\, m = k\\ m & \text{for} \,\, m > k\end{cases}
θ
k
(
m
)
=
⎩
⎨
⎧
m
+
1
1
m
for
m
<
k
for
m
=
k
for
m
>
k
Call a finite sequence
{
a
i
}
i
=
1
j
\{a_i\}^{j}_{i=1}
{
a
i
}
i
=
1
j
a disentanglement of
σ
\sigma
σ
if
θ
a
j
∘
.
.
.
∘
θ
a
j
∘
σ
\theta_{a_j} \circ ...\circ \theta_{a_j} \circ \sigma
θ
a
j
∘
...
∘
θ
a
j
∘
σ
is the identity permutation. For example, when
σ
=
(
3
,
2
,
1
)
\sigma = (3, 2, 1)
σ
=
(
3
,
2
,
1
)
, then
{
2
,
3
}
\{2, 3\}
{
2
,
3
}
is a disentaglement of
σ
\sigma
σ
. Let
f
(
σ
)
f(\sigma)
f
(
σ
)
denote the minimum number
k
k
k
such that there is a disentanglement of
σ
\sigma
σ
of length
k
k
k
. Let
g
(
n
)
g(n)
g
(
n
)
be the expected value for
f
(
σ
)
f(\sigma)
f
(
σ
)
if
σ
\sigma
σ
is a random permutation of
{
1
,
.
.
.
,
n
}
\{1, ... , n\}
{
1
,
...
,
n
}
. What is
g
(
6
)
g(6)
g
(
6
)
?
8
2
Hide problems
Spring Round (2012) #8
You are tossing an unbiased coin. The last
28
28
28
consecutive flips have all resulted in heads. Let
x
x
x
be the expected number of additional tosses you must make before you get
60
60
60
consecutive heads. Find the sum of all distinct prime factors in
x
x
x
.
2012 BMT Team 8
Let
ϕ
\phi
ϕ
be the Euler totient function. Let
ϕ
k
(
n
)
=
(
ϕ
∘
.
.
.
∘
ϕ
⏟
k
)
(
n
)
\phi^k (n) = (\underbrace{\phi \circ ... \circ \phi}_{k})(n)
ϕ
k
(
n
)
=
(
k
ϕ
∘
...
∘
ϕ
)
(
n
)
be
ϕ
\phi
ϕ
composed with itself
k
k
k
times. Define \theta (n) = min \{k \in N | \phi^k (n)=1 \} . For example,
ϕ
1
(
13
)
=
ϕ
(
13
)
=
12
\phi^1 (13) = \phi(13) = 12
ϕ
1
(
13
)
=
ϕ
(
13
)
=
12
ϕ
2
(
13
)
=
ϕ
(
ϕ
(
13
)
)
=
4
\phi^2 (13) = \phi (\phi (13)) = 4
ϕ
2
(
13
)
=
ϕ
(
ϕ
(
13
))
=
4
ϕ
3
(
13
)
=
ϕ
(
ϕ
(
ϕ
(
13
)
)
)
=
2
\phi^3 (13) = \phi(\phi(\phi(13))) = 2
ϕ
3
(
13
)
=
ϕ
(
ϕ
(
ϕ
(
13
)))
=
2
ϕ
4
(
13
)
=
ϕ
(
ϕ
(
ϕ
(
ϕ
(
13
)
)
)
)
=
1
\phi^4 (13) = \phi(\phi(\phi(\phi(13)))) = 1
ϕ
4
(
13
)
=
ϕ
(
ϕ
(
ϕ
(
ϕ
(
13
))))
=
1
so
θ
(
13
)
=
4
\theta (13) = 4
θ
(
13
)
=
4
. Let
f
(
r
)
=
θ
(
1
3
r
)
f(r) = \theta (13^r)
f
(
r
)
=
θ
(
1
3
r
)
. Determine
f
(
2012
)
f(2012)
f
(
2012
)
.
7
2
Hide problems
Spring Round (2012) #7
Let
a
a
a
,
b
b
b
,
c
c
c
,
d
d
d
,
(
a
+
b
+
c
+
18
+
d
)
(a + b + c + 18 + d)
(
a
+
b
+
c
+
18
+
d
)
,
(
a
+
b
+
c
+
18
−
d
)
(a + b + c + 18 - d)
(
a
+
b
+
c
+
18
−
d
)
,
(
b
+
c
)
(b + c)
(
b
+
c
)
, and
(
c
+
d
)
(c + d)
(
c
+
d
)
be distinct prime numbers such that
a
+
b
+
c
=
2010
a + b + c = 2010
a
+
b
+
c
=
2010
,
a
a
a
,
b
b
b
,
c
c
c
,
d
≠
3
d \neq 3
d
=
3
, and
d
≤
50
d \le 50
d
≤
50
. Find the maximum value of the difference between two of these prime numbers.
2012 BMT Team 7
Suppose Bob begins walking at a constant speed from point
N
N
N
to point
S
S
S
along the path indicated by the following figure. https://cdn.artofproblemsolving.com/attachments/6/2/f5819267020f2bd38e52c6e873a2cf91ce8c49.png After Bob has walked a distance of
x
x
x
, Alice begins walking at point
N
N
N
, heading towards point
S
S
S
along the same path. Alice walks
1.28
1.28
1.28
times as fast as Bob when they are on the same line segment and
1.06
1.06
1.06
times as fast as Bob otherwise. For what value of
x
x
x
do Alice and Bob meet at point
S
S
S
?
6
2
Hide problems
Spring Round (2012) #6
Let
ABCD
\text{ABCD}
ABCD
be a cyclic quadrilateral, with
AB
=
7
\text{AB} = 7
AB
=
7
,
BC
=
11
\text{BC} = 11
BC
=
11
,
CD
=
13
\text{CD} = 13
CD
=
13
, and
DA
=
17
\text{DA} = 17
DA
=
17
. Let the incircle of
ABD
\text{ABD}
ABD
hit
BD
\text{BD}
BD
at
R
\text{R}
R
and the incircle of
CBD
\text{CBD}
CBD
hit
BD
\text{BD}
BD
at
S
\text{S}
S
. What is
RS
\text{RS}
RS
?
(A(s)+A(t))/(A(a)+A(b)), areas of circles 2012 BMT Team 6
A circle with diameter
A
B
AB
A
B
is drawn, and the point
P
P
P
is chosen on segment
A
B
AB
A
B
so that
A
P
A
B
=
1
42
\frac{AP}{AB} =\frac{1}{42}
A
B
A
P
=
42
1
. Two new circles
a
a
a
and
b
b
b
are drawn with diameters
A
P
AP
A
P
and
P
B
PB
PB
respectively. The perpendicular line to
A
B
AB
A
B
passing through
P
P
P
intersects the circle twice at points
S
S
S
and
T
T
T
. Two more circles
s
s
s
and
t
t
t
are drawn with diameters
S
P
SP
SP
and
S
T
ST
ST
respectively. For any circle
ω
\omega
ω
let
A
(
ω
)
A(\omega)
A
(
ω
)
denote the area of the circle. What is
A
(
s
)
+
A
(
t
)
A
(
a
)
+
A
(
b
)
\frac{A(s)+A(t)}{A(a)+A(b)}
A
(
a
)
+
A
(
b
)
A
(
s
)
+
A
(
t
)
?
5
2
Hide problems
Spring Round (2012) #5
Let
a
↑
↑
b
=
a
a
a
…
a
a
{ a\uparrow\uparrow b = {{{{{a^{a}}^a}^{\dots}}}^{a}}^{a}}
a
↑↑
b
=
a
a
a
…
a
a
, where there are
b
b
b
a's in total. That is
a
↑
↑
b
a\uparrow\uparrow b
a
↑↑
b
is given by the recurrence
a
↑
↑
b
=
{
a
b
=
1
a
a
↑
↑
(
b
−
1
)
b
≥
2
a\uparrow\uparrow b = \begin{cases} a & b=1\\ a^{a\uparrow\uparrow (b-1)} & b\ge2\end{cases}
a
↑↑
b
=
{
a
a
a
↑↑
(
b
−
1
)
b
=
1
b
≥
2
What is the remainder of
3
↑
↑
(
3
↑
↑
(
3
↑
↑
3
)
)
3\uparrow\uparrow( 3\uparrow\uparrow ( 3\uparrow\uparrow 3))
3
↑↑
(
3
↑↑
(
3
↑↑
3
))
when divided by
60
60
60
?
2012 BMT Team 5
Let
p
>
1
p > 1
p
>
1
be relatively prime to
10
10
10
. Let
n
n
n
be any positive number and
d
d
d
be the last digit of
n
n
n
. Define
f
(
n
)
=
⌊
n
10
⌋
+
d
⋅
m
f(n) = \lfloor \frac{n}{10} \rfloor + d \cdot m
f
(
n
)
=
⌊
10
n
⌋
+
d
⋅
m
. Then, we can call
m
m
m
a divisibility multiplier for
p
p
p
, if
f
(
n
)
f(n)
f
(
n
)
is divisible by
p
p
p
if and only if
n
n
n
is divisible by
p
p
p
. Find a divisibility multiplier for
2013
2013
2013
.
4
2
Hide problems
Spring Round (2012) #4
Tyler rolls two
4025
4025
4025
sided fair dice with sides numbered
1
,
…
,
4025
1, \dots , 4025
1
,
…
,
4025
. Given that the number on the first die is greater than or equal to the number on the second die, what is the probability that the number on the first die is less than or equal to
2012
2012
2012
?
2012 BMT Team 4
There are 1
2
2
2
people labeled
1
,
.
.
.
,
12
1, ..., 12
1
,
...
,
12
working together on
12
12
12
missions, with people
1
,
.
.
.
,
i
1, ... , i
1
,
...
,
i
working on the
i
i
i
th mission. There is exactly one spy among them. If the spy is not working on a mission, it will be a huge success, but if the spy is working on the mission, it will fail with probability
1
/
2
1/2
1/2
. Given that the first
11
11
11
missions succeed, and the
12
12
12
th mission fails, what is the probability that person
12
12
12
is the spy?
3
2
Hide problems
Sprind Round (2012) #3
Find the largest prime factor of
1
∑
n
=
1
∞
2012
n
(
n
+
1
)
(
n
+
2
)
(
n
+
3
)
…
(
n
+
2012
)
\frac{1}{\sum_{n=1}^\infty \frac{2012}{n(n+1)(n+2)(n+3)\dots(n+2012)}}
∑
n
=
1
∞
n
(
n
+
1
)
(
n
+
2
)
(
n
+
3
)
…
(
n
+
2012
)
2012
1
ratio of sums of areas of squares 2012 BMT Team 3
Let
A
B
C
ABC
A
BC
be a triangle with side lengths
A
B
=
2011
AB = 2011
A
B
=
2011
,
B
C
=
2012
BC = 2012
BC
=
2012
,
A
C
=
2013
AC = 2013
A
C
=
2013
. Create squares
S
1
=
A
B
B
′
A
′
′
S_1 =ABB'A''
S
1
=
A
B
B
′
A
′′
,
S
2
=
A
C
C
′
′
A
′
S_2 = ACC''A'
S
2
=
A
C
C
′′
A
′
, and
S
3
=
C
B
B
′
′
C
′
S_3 = CBB''C'
S
3
=
CB
B
′′
C
′
using the sides
A
B
AB
A
B
,
A
C
AC
A
C
,
B
C
BC
BC
respectively, so that the side
B
′
A
′
′
B'A''
B
′
A
′′
is on the opposite side of
A
B
AB
A
B
from
C
C
C
, and so forth. Let square
S
4
S_4
S
4
have side length
A
′
′
A
′
A''A'
A
′′
A
′
, square
S
5
S_5
S
5
have side length
C
′
′
C
′
C''C'
C
′′
C
′
, and square
S
6
S_6
S
6
have side length
B
′
′
B
′
B''B'
B
′′
B
′
. Let
A
(
S
i
)
A(S_i)
A
(
S
i
)
be the area of square
S
i
S_i
S
i
. Compute
A
(
S
4
)
+
A
(
S
5
)
+
A
(
S
6
)
A
(
S
1
)
+
A
(
S
2
)
+
A
(
S
3
)
\frac{A(S_4)+A(S_5)+A(S_6)}{A(S_1)+A(S_2)+A(S_3)}
A
(
S
1
)
+
A
(
S
2
)
+
A
(
S
3
)
A
(
S
4
)
+
A
(
S
5
)
+
A
(
S
6
)
?
2
2
Hide problems
Spring Round (2012) #2
Find the smallest number with exactly 28 divisors.
2012 BMT Team 2
Evaluate
∏
k
=
1
254
log
k
+
1
(
k
+
2
)
u
k
\prod_{k=1}^{254}\log_{k+1}(k + 2)^{u_k}
∏
k
=
1
254
lo
g
k
+
1
(
k
+
2
)
u
k
, where
u
k
=
{
−
k
if
k
is odd
1
k
−
1
if
k
is even
u_k = \begin{cases}- k & \text{if} \,\, k \,\, \text{is odd}\\ \frac{1}{k-1} & \text{if} \,\, k \,\, \text{is even} \end{cases}
u
k
=
{
−
k
k
−
1
1
if
k
is odd
if
k
is even
1
2
Hide problems
Sprind Round (2012) #1
Let
{
a
n
}
n
=
1
∞
\{a_n\}_{n=1}^\infty
{
a
n
}
n
=
1
∞
be an arithmetic progression with
a
1
>
0
a_1 > 0
a
1
>
0
and
5
⋅
a
13
=
6
⋅
a
19
5\cdot a_{13} = 6\cdot a_{19}
5
⋅
a
13
=
6
⋅
a
19
. What is the smallest integer
n
n
n
such that
a
n
<
0
a_n<0
a
n
<
0
?
2012 BMT Team 1
Let
S
S
S
be the set of all rational numbers
x
∈
[
0
,
1
]
x \in [0, 1]
x
∈
[
0
,
1
]
with repeating base
6
6
6
expansion
x
=
0.
a
1
a
2
.
.
.
a
k
‾
=
0.
a
1
a
2
.
.
.
a
k
a
1
a
2
.
.
.
a
k
.
.
.
x = 0.\overline{a_1a_2 ... a_k} = 0.a_1a_2...a_ka_1a_2...a_k...
x
=
0.
a
1
a
2
...
a
k
=
0.
a
1
a
2
...
a
k
a
1
a
2
...
a
k
...
for some finite sequence
{
a
i
}
i
=
1
k
\{a_i\}^{k}_{i=1}
{
a
i
}
i
=
1
k
of distinct nonnegative integers less than
6
6
6
. What is the sum of all numbers that can be written in this form? (Put your answer in base
10
10
10
.)