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Harvard-MIT Mathematics Tournament
2014 Harvard-MIT Mathematics Tournament
2014 Harvard-MIT Mathematics Tournament
Part of
Harvard-MIT Mathematics Tournament
Subcontests
(32)
32
1
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2014 Guts #32: A Classic System
Find all ordered pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
of complex numbers with
a
2
+
b
2
≠
0
a^2+b^2\neq 0
a
2
+
b
2
=
0
,
a
+
10
b
a
2
+
b
2
=
5
a+\tfrac{10b}{a^2+b^2}=5
a
+
a
2
+
b
2
10
b
=
5
, and
b
+
10
a
a
2
+
b
2
=
4
b+\tfrac{10a}{a^2+b^2}=4
b
+
a
2
+
b
2
10
a
=
4
.
31
1
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2014 Guts #31: Sum of 2014th Powers of Cosines
Compute
∑
k
=
1
1007
(
cos
(
π
k
1007
)
)
2014
.
\sum_{k=1}^{1007}\left(\cos\left(\dfrac{\pi k}{1007}\right)\right)^{2014}.
k
=
1
∑
1007
(
cos
(
1007
πk
)
)
2014
.
30
1
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2014 Guts #30: OI parallel to side
Let
A
B
C
ABC
A
BC
be a triangle with circumcenter
O
O
O
, incenter
I
I
I
,
∠
B
=
4
5
∘
\angle B=45^\circ
∠
B
=
4
5
∘
, and
O
I
∥
B
C
OI\parallel BC
O
I
∥
BC
. Find
cos
∠
C
\cos\angle C
cos
∠
C
.
29
1
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2014 Guts #29: Coloring the Unit Interval Black
Natalie has a copy of the unit interval
[
0
,
1
]
[0,1]
[
0
,
1
]
that is colored white. She also has a black marker, and she colors the interval in the following manner: at each step, she selects a value
x
∈
[
0
,
1
]
x\in [0,1]
x
∈
[
0
,
1
]
uniformly at random, and (a) If
x
≤
1
2
x\leq\tfrac12
x
≤
2
1
she colors the interval
[
x
,
x
+
1
2
]
[x,x+\tfrac12]
[
x
,
x
+
2
1
]
with her marker.(b) If
x
>
1
2
x>\tfrac12
x
>
2
1
she colors the intervals
[
x
,
1
]
[x,1]
[
x
,
1
]
and
[
0
,
x
−
1
2
]
[0,x-\tfrac12]
[
0
,
x
−
2
1
]
with her marker.What is the expected value of the number of steps Natalie will need to color the entire interval black?
28
1
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2014 Guts #28: Extension of Standard Polynomial Problem
Let
f
(
n
)
f(n)
f
(
n
)
and
g
(
n
)
g(n)
g
(
n
)
be polynomials of degree
2014
2014
2014
such that
f
(
n
)
+
(
−
1
)
n
g
(
n
)
=
2
n
f(n)+(-1)^ng(n)=2^n
f
(
n
)
+
(
−
1
)
n
g
(
n
)
=
2
n
for
n
=
1
,
2
,
…
,
4030
n=1,2,\ldots,4030
n
=
1
,
2
,
…
,
4030
. Find the coefficient of
x
2014
x^{2014}
x
2014
in
g
(
x
)
g(x)
g
(
x
)
.
27
1
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2014 Guts #27: Maximum Value of Sum of Minima
Suppose that
(
a
1
,
…
,
a
20
)
(a_1,\ldots,a_{20})
(
a
1
,
…
,
a
20
)
and
(
b
1
,
…
,
b
20
)
(b_1,\ldots,b_{20})
(
b
1
,
…
,
b
20
)
are two sequences of integers such that the sequence
(
a
1
,
…
,
a
20
,
b
1
,
…
,
b
20
)
(a_1,\ldots,a_{20},b_1,\ldots,b_{20})
(
a
1
,
…
,
a
20
,
b
1
,
…
,
b
20
)
contains each of the numbers
1
,
…
,
40
1,\ldots,40
1
,
…
,
40
exactly once. What is the maximum possible value of the sum
∑
i
=
1
20
∑
j
=
1
20
min
(
a
i
,
b
j
)
?
\sum_{i=1}^{20}\sum_{j=1}^{20}\min(a_i,b_j)?
i
=
1
∑
20
j
=
1
∑
20
min
(
a
i
,
b
j
)?
26
1
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2014 Guts #26: Sum of Reciprocals of Sequence
For
1
≤
j
≤
2014
1\leq j\leq 2014
1
≤
j
≤
2014
, define
b
j
=
j
2014
∏
i
=
1
,
i
≠
j
2014
(
i
2014
−
j
2014
)
b_j=j^{2014}\prod_{i=1, i\neq j}^{2014}(i^{2014}-j^{2014})
b
j
=
j
2014
i
=
1
,
i
=
j
∏
2014
(
i
2014
−
j
2014
)
where the product is over all
i
∈
{
1
,
…
,
2014
}
i\in\{1,\ldots,2014\}
i
∈
{
1
,
…
,
2014
}
except
i
=
j
i=j
i
=
j
. Evaluate
1
b
1
+
1
b
2
+
⋯
+
1
b
2014
.
\dfrac1{b_1}+\dfrac1{b_2}+\cdots+\dfrac1{b_{2014}}.
b
1
1
+
b
2
1
+
⋯
+
b
2014
1
.
25
1
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2014 Guts #25: Extending Trisectors to form a Hexagon
Let
A
B
C
ABC
A
BC
be an equilateral triangle of side length
6
6
6
inscribed in a circle
ω
\omega
ω
. Let
A
1
,
A
2
A_1,A_2
A
1
,
A
2
be the points (distinct from
A
A
A
) where the lines through
A
A
A
passing through the two trisection points of
B
C
BC
BC
meet
ω
\omega
ω
. Define
B
1
,
B
2
,
C
1
,
C
2
B_1,B_2,C_1,C_2
B
1
,
B
2
,
C
1
,
C
2
similarly. Given that
A
1
,
A
2
,
B
1
,
B
2
,
C
1
,
C
2
A_1,A_2,B_1,B_2,C_1,C_2
A
1
,
A
2
,
B
1
,
B
2
,
C
1
,
C
2
appear on
ω
\omega
ω
in that order, find the area of hexagon
A
1
A
2
B
1
B
2
C
1
C
2
A_1A_2B_1B_2C_1C_2
A
1
A
2
B
1
B
2
C
1
C
2
.
24
1
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2014 Guts #24: Mean of Elements in Subset is an Integer
Let
A
=
{
a
1
,
a
2
,
…
,
a
7
}
A=\{a_1,a_2,\ldots,a_7\}
A
=
{
a
1
,
a
2
,
…
,
a
7
}
be a set of distinct positive integers such that the mean of the elements of any nonempty subset of
A
A
A
is an integer. Find the smallest possible value of the sum of the elements in
A
A
A
.
23
1
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2014 Guts #23: Expected Number of Distinct Absolute Values
Let
S
=
{
−
100
,
−
99
,
−
98
,
…
,
99
,
100
}
S=\{-100,-99,-98,\ldots,99,100\}
S
=
{
−
100
,
−
99
,
−
98
,
…
,
99
,
100
}
. Choose a
50
50
50
-element subset
T
T
T
of
S
S
S
at random. Find the expected number of elements of the set
{
∣
x
∣
:
x
∈
T
}
\{|x|:x\in T\}
{
∣
x
∣
:
x
∈
T
}
.
22
1
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2014 Guts #22: Tangent to a Cyclic Quad
Let
ω
\omega
ω
be a circle, and let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral inscribed in
ω
\omega
ω
. Suppose that
B
D
BD
B
D
and
A
C
AC
A
C
intersect at a point
E
E
E
. The tangent to
ω
\omega
ω
at
B
B
B
meets line
A
C
AC
A
C
at a point
F
F
F
, so that
C
C
C
lies between
E
E
E
and
F
F
F
. Given that
A
E
=
6
AE=6
A
E
=
6
,
E
C
=
4
EC=4
EC
=
4
,
B
E
=
2
BE=2
BE
=
2
, and
B
F
=
12
BF=12
BF
=
12
, find
D
A
DA
D
A
.
21
1
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2014 Guts #21: Sum of Powers of Two Divisible by Five
Compute the number of ordered quintuples of nonnegative integers
(
a
1
,
a
2
,
a
3
,
a
4
,
a
5
)
(a_1,a_2,a_3,a_4,a_5)
(
a
1
,
a
2
,
a
3
,
a
4
,
a
5
)
such that
0
≤
a
1
,
a
2
,
a
3
,
a
4
,
a
5
≤
7
0\leq a_1,a_2,a_3,a_4,a_5\leq 7
0
≤
a
1
,
a
2
,
a
3
,
a
4
,
a
5
≤
7
and
5
5
5
divides
2
a
1
+
2
a
2
+
2
a
3
+
2
a
4
+
2
a
5
2^{a_1}+2^{a_2}+2^{a_3}+2^{a_4}+2^{a_5}
2
a
1
+
2
a
2
+
2
a
3
+
2
a
4
+
2
a
5
.
20
1
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2014 Guts #20: Odd Number of Ranks
A deck of
8056
8056
8056
cards has
2014
2014
2014
ranks numbered
1
1
1
–
2014
2014
2014
. Each rank has four suits - hearts, diamonds, clubs, and spades. Each card has a rank and a suit, and no two cards have the same rank and the same suit. How many subsets of the set of cards in this deck have cards from an odd number of distinct ranks?
19
1
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2014 Guts #19: Bisectors in a Trapezoid
Let
A
B
C
D
ABCD
A
BC
D
be a trapezoid with
A
B
∥
C
D
AB\parallel CD
A
B
∥
C
D
. The bisectors of
∠
C
D
A
\angle CDA
∠
C
D
A
and
∠
D
A
B
\angle DAB
∠
D
A
B
meet at
E
E
E
, the bisectors of
∠
A
B
C
\angle ABC
∠
A
BC
and
∠
B
C
D
\angle BCD
∠
BC
D
meet at
F
F
F
, the bisectors of
∠
B
C
D
\angle BCD
∠
BC
D
and
∠
C
D
A
\angle CDA
∠
C
D
A
meet at
G
G
G
, and the bisectors of
∠
D
A
B
\angle DAB
∠
D
A
B
and
∠
A
B
C
\angle ABC
∠
A
BC
meet at
H
H
H
. Quadrilaterals
E
A
B
F
EABF
E
A
BF
and
E
D
C
F
EDCF
E
D
CF
have areas
24
24
24
and
36
36
36
, respectively, and triangle
A
B
H
ABH
A
B
H
has area
25
25
25
. Find the area of triangle
C
D
G
CDG
C
D
G
.
18
1
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2014 Guts #18: Product of Factors of 30 Exceeds 900
Find the number of ordered quadruples of positive integers
(
a
,
b
,
c
,
d
)
(a,b,c,d)
(
a
,
b
,
c
,
d
)
such that
a
,
b
,
c
,
a,b,c,
a
,
b
,
c
,
and
d
d
d
are all (not necessarily distinct) factors of
30
30
30
and
a
b
c
d
>
900
abcd>900
ab
c
d
>
900
.
17
1
Hide problems
2014 Guts #17: Integer-Valued Function
Let
f
:
N
→
N
f:\mathbb{N}\to\mathbb{N}
f
:
N
→
N
be a function satisfying the following conditions:(a)
f
(
1
)
=
1
f(1)=1
f
(
1
)
=
1
. (b)
f
(
a
)
≤
f
(
b
)
f(a)\leq f(b)
f
(
a
)
≤
f
(
b
)
whenever
a
a
a
and
b
b
b
are positive integers with
a
≤
b
a\leq b
a
≤
b
. (c)
f
(
2
a
)
=
f
(
a
)
+
1
f(2a)=f(a)+1
f
(
2
a
)
=
f
(
a
)
+
1
for all positive integers
a
a
a
.How many possible values can the
2014
2014
2014
-tuple
(
f
(
1
)
,
f
(
2
)
,
…
,
f
(
2014
)
)
(f(1),f(2),\ldots,f(2014))
(
f
(
1
)
,
f
(
2
)
,
…
,
f
(
2014
))
take?
16
1
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2014 Guts #16: Maximum Given Constraint
Suppose that
x
x
x
and
y
y
y
are positive real numbers such that
x
2
−
x
y
+
2
y
2
=
8
x^2-xy+2y^2=8
x
2
−
x
y
+
2
y
2
=
8
. Find the maximum possible value of
x
2
+
x
y
+
2
y
2
x^2+xy+2y^2
x
2
+
x
y
+
2
y
2
.
15
1
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2014 Guts #15: Pivot Lines in a Pentagon
Given a regular pentagon of area
1
1
1
, a pivot line is a line not passing through any of the pentagon's vertices such that there are
3
3
3
vertices of the pentagon on one side of the line and
2
2
2
on the other. A pivot point is a point inside the pentagon with only finitely many non-pivot lines passing through it. Find the area of the region of pivot points.
14
1
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2014 Guts #14: Similar, but not Congruent
Let
A
B
C
D
ABCD
A
BC
D
be a trapezoid with
A
B
∥
C
D
AB\parallel CD
A
B
∥
C
D
and
∠
D
=
9
0
∘
\angle D=90^\circ
∠
D
=
9
0
∘
. Suppose that there is a point
E
E
E
on
C
D
CD
C
D
such that
A
E
=
B
E
AE=BE
A
E
=
BE
and that triangles
A
E
D
AED
A
E
D
and
C
E
B
CEB
CEB
are similar, but not congruent. Given that
C
D
A
B
=
2014
\tfrac{CD}{AB}=2014
A
B
C
D
=
2014
, find
B
C
A
D
\tfrac{BC}{AD}
A
D
BC
.
13
1
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2014 Guts #13: Seating an Auditorium
An auditorium has two rows of seats, with
50
50
50
seats in each row.
100
100
100
indistinguishable people sit in the seats one at a time, subject to the condition that each person, except for the first person to sit in each row, must sit to the left or right of an occupied seat, and no two people can sit in the same seat. In how many ways can this process occur?
12
1
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2014 Guts #12: Find the Monic Polynomial
Find a nonzero monic polynomial
P
(
x
)
P(x)
P
(
x
)
with integer coefficients and minimal degree such that
P
(
1
−
2
3
+
4
3
)
=
0
P(1-\sqrt[3]2+\sqrt[3]4)=0
P
(
1
−
3
2
+
3
4
)
=
0
. (A polynomial is called
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
m
o
n
i
c
<
/
s
p
a
n
>
<span class='latex-italic'>monic</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
m
o
ni
c
<
/
s
p
an
>
if its leading coefficient is
1
1
1
.)
11
1
Hide problems
2014 Guts #11: Expected Remainder of Product of Dice
Two fair octahedral dice, each with the numbers
1
1
1
through
8
8
8
on their faces, are rolled. Let
N
N
N
be the remainder when the product of the numbers showing on the two dice is divided by
8
8
8
. Find the expected value of
N
N
N
.
10
5
Show problems
9
5
Show problems
8
5
Show problems
7
5
Show problems
6
5
Show problems
5
5
Show problems
4
5
Show problems
3
5
Show problems
2
5
Show problems
1
5
Show problems