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2013 Harvard-MIT Mathematics Tournament
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Subcontests
(36)
36
1
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2013 HMMT Guts #36: Mathematicians Born Apart
(Mathematicians A to Z) Below are the names of 26 mathematicians, one for each letter of the alphabet. Your answer to this question should be a subset of
{
A
,
B
,
⋯
,
Z
}
\{A,B,\cdots,Z\}
{
A
,
B
,
⋯
,
Z
}
, where each letter represents the corresponding mathematician. If two mathematicians in your subset have birthdates that are within
20
20
20
years of each other, then your score is
0
0
0
. Otherwise, your score is
max
(
3
(
k
−
3
)
,
0
)
\max(3(k-3),0)
max
(
3
(
k
−
3
)
,
0
)
where
k
k
k
is the number of elements in your set.\begin{tabular}{cc}Niels Abel & Isaac Newton\\Etienne Bezout & Nicole Oresme \\ Augustin-Louis Cauchy & Blaise Pascal \\ Rene Descartes & Daniel Quillen \\ Leonhard Euler & Bernhard Riemann\\ Pierre Fatou & Jean-Pierre Serre \\ Alexander Grothendieck & Alan Turing \\ David Hilbert & Stanislaw Ulam \\ Kenkichi Iwasawa & John Venn \\ Carl Jacobi & Andrew Wiles \\ Andrey Kolmogorov & Leonardo Ximenes \\ Joseph-Louis Lagrange & Shing-Tung Yau \\ John Milnor & Ernst Zermelo\end{tabular}
35
1
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2013 HMMT Guts #35: Partitioning 2013 into Prime Numbers
Let
P
P
P
be the number of ways to partition
2013
2013
2013
into an ordered tuple of prime numbers. What is
log
2
(
P
)
\log_2 (P)
lo
g
2
(
P
)
? If your answer is
A
A
A
and the correct answer is
C
C
C
, then your score on this problem will be
⌊
125
2
(
min
(
C
A
,
A
C
)
−
3
5
)
⌋
\left\lfloor\frac{125}2\left(\min\left(\frac CA,\frac AC\right)-\frac35\right)\right\rfloor
⌊
2
125
(
min
(
A
C
,
C
A
)
−
5
3
)
⌋
or zero, whichever is larger.
34
1
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2013 HMMT Guts #34: Sums of Multiples of Powers of -1
For how many unordered sets
{
a
,
b
,
c
,
d
}
\{a,b,c,d\}
{
a
,
b
,
c
,
d
}
of positive integers, none of which exceed
168
168
168
, do there exist integers
w
,
x
,
y
,
z
w,x,y,z
w
,
x
,
y
,
z
such that
(
−
1
)
w
a
+
(
−
1
)
x
b
+
(
−
1
)
y
c
+
(
−
1
)
z
d
=
168
(-1)^wa+(-1)^xb+(-1)^yc+(-1)^zd=168
(
−
1
)
w
a
+
(
−
1
)
x
b
+
(
−
1
)
y
c
+
(
−
1
)
z
d
=
168
? If your answer is
A
A
A
and the correct answer is
C
C
C
, then your score on this problem will be
⌊
25
e
−
3
∣
C
−
A
∣
C
⌋
\left\lfloor25e^{-3\frac{|C-A|}C}\right\rfloor
⌊
25
e
−
3
C
∣
C
−
A
∣
⌋
.
33
1
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2013 HMMT Guts #33: Sums of Powers
Compute the value of
1
25
+
2
24
+
3
23
+
…
+
2
4
2
+
2
5
1
1^{25}+2^{24}+3^{23}+\ldots+24^2+25^1
1
25
+
2
24
+
3
23
+
…
+
2
4
2
+
2
5
1
. If your answer is
A
A
A
and the correct answer is
C
C
C
, then your score on this problem will be
⌊
25
min
(
(
A
C
)
2
,
(
C
A
)
2
)
⌋
\left\lfloor25\min\left(\left(\frac AC\right)^2,\left(\frac CA\right)^2\right)\right\rfloor
⌊
25
min
(
(
C
A
)
2
,
(
A
C
)
2
)
⌋
.
32
1
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2013 HMMT Guts #32: Kevin's Stone Solitare Game
For an even positive integer
n
n
n
Kevin has a tape of length
4
n
4n
4
n
with marks at
−
2
n
,
−
2
n
+
1
,
…
,
2
n
−
1
,
2
n
-2n,-2n+1,\ldots,2n-1,2n
−
2
n
,
−
2
n
+
1
,
…
,
2
n
−
1
,
2
n
. He then randomly picks
n
n
n
points in the set
−
n
,
−
n
+
1
,
−
n
+
2
,
…
,
n
−
1
,
n
-n,-n+1,-n+2,\ldots,n-1,n
−
n
,
−
n
+
1
,
−
n
+
2
,
…
,
n
−
1
,
n
and places a stone on each of these points. We call a stone 'stuck' if it is on
2
n
2n
2
n
or
−
2
n
-2n
−
2
n
, or either all the points to the right, or all the points to the left, all contain stones. Then, every minute, Kevin shifts the unstruck stones in the following manner:[*]He picks an unstuck stone uniformly at random and then flips a fair coin. [*]If the coin came up heads, he then moves that stone and every stone in the largest contiguous set containing that stone one point to the left. If the coin came up tails, he moves every stone in that set one point right instead. [*]He repeats until all the stones are stuck. Let
p
n
p_n
p
n
be the probability that at the end of the process there are exactly
k
k
k
stones in the right half. Evaluate
p
n
−
1
−
p
n
−
2
+
p
n
−
3
+
…
+
p
3
−
p
2
+
p
1
p
n
−
1
+
p
n
−
2
+
p
n
−
3
+
…
+
p
3
+
p
2
+
p
1
\dfrac{p_{n-1}-p_{n-2}+p_{n-3}+\ldots+p_3-p_2+p_1}{p_{n-1}+p_{n-2}+p_{n-3}+\ldots+p_3+p_2+p_1}
p
n
−
1
+
p
n
−
2
+
p
n
−
3
+
…
+
p
3
+
p
2
+
p
1
p
n
−
1
−
p
n
−
2
+
p
n
−
3
+
…
+
p
3
−
p
2
+
p
1
in terms of
n
n
n
.
31
1
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2013 HMMT Guts #31: Quadrilateral in Unit Circle
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral inscribed in a unit circle with center
O
O
O
. Suppose that
∠
A
O
B
=
∠
C
O
D
=
13
5
∘
\angle AOB = \angle COD = 135^\circ
∠
A
OB
=
∠
CO
D
=
13
5
∘
,
B
C
=
1
BC=1
BC
=
1
. Let
B
′
B^\prime
B
′
and
C
′
C^\prime
C
′
be the reflections of
A
A
A
across
B
O
BO
BO
and
C
O
CO
CO
respectively. Let
H
1
H_1
H
1
and
H
2
H_2
H
2
be the orthocenters of
A
B
′
C
′
AB^\prime C^\prime
A
B
′
C
′
and
B
C
D
BCD
BC
D
, respectively. If
M
M
M
is the midpoint of
O
H
1
OH_1
O
H
1
, and
O
′
O^\prime
O
′
is the reflection of
O
O
O
about the midpoint of
M
H
2
MH_2
M
H
2
, compute
O
O
′
OO^\prime
O
O
′
.
30
1
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2013 HMMT Guts #30: Solutions to LCM Equation
How many positive integers
k
k
k
are there such that
k
2013
(
a
+
b
)
=
l
c
m
(
a
,
b
)
\dfrac k{2013}(a+b)=lcm(a,b)
2013
k
(
a
+
b
)
=
l
c
m
(
a
,
b
)
has a solution in positive integers
(
a
,
b
)
(a,b)
(
a
,
b
)
?
29
1
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2013 HMMT Guts #29: (Almost) Always Intersecting Sets
Let
A
1
,
A
2
,
…
,
A
m
A_1,A_2,\ldots,A_m
A
1
,
A
2
,
…
,
A
m
be finite sets of size
2012
2012
2012
and let
B
1
,
B
2
,
…
,
B
m
B_1,B_2,\ldots,B_m
B
1
,
B
2
,
…
,
B
m
be finite sets of size
2013
2013
2013
such that
A
i
∩
B
j
=
∅
A_i\cap B_j=\emptyset
A
i
∩
B
j
=
∅
if and only if
i
=
j
i=j
i
=
j
. Find the maximum value of
m
m
m
.
28
1
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2013 HMMT Guts #28: Sum of Sums of Series
Let
z
0
+
z
1
+
z
2
+
⋯
z_0+z_1+z_2+\cdots
z
0
+
z
1
+
z
2
+
⋯
be an infinite complex geometric series such that
z
0
=
1
z_0=1
z
0
=
1
and
z
2013
=
1
201
3
2013
z_{2013}=\dfrac 1{2013^{2013}}
z
2013
=
201
3
2013
1
. Find the sum of all possible sums of this series.
27
1
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2013 HMMT Guts #27: Intersection of Hypercube and Hyperplane
Let
W
W
W
be the hypercube
{
(
x
1
,
x
2
,
x
3
,
x
4
)
∣
0
≤
x
1
,
x
2
,
x
3
,
x
4
≤
1
}
\{(x_1,x_2,x_3,x_4)\,|\,0\leq x_1,x_2,x_3,x_4\leq 1\}
{(
x
1
,
x
2
,
x
3
,
x
4
)
∣
0
≤
x
1
,
x
2
,
x
3
,
x
4
≤
1
}
. The intersection of
W
W
W
and a hyperplane parallel to
x
1
+
x
2
+
x
3
+
x
4
=
0
x_1+x_2+x_3+x_4=0
x
1
+
x
2
+
x
3
+
x
4
=
0
is a non-degenerate
3
3
3
-dimensional polyhedron. What is the maximum number of faces of this polyhedron?
26
1
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2013 HMMT Guts #26: Altitudes form Triangles, too!
Triangle
A
B
C
ABC
A
BC
has perimeter
1
1
1
. Its three altitudes form the side lengths of a triangle. Find the set of all possible values of
min
(
A
B
,
B
C
,
C
A
)
\min(AB,BC,CA)
min
(
A
B
,
BC
,
C
A
)
.
25
1
Hide problems
2013 HMMT Guts #25: Sequence of Complex Numbers
The sequence
(
z
n
)
(z_n)
(
z
n
)
of complex numbers satisfies the following properties:[*]
z
1
z_1
z
1
and
z
2
z_2
z
2
are not real. [*]
z
n
+
2
=
z
n
+
1
2
z
n
z_{n+2}=z_{n+1}^2z_n
z
n
+
2
=
z
n
+
1
2
z
n
for all integers
n
≥
1
n\geq 1
n
≥
1
. [*]
z
n
+
3
z
n
2
\dfrac{z_{n+3}}{z_n^2}
z
n
2
z
n
+
3
is real for all integers
n
≥
1
n\geq 1
n
≥
1
. [*]
∣
z
3
z
4
∣
=
∣
z
4
z
5
∣
=
2
\left|\dfrac{z_3}{z_4}\right|=\left|\dfrac{z_4}{z_5}\right|=2
z
4
z
3
=
z
5
z
4
=
2
. Find the product of all possible values of
z
1
z_1
z
1
.
24
1
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2013 HMMT Guts #24: Function in terms of Distance
Given a point
p
p
p
and a line segment
l
l
l
, let
d
(
p
,
l
)
d(p,l)
d
(
p
,
l
)
be the distance between them. Let
A
A
A
,
B
B
B
, and
C
C
C
be points in the plane such that
A
B
=
6
AB=6
A
B
=
6
,
B
C
=
8
BC=8
BC
=
8
,
A
C
=
10
AC=10
A
C
=
10
. What is the area of the region in the
(
x
,
y
)
(x,y)
(
x
,
y
)
-plane formed by the ordered pairs
(
x
,
y
)
(x,y)
(
x
,
y
)
such that there exists a point
P
P
P
inside triangle
A
B
C
ABC
A
BC
with
d
(
P
,
A
B
)
+
x
=
d
(
P
,
B
C
)
+
y
=
d
(
P
,
A
C
)
?
d(P,AB)+x=d(P,BC)+y=d(P,AC)?
d
(
P
,
A
B
)
+
x
=
d
(
P
,
BC
)
+
y
=
d
(
P
,
A
C
)?
23
1
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2013 HMMT Guts #23: Concurrent Lines in a Parallelogram
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram with
A
B
=
8
AB=8
A
B
=
8
,
A
D
=
11
AD=11
A
D
=
11
, and
∠
B
A
D
=
6
0
∘
\angle BAD=60^\circ
∠
B
A
D
=
6
0
∘
. Let
X
X
X
be on segment
C
D
CD
C
D
with
C
X
/
X
D
=
1
/
3
CX/XD=1/3
CX
/
X
D
=
1/3
and
Y
Y
Y
be on segment
A
D
AD
A
D
with
A
Y
/
Y
D
=
1
/
2
AY/YD=1/2
A
Y
/
Y
D
=
1/2
. Let
Z
Z
Z
be on segment
A
B
AB
A
B
such that
A
X
AX
A
X
,
B
Y
BY
B
Y
, and
D
Z
DZ
D
Z
are concurrent. Determine the area of triangle
X
Y
Z
XYZ
X
Y
Z
.
22
1
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2013 HMMT Guts #22: Guessing the Cards
Sherry and Val are playing a game. Sherry has a deck containing
2011
2011
2011
red cards and
2012
2012
2012
black cards, shuffled randomly. Sherry flips these cards over one at a time, and before she flips each card over, Val guesses whether it is red or black. If Val guesses correctly, she wins
1
1
1
dollar; otherwise, she loses
1
1
1
dollar. In addition, Val must guess red exactly
2011
2011
2011
times. If Val plays optimally, what is her expected profit from this game?
21
1
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2013 HMMT Guts #21: Summation of Powers of 3
Find the number of positive integers
j
≤
3
2013
j\leq 3^{2013}
j
≤
3
2013
such that
j
=
∑
k
=
0
m
(
(
−
1
)
k
⋅
3
a
k
)
j=\sum_{k=0}^m\left((-1)^k\cdot 3^{a_k}\right)
j
=
k
=
0
∑
m
(
(
−
1
)
k
⋅
3
a
k
)
for some strictly increasing sequence of nonnegative integers
{
a
k
}
\{a_k\}
{
a
k
}
. For example, we may write
3
=
3
1
3=3^1
3
=
3
1
and
55
=
3
0
−
3
3
+
3
4
55=3^0-3^3+3^4
55
=
3
0
−
3
3
+
3
4
, but
4
4
4
cannot be written in this form.
20
1
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2013 HMMT Guts #20: Polynomial Transformation of Roots
The polynomial
f
(
x
)
=
x
3
−
3
x
2
−
4
x
+
4
f(x)=x^3-3x^2-4x+4
f
(
x
)
=
x
3
−
3
x
2
−
4
x
+
4
has three real roots
r
1
r_1
r
1
,
r
2
r_2
r
2
, and
r
3
r_3
r
3
. Let
g
(
x
)
=
x
3
+
a
x
2
+
b
x
+
c
g(x)=x^3+ax^2+bx+c
g
(
x
)
=
x
3
+
a
x
2
+
b
x
+
c
be the polynomial which has roots
s
1
s_1
s
1
,
s
2
s_2
s
2
, and
s
3
s_3
s
3
, where
s
1
=
r
1
+
r
2
z
+
r
3
z
2
s_1=r_1+r_2z+r_3z^2
s
1
=
r
1
+
r
2
z
+
r
3
z
2
,
s
2
=
r
1
z
+
r
2
z
2
+
r
3
s_2=r_1z+r_2z^2+r_3
s
2
=
r
1
z
+
r
2
z
2
+
r
3
,
s
3
=
r
1
z
2
+
r
2
+
r
3
z
s_3=r_1z^2+r_2+r_3z
s
3
=
r
1
z
2
+
r
2
+
r
3
z
, and
z
=
−
1
+
i
3
2
z=\frac{-1+i\sqrt3}2
z
=
2
−
1
+
i
3
. Find the real part of the sum of the coefficients of
g
(
x
)
g(x)
g
(
x
)
.
19
1
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2013 HMMT Guts #19: Circumscribed Triangles
An isosceles trapezoid
A
B
C
D
ABCD
A
BC
D
with bases
A
B
AB
A
B
and
C
D
CD
C
D
has
A
B
=
13
AB=13
A
B
=
13
,
C
D
=
17
CD=17
C
D
=
17
, and height
3
3
3
. Let
E
E
E
be the intersection of
A
C
AC
A
C
and
B
D
BD
B
D
. Circles
Ω
\Omega
Ω
and
ω
\omega
ω
are circumscribed about triangles
A
B
E
ABE
A
BE
and
C
D
E
CDE
C
D
E
. Compute the sum of the radii of
Ω
\Omega
Ω
and
ω
\omega
ω
.
18
1
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2013 HMMT Guts #18: Multi-Base Recursion
Define the sequence of positive integers
{
a
n
}
\{a_n\}
{
a
n
}
as follows. Let
a
1
=
1
a_1=1
a
1
=
1
,
a
2
=
3
a_2=3
a
2
=
3
, and for each
n
>
2
n>2
n
>
2
, let
a
n
a_n
a
n
be the result of expressing
a
n
−
1
a_{n-1}
a
n
−
1
in base
n
−
1
n-1
n
−
1
, then reading the resulting numeral in base
n
n
n
, then adding
2
2
2
(in base
n
n
n
). For example,
a
2
=
3
10
=
1
1
2
a_2=3_{10}=11_2
a
2
=
3
10
=
1
1
2
, so
a
3
=
1
1
3
+
2
3
=
6
10
a_3=11_3+2_3=6_{10}
a
3
=
1
1
3
+
2
3
=
6
10
. Express
a
2013
a_{2013}
a
2013
in base
10
10
10
.
17
1
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2013 HMMT Guts #17: Medians Along Lines
The lines
y
=
x
y=x
y
=
x
,
y
=
2
x
y=2x
y
=
2
x
, and
y
=
3
x
y=3x
y
=
3
x
are the three medians of a triangle with perimeter
1
1
1
. Find the length of the longest side of the triangle.
16
1
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2013 HMMT Guts #16: Rolling a Ball out of Boredom
The walls of a room are in the shape of a triangle
A
B
C
ABC
A
BC
with
∠
A
B
C
=
9
0
∘
\angle ABC = 90^\circ
∠
A
BC
=
9
0
∘
,
∠
B
A
C
=
6
0
∘
\angle BAC = 60^\circ
∠
B
A
C
=
6
0
∘
, and
A
B
=
6
AB=6
A
B
=
6
. Chong stands at the midpoint of
B
C
BC
BC
and rolls a ball toward
A
B
AB
A
B
. Suppose that the ball bounces off
A
B
AB
A
B
, then
A
C
AC
A
C
, then returns exactly to Chong. Find the length of the path of the ball.
15
1
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2013 HMMT Guts #15: Expected Amount of Tenus
Tim and Allen are playing a match of tenus. In a match of tenus, the two players play a series of games, each of which is won by one of the two players. The match ends when one player has won exactly two more games than the other player, at which point the player who has won more games wins the match. In odd-numbered games, Tim wins with probability
3
/
4
3/4
3/4
, and in the even-numbered games, Allen wins with probability
3
/
4
3/4
3/4
. What is the expected number of games in a match?
14
1
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2013 HMMT Guts #14: Angle Bisectors Galore
Consider triangle
A
B
C
ABC
A
BC
with
∠
A
=
2
∠
B
\angle A=2\angle B
∠
A
=
2∠
B
. The angle bisectors from
A
A
A
and
C
C
C
intersect at
D
D
D
, and the angle bisector from
C
C
C
intersects
A
B
‾
\overline{AB}
A
B
at
E
E
E
. If
D
E
D
C
=
1
3
\dfrac{DE}{DC}=\dfrac13
D
C
D
E
=
3
1
, compute
A
B
A
C
\dfrac{AB}{AC}
A
C
A
B
.
13
1
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2013 HMMT Guts #13: Smallest Solution to Inequality
Find the smallest positive integer
n
n
n
such that
5
n
+
1
+
2
n
+
1
5
n
+
2
n
>
4.99
\dfrac{5^{n+1}+2^{n+1}}{5^n+2^n}>4.99
5
n
+
2
n
5
n
+
1
+
2
n
+
1
>
4.99
.
12
1
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2013 HMMT Guts #12: k^k ends in a 1
For how many integers
1
≤
k
≤
2013
1\leq k\leq 2013
1
≤
k
≤
2013
does the decimal representation of
k
k
k^k
k
k
end with a
1
1
1
?
11
1
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2013 HMMT Guts #11: Prime Factors of Huge Number
Compute the prime factorization of
1007021035035021007001
1007021035035021007001
1007021035035021007001
. (You should write your answer in the form
p
1
e
1
p
2
e
2
…
p
k
e
k
p_1^{e_1}p_2^{e_2}\ldots p_k^{e_k}
p
1
e
1
p
2
e
2
…
p
k
e
k
where
p
1
,
…
,
p
k
p_1,\ldots,p_k
p
1
,
…
,
p
k
are distinct prime numbers and
e
1
,
…
,
e
k
e_1,\ldots,e_k
e
1
,
…
,
e
k
are positive integers.)
10
4
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9
4
Show problems
8
4
Show problems
7
4
Show problems
6
4
Show problems
5
4
Show problems
4
4
Show problems
3
4
Show problems
2
4
Show problems
1
4
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