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Duke Math Meet (DMM)
2012 Duke Math Meet
2012 Duke Math Meet
Part of
Duke Math Meet (DMM)
Subcontests
(1)
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2012 DMM Tiebreaker Round - Duke Math Meet
p1. An
8
8
8
-inch by
11
11
11
-inch sheet of paper is laid flat so that the top and bottom edges are
8
8
8
inches long. The paper is then folded so that the top left corner touches the right edge. What is the minimum possible length of the fold? p2. Triangle
A
B
C
ABC
A
BC
is equilateral, with
A
B
=
6
AB = 6
A
B
=
6
. There are points
D
D
D
,
E
E
E
on segment AB (in the order
A
A
A
,
D
D
D
,
E
E
E
,
B
B
B
), points
F
F
F
,
G
G
G
on segment
B
C
BC
BC
(in the order
B
B
B
,
F
F
F
,
G
G
G
,
C
C
C
), and points
H
H
H
,
I
I
I
on segment
C
A
CA
C
A
(in the order
C
C
C
,
H
H
H
,
I
I
I
,
A
A
A
) such that
D
E
=
F
G
=
H
I
=
2
DE = F G = HI = 2
D
E
=
FG
=
H
I
=
2
. Considering all such configurations of
D
D
D
,
E
E
E
,
F
F
F
,
G
G
G
,
H
H
H
,
I
I
I
, let
A
1
A_1
A
1
be the maximum possible area of (possibly degenerate) hexagon
D
E
F
G
H
I
DEF GHI
D
EFG
H
I
and let
A
2
A_2
A
2
be the minimum possible area. Find
A
1
−
A
2
A_1 - A_2
A
1
−
A
2
. p3. Find
tan
π
7
tan
2
π
7
tan
3
π
7
\tan \frac{\pi}{7} \tan \frac{2\pi}{7} \tan \frac{3\pi}{7}
tan
7
π
tan
7
2
π
tan
7
3
π
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
2012 DMM Team Round - Duke Math Meet
p1. Let
2
k
2^k
2
k
be the largest power of
2
2
2
dividing
30
!
=
30
⋅
29
⋅
28...2
⋅
1
30! = 30 \cdot 29 \cdot 28 ... 2 \cdot 1
30
!
=
30
⋅
29
⋅
28...2
⋅
1
. Find
k
k
k
. p2. Let
d
(
n
)
d(n)
d
(
n
)
be the total number of digits needed to write all the numbers from
1
1
1
to
n
n
n
in base
10
10
10
, for example,
d
(
5
)
=
5
d(5) = 5
d
(
5
)
=
5
and
d
(
20
)
=
31
d(20) = 31
d
(
20
)
=
31
. Find
d
(
2012
)
d(2012)
d
(
2012
)
. p3. Jim and TongTong play a game. Jim flips
10
10
10
coins and TongTong flips
11
11
11
coins, whoever gets the most heads wins. If they get the same number of heads, there is a tie. What is the probability that TongTong wins? p4. There are a certain number of potatoes in a pile. When separated into mounds of three, two remain. When divided into mounds of four, three remain. When divided into mounds of five, one remain. It is clear there are at least
150
150
150
potatoes in the pile. What is the least number of potatoes there can be in the pile? p5. Call an ordered triple of sets
(
A
,
B
,
C
)
(A, B, C)
(
A
,
B
,
C
)
nice if
∣
A
∩
B
∣
=
∣
B
∩
C
∣
=
∣
C
∩
A
∣
=
2
|A \cap B| = |B \cap C| = |C \cap A| = 2
∣
A
∩
B
∣
=
∣
B
∩
C
∣
=
∣
C
∩
A
∣
=
2
and
∣
A
∩
B
∩
C
∣
=
0
|A \cap B \cap C| = 0
∣
A
∩
B
∩
C
∣
=
0
. How many ordered triples of subsets of
{
1
,
2
,
⋅
⋅
⋅
,
9
}
\{1, 2, · · · , 9\}
{
1
,
2
,⋅⋅⋅,
9
}
are nice? p6. Brett has an
n
×
n
×
n
n \times n \times n
n
×
n
×
n
cube (where
n
n
n
is an integer) which he dips into blue paint. He then cuts the cube into a bunch of
1
×
1
×
1
1 \times 1 \times 1
1
×
1
×
1
cubes, and notices that the number of un-painted cubes (which is positive) evenly divides the number of painted cubes. What is the largest possible side length of Brett’s original cube?Note that
⌊
x
⌋
\lfloor x\rfloor
⌊
x
⌋
denotes the largest integer less than or equal to
x
x
x
. p7. Choose two real numbers
x
x
x
and
y
y
y
uniformly at random from the interval
[
0
,
1
]
[0, 1]
[
0
,
1
]
. What is the probability that
x
x
x
is closer to
1
/
4
1/4
1/4
than
y
y
y
is to
1
/
2
1/2
1/2
? p8. In triangle
A
B
C
ABC
A
BC
, we have
∠
B
A
C
=
2
0
o
\angle BAC = 20^o
∠
B
A
C
=
2
0
o
and
A
B
=
A
C
AB = AC
A
B
=
A
C
.
D
D
D
is a point on segment
A
B
AB
A
B
such that
A
D
=
B
C
AD = BC
A
D
=
BC
. What is
∠
A
D
C
\angle ADC
∠
A
D
C
, in degree. p9. Let
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
be real numbers such that
a
b
+
c
+
d
=
2012
ab + c + d = 2012
ab
+
c
+
d
=
2012
,
b
c
+
d
+
a
=
2010
bc + d + a = 2010
b
c
+
d
+
a
=
2010
,
c
d
+
a
+
b
=
2013
cd + a + b = 2013
c
d
+
a
+
b
=
2013
,
d
a
+
b
+
c
=
2009
da + b + c = 2009
d
a
+
b
+
c
=
2009
. Find
d
d
d
. p10. Let
θ
∈
[
0
,
2
π
)
\theta \in [0, 2\pi)
θ
∈
[
0
,
2
π
)
such that
cos
θ
=
2
/
3
\cos \theta = 2/3
cos
θ
=
2/3
. Find
∑
n
=
0
∞
1
2
n
cos
(
n
θ
)
\sum_{n=0}^{\infty}\frac{1}{2^n}\cos(n \theta)
∑
n
=
0
∞
2
n
1
cos
(
n
θ
)
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
2012 DMM Individual Round - Duke Math Meet
p1. Vivek has three letters to send out. Unfortunately, he forgets which letter is which after sealing the envelopes and before putting on the addresses. He puts the addresses on at random sends out the letters anyways. What are the chances that none of the three recipients get their intended letter? p2. David is a horrible bowler. Luckily, Logan and Christy let him use bumpers. The bowling lane is
2
2
2
meters wide, and David's ball travels a total distance of
24
24
24
meters. How many times did David's bowling ball hit the bumpers, if he threw it from the middle of the lane at a
6
0
o
60^o
6
0
o
degree angle to the horizontal? p3. Find
gcd
(
212106
,
106212
)
\gcd \,(212106, 106212)
g
cd
(
212106
,
106212
)
. p4. Michael has two fair dice, one six-sided (with sides marked
1
1
1
through
6
6
6
) and one eight-sided (with sides marked
1
−
8
1-8
1
−
8
). Michael play a game with Alex: Alex calls out a number, and then Michael rolls the dice. If the sum of the dice is equal to Alex's number, Michael gives Alex the amount of the sum. Otherwise Alex wins nothing. What number should Alex call to maximize his expected gain of money? p5. Suppose that
x
x
x
is a real number with
log
5
sin
x
+
log
5
cos
x
=
−
1
\log_5 \sin x + \log_5 \cos x = -1
lo
g
5
sin
x
+
lo
g
5
cos
x
=
−
1
. Find
∣
sin
2
x
cos
x
+
cos
2
x
sin
x
∣
.
|\sin^2 x \cos x + \cos^2 x \sin x|.
∣
sin
2
x
cos
x
+
cos
2
x
sin
x
∣.
p6. What is the volume of the largest sphere that FIts inside a regular tetrahedron of side length
6
6
6
? p7. An ant is wandering on the edges of a cube. At every second, the ant randomly chooses one of the three edges incident at one vertex and walks along that edge, arriving at the other vertex at the end of the second. What is the probability that the ant is at its starting vertex after exactly
6
6
6
seconds? p8. Determine the smallest positive integer
k
k
k
such that there exist
m
,
n
m, n
m
,
n
non-negative integers with
m
>
1
m > 1
m
>
1
satisfying
k
=
2
2
m
+
1
−
n
2
.
k = 2^{2m+1} - n^2.
k
=
2
2
m
+
1
−
n
2
.
p9. For
A
,
B
⊂
Z
A,B \subset Z
A
,
B
⊂
Z
with
A
,
B
≠
∅
A,B \ne \emptyset
A
,
B
=
∅
, define
A
+
B
=
{
a
+
b
∣
a
∈
A
,
b
∈
B
}
A + B = \{a + b|a \in A, b \in B\}
A
+
B
=
{
a
+
b
∣
a
∈
A
,
b
∈
B
}
. Determine the least
n
n
n
such that there exist sets
A
,
B
A,B
A
,
B
with
∣
A
∣
=
∣
B
∣
=
n
|A| = |B| = n
∣
A
∣
=
∣
B
∣
=
n
and
A
+
B
=
{
0
,
1
,
2
,
.
.
.
,
2012
}
A + B = \{0, 1, 2,..., 2012\}
A
+
B
=
{
0
,
1
,
2
,
...
,
2012
}
. p10. For positive integers
n
≥
1
n \ge 1
n
≥
1
, let
τ
(
n
)
\tau (n)
τ
(
n
)
and
σ
(
n
)
\sigma (n)
σ
(
n
)
be, respectively, the number of and sum of the positive integer divisors of
n
n
n
(including
1
1
1
and
n
n
n
). For example,
τ
(
1
)
=
σ
(
1
)
=
1
\tau (1) = \sigma (1) = 1
τ
(
1
)
=
σ
(
1
)
=
1
and
τ
(
6
)
=
4
\tau (6) = 4
τ
(
6
)
=
4
,
σ
(
6
)
=
12
\sigma (6) = 12
σ
(
6
)
=
12
. Find the number of positive integers
n
≤
100
n \le 100
n
≤
100
such that
σ
(
n
)
≤
(
n
−
1
)
2
+
τ
(
n
)
n
.
\sigma (n) \le (\sqrt{n} - 1)^2 +\tau (n)\sqrt{n}.
σ
(
n
)
≤
(
n
−
1
)
2
+
τ
(
n
)
n
.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.