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Harvard-MIT Mathematics Tournament
2007 Harvard-MIT Mathematics Tournament
2007 Harvard-MIT Mathematics Tournament
Part of
Harvard-MIT Mathematics Tournament
Subcontests
(36)
36
1
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2007 Guts #36: The Marathon
The Marathon. Let
ω
\omega
ω
denote the incircle of triangle
A
B
C
ABC
A
BC
. The segments
B
C
BC
BC
,
C
A
CA
C
A
, and
A
B
AB
A
B
are tangent to
ω
\omega
ω
at
D
D
D
,
E
E
E
and
F
F
F
, respectively. Point
P
P
P
lies on
E
F
EF
EF
such that segment
P
D
PD
P
D
is perpendicular to
B
C
BC
BC
. The line
A
P
AP
A
P
intersects
B
C
BC
BC
at
Q
Q
Q
. The circles
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
pass through
B
B
B
and
C
C
C
, respectively, and are tangent to
A
Q
AQ
A
Q
at
Q
Q
Q
; the former meets
A
B
AB
A
B
again at
X
X
X
, and the latter meets
A
C
AC
A
C
again at
Y
Y
Y
. The line
X
Y
XY
X
Y
intersects
B
C
BC
BC
at
Z
Z
Z
. Given that
A
B
=
15
AB=15
A
B
=
15
,
B
C
=
14
BC=14
BC
=
14
, and
C
A
=
13
CA=13
C
A
=
13
, find
⌊
X
Z
⋅
Y
Z
⌋
\lfloor XZ\cdot YZ\rfloor
⌊
XZ
⋅
Y
Z
⌋
.
35
1
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2007 Guts #35: The Algorithm
The Algorithm. There are thirteen broken computers situated at the following set
S
S
S
of thirteen points in the plane:
A
=
(
1
,
10
)
B
=
(
976
,
9
)
C
=
(
666
,
87
)
D
=
(
377
,
422
)
E
=
(
535
,
488
)
F
=
(
775
,
488
)
G
=
(
941
,
500
)
H
=
(
225
,
583
)
I
=
(
388
,
696
)
J
=
(
3
,
713
)
K
=
(
504
,
872
)
L
=
(
560
,
934
)
M
=
(
22
,
997
)
\begin{array}{ccc}A=(1,10)&B=(976,9)&C=(666,87)\\D=(377,422)&E=(535,488)&F=(775,488) \\ G=(941,500) & H=(225,583)&I=(388,696)\\J=(3,713)&K=(504,872)&L=(560,934)\\&M=(22,997)&\end{array}
A
=
(
1
,
10
)
D
=
(
377
,
422
)
G
=
(
941
,
500
)
J
=
(
3
,
713
)
B
=
(
976
,
9
)
E
=
(
535
,
488
)
H
=
(
225
,
583
)
K
=
(
504
,
872
)
M
=
(
22
,
997
)
C
=
(
666
,
87
)
F
=
(
775
,
488
)
I
=
(
388
,
696
)
L
=
(
560
,
934
)
At time
t
=
0
t=0
t
=
0
, a repairman begins moving from one computer to the next, traveling continuously in straight lines at unit speed. Assuming the repairman begins and
A
A
A
and fixes computers instantly, what path does he take to minimize the total downtime of the computers? List the points he visits in order. Your score will be
⌊
N
40
⌋
\left\lfloor \dfrac{N}{40}\right\rfloor
⌊
40
N
⌋
, where
N
=
1000
+
⌊
the optimal downtime
⌋
−
⌊
your downtime
⌋
,
N=1000+\lfloor\text{the optimal downtime}\rfloor - \lfloor \text{your downtime}\rfloor ,
N
=
1000
+
⌊
the optimal downtime
⌋
−
⌊
your downtime
⌋
,
or
0
0
0
, whichever is greater. By total downtime we mean the sum
∑
P
∈
S
t
P
,
\sum_{P\in S}t_P,
P
∈
S
∑
t
P
,
where
t
P
t_P
t
P
is the time at which the repairman reaches
P
P
P
.
34
1
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2007 Guts #34: The Game
The Game. Eric and Greg are watching their new favorite TV show, The Price is Right. Bob Barker recently raised the intellectual level of his program, and he begins the latest installment with bidding on following question: How many Carmichael numbers are there less than
100
,
000
100,000
100
,
000
?Each team is to list one nonnegative integer not greater than
100
,
000
100,000
100
,
000
. Let
X
X
X
denote the answer to Bob’s question. The teams listing
N
N
N
, a maximal bid (of those submitted) not greater than
X
X
X
, will receive
N
N
N
points, and all other teams will neither receive nor lose points. (A Carmichael number is an odd composite integer
n
n
n
such that
n
n
n
divides
a
n
−
1
−
1
a^{n-1}-1
a
n
−
1
−
1
for all integers
a
a
a
relatively prime to
n
n
n
with
1
<
a
<
n
1<a<n
1
<
a
<
n
.)
33
1
Hide problems
2007 Guts #33: Interesting Integral
Compute
∫
1
2
9
x
+
4
x
5
+
3
x
2
+
x
d
x
.
\int_1^2\dfrac{9x+4}{x^5+3x^2+x}dx.
∫
1
2
x
5
+
3
x
2
+
x
9
x
+
4
d
x
.
(No, your TI-89 doesn’t know how to do this one. Yes, the end is near.)
32
1
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2007 Guts #32: Circumcircles and Triangles
Triangle
A
B
C
ABC
A
BC
has
A
B
=
4
AB=4
A
B
=
4
,
B
C
=
6
BC=6
BC
=
6
, and
A
C
=
5
AC=5
A
C
=
5
. Let
O
O
O
denote the circumcenter of
A
B
C
ABC
A
BC
. The circle
Γ
\Gamma
Γ
is tangent to and surrounds the circumcircles of triangle
A
O
B
AOB
A
OB
,
B
O
C
BOC
BOC
, and
A
O
C
AOC
A
OC
. Determine the diameter of
Γ
\Gamma
Γ
.
31
1
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2007 Guts #31: Recursion without Starting Value
A sequence
{
a
n
}
n
≥
0
\{a_n\}_{n\geq 0}
{
a
n
}
n
≥
0
of real numbers satisfies the recursion
a
n
+
1
=
a
n
3
−
3
a
n
2
+
3
a_{n+1}=a_n^3-3a_n^2+3
a
n
+
1
=
a
n
3
−
3
a
n
2
+
3
for all positive integers
n
n
n
. For how many values of
a
0
a_0
a
0
does
a
2007
=
a
0
a_{2007}=a_0
a
2007
=
a
0
?
30
1
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2007 Guts #30: Feet of the Perpendiculars of the Cyclic Quad
A
B
C
D
ABCD
A
BC
D
is a cyclic quadrilateral in which
A
B
=
3
AB=3
A
B
=
3
,
B
C
=
5
BC=5
BC
=
5
,
C
D
=
6
CD=6
C
D
=
6
, and
A
D
=
10
AD=10
A
D
=
10
.
M
M
M
,
I
I
I
, and
T
T
T
are the feet of the perpendiculars from
D
D
D
to lines
A
B
AB
A
B
,
A
C
AC
A
C
, and
B
C
BC
BC
respectively. Determine the value of
M
I
/
I
T
MI/IT
M
I
/
I
T
.
29
1
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2007 Guts #29: Infinite Roots of Recursive Sequence
A sequence
{
a
n
}
n
≥
1
\{a_n\}_{n\geq 1}
{
a
n
}
n
≥
1
of positive reals is defined by the rule
a
n
+
1
a
n
−
1
5
=
a
n
4
a
n
−
2
2
a_{n+1}a_{n-1}^5=a_n^4a_{n-2}^2
a
n
+
1
a
n
−
1
5
=
a
n
4
a
n
−
2
2
for integers
n
>
2
n>2
n
>
2
together with the initial values
a
1
=
8
a_1=8
a
1
=
8
and
a
2
=
64
a_2=64
a
2
=
64
and
a
3
=
1024
a_3=1024
a
3
=
1024
. Compute
a
1
+
a
2
+
a
3
+
⋯
\sqrt{a_1+\sqrt{a_2+\sqrt{a_3+\cdots}}}
a
1
+
a
2
+
a
3
+
⋯
28
1
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2007 Guts #28: Cyclic Hexagon
Compute the circumradius of cyclic hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
, which has side lengths
A
B
=
B
C
=
2
AB=BC=2
A
B
=
BC
=
2
,
C
D
=
D
E
=
9
CD=DE=9
C
D
=
D
E
=
9
, and
E
F
=
F
A
=
12
EF=FA=12
EF
=
F
A
=
12
.
27
1
Hide problems
2007 Guts #27: Sums of Sixth Powers
Find the number of
7
7
7
-tuples
(
n
1
,
…
,
n
7
)
(n_1,\ldots,n_7)
(
n
1
,
…
,
n
7
)
of integers such that
∑
i
=
1
7
n
i
6
=
96957.
\sum_{i=1}^7 n_i^6=96957.
i
=
1
∑
7
n
i
6
=
96957.
26
1
Hide problems
2007 Guts #26: Cyclic Quadrilateral Diagonals
A
B
C
D
ABCD
A
BC
D
is a cyclic quadrilateral in which
A
B
=
4
AB=4
A
B
=
4
,
B
C
=
3
BC=3
BC
=
3
,
C
D
=
2
CD=2
C
D
=
2
, and
A
D
=
5
AD=5
A
D
=
5
. Diagonals
A
C
AC
A
C
and
B
D
BD
B
D
intersect at
X
X
X
. A circle
ω
\omega
ω
passes through
A
A
A
and is tangent to
B
D
BD
B
D
at
X
X
X
.
ω
\omega
ω
intersects
A
B
AB
A
B
and
A
D
AD
A
D
at
Y
Y
Y
and
Z
Z
Z
respectively. Compute
Y
Z
/
B
D
YZ/BD
Y
Z
/
B
D
.
23
1
Hide problems
2007 Guts #23: Obtuse Triangle Angle Bisector
In triangle
A
B
C
ABC
A
BC
,
∠
A
B
C
\angle ABC
∠
A
BC
is obtuse. Point
D
D
D
lies on side
A
C
AC
A
C
such that
∠
A
B
D
\angle ABD
∠
A
B
D
is right, and point
E
E
E
lies on side
A
C
AC
A
C
between
A
A
A
and
D
D
D
such that
B
D
BD
B
D
bisects
∠
E
B
C
\angle EBC
∠
EBC
. Find
C
E
CE
CE
given that
A
C
=
35
AC=35
A
C
=
35
,
B
C
=
7
BC=7
BC
=
7
, and
B
E
=
5
BE=5
BE
=
5
.
22
1
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2007 Guts #22: Recursive Sequences GCD
The sequence
{
a
n
}
n
≥
1
\{a_n\}_{n\geq 1}
{
a
n
}
n
≥
1
is defined by
a
n
+
2
=
7
a
n
+
1
−
a
n
a_{n+2}=7a_{n+1}-a_n
a
n
+
2
=
7
a
n
+
1
−
a
n
for positive integers
n
n
n
with initial values
a
1
=
1
a_1=1
a
1
=
1
and
a
2
=
8
a_2=8
a
2
=
8
. Another sequence,
{
b
n
}
\{b_n\}
{
b
n
}
, is defined by the rule
b
n
+
2
=
3
b
n
+
1
−
b
n
b_{n+2}=3b_{n+1}-b_n
b
n
+
2
=
3
b
n
+
1
−
b
n
for positive integers
n
n
n
together with the values
b
1
=
1
b_1=1
b
1
=
1
and
b
2
=
2
b_2=2
b
2
=
2
. Find
gcd
(
a
5000
,
b
501
)
\gcd(a_{5000},b_{501})
g
cd
(
a
5000
,
b
501
)
.
20
1
Hide problems
2007 Guts #20: Roots of an Equation
For
a
a
a
a positive real number, let
x
1
x_1
x
1
,
x
2
x_2
x
2
,
x
3
x_3
x
3
be the roots of the equation
x
3
−
a
x
2
+
a
x
−
a
=
0
x^3-ax^2+ax-a=0
x
3
−
a
x
2
+
a
x
−
a
=
0
. Determine the smallest possible value of
x
1
3
+
x
2
3
+
x
3
3
−
3
x
1
x
2
x
3
x_1^3+x_2^3+x_3^3-3x_1x_2x_3
x
1
3
+
x
2
3
+
x
3
3
−
3
x
1
x
2
x
3
.
19
1
Hide problems
2007 Guts #19: Wacky Multivariable Function
Define
x
⋆
y
=
x
2
+
3
x
y
+
y
2
−
2
x
−
2
y
+
4
x
y
+
4
x\star y=\frac{\sqrt{x^2+3xy+y^2-2x-2y+4}}{xy+4}
x
⋆
y
=
x
y
+
4
x
2
+
3
x
y
+
y
2
−
2
x
−
2
y
+
4
. Compute
(
(
⋯
(
(
2007
⋆
2006
)
⋆
2005
)
⋆
⋯
)
⋆
1
)
.
((\cdots ((2007\star 2006)\star 2005)\star\cdots )\star 1).
((
⋯
((
2007
⋆
2006
)
⋆
2005
)
⋆
⋯
)
⋆
1
)
.
18
1
Hide problems
2007 Guts #18: Diagonals of Convex Quadrilateral
Convex quadrilateral
A
B
C
D
ABCD
A
BC
D
has right angles
∠
A
\angle A
∠
A
and
∠
C
\angle C
∠
C
and is such that
A
B
=
B
C
AB=BC
A
B
=
BC
and
A
D
=
C
D
AD=CD
A
D
=
C
D
. The diagonals
A
C
AC
A
C
and
B
D
BD
B
D
intersect at point
M
M
M
. Points
P
P
P
and
Q
Q
Q
lie on the circumcircle of triangle
A
M
B
AMB
A
MB
and segment
C
D
CD
C
D
, respectively, such that points
P
P
P
,
M
M
M
, and
Q
Q
Q
are collinear. Suppose that
m
∠
A
B
C
=
16
0
∘
m\angle ABC=160^\circ
m
∠
A
BC
=
16
0
∘
and
m
∠
Q
M
C
=
4
0
∘
m\angle QMC=40^\circ
m
∠
QMC
=
4
0
∘
. Find
M
P
⋅
M
Q
MP\cdot MQ
MP
⋅
MQ
, given that
M
C
=
6
MC=6
MC
=
6
.
17
1
Hide problems
2007 Guts #17: Redskins Wins and Losses
During the regular season, Washington Redskins achieve a record of
10
10
10
wins and
6
6
6
losses. Compute the probability that their wins came in three streaks of consecutive wins, assuming that all possible arrangements of wins and losses are equally likely. (For example, the record
L
L
W
W
W
W
W
L
W
W
L
W
W
W
L
L
LLWWWWWLWWLWWWLL
LL
WWWWW
L
WW
L
WWW
LL
contains three winning streaks, while
W
W
W
W
W
W
W
L
L
L
L
L
L
W
W
W
WWWWWWWLLLLLLWWW
WWWWWWW
LLLLLL
WWW
has just two.)
16
1
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2007 Guts #16: Triangles and Parallel Lines
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
=
7
AB=7
A
B
=
7
,
B
C
=
9
BC=9
BC
=
9
, and
C
A
=
4
CA=4
C
A
=
4
. Let
D
D
D
be the point such that
A
B
∥
C
D
AB\parallel CD
A
B
∥
C
D
and
C
A
∥
B
D
CA\parallel BD
C
A
∥
B
D
. Let
R
R
R
be a point within triangle
B
C
D
BCD
BC
D
. Lines
ℓ
\ell
ℓ
and
m
m
m
going through
R
R
R
are parallel to
C
A
CA
C
A
and
A
B
AB
A
B
respectively. Line
ℓ
\ell
ℓ
meets
A
B
AB
A
B
and
B
C
BC
BC
at
P
P
P
and
P
′
P^\prime
P
′
respectively, and
m
m
m
meets
C
A
CA
C
A
and
B
C
BC
BC
at
Q
Q
Q
and
Q
′
Q^\prime
Q
′
respectively. If
S
S
S
denotes the largest possible sum of the areas of triangle
B
P
P
′
BPP^\prime
BP
P
′
,
R
P
′
Q
′
RP^\prime Q^\prime
R
P
′
Q
′
, and
C
Q
Q
′
CQQ^\prime
CQ
Q
′
, determine the value of
S
2
S^2
S
2
.
15
1
Hide problems
2007 Guts #15: Largest Angle Possible
Points
A
A
A
,
B
B
B
, and
C
C
C
lie in that order on line
ℓ
\ell
ℓ
such that
A
B
=
3
AB=3
A
B
=
3
and
B
C
=
2
BC=2
BC
=
2
. Point
H
H
H
is such that
C
H
CH
C
H
is perpendicular to
ℓ
\ell
ℓ
. Determine the length
C
H
CH
C
H
such that
∠
A
H
B
\angle AHB
∠
A
H
B
is as large as possible.
14
1
Hide problems
2007 Guts #14: Similar Triangles
We are given some similar triangles. Their areas are
1
2
,
3
2
,
5
2
,
⋯
,
1^2,3^2,5^2,\cdots,
1
2
,
3
2
,
5
2
,
⋯
,
and
4
9
2
49^2
4
9
2
. If the smallest triangle has a perimeter of
4
4
4
, what is the sum of all the triangles' perimeters?
13
1
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2007 Guts #13: Largest Integer Division
Determine the largest integer
n
n
n
such that
7
2048
−
1
7^{2048}-1
7
2048
−
1
is divisible by
2
n
2^n
2
n
.
12
1
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2007 Guts #12: A Sequence of Primes
Let
A
11
A_{11}
A
11
denote the answer to problem
11
11
11
. Determine the smallest prime
p
p
p
such that the arithmetic sequence
p
,
p
+
A
11
,
p
+
2
A
11
,
⋯
p,p+A_{11},p+2A_{11},\cdots
p
,
p
+
A
11
,
p
+
2
A
11
,
⋯
begins with the largest number of primes.There is just one triple of possible
(
A
10
,
A
11
,
A
12
)
(A_{10},A_{11},A_{12})
(
A
10
,
A
11
,
A
12
)
of answers to these three problems. Your team will receive credit only for answers matching these. (So, for example, submitting a wrong answer for problem
11
11
11
will not alter the correctness of your answer to problem
12
12
12
.)
21
1
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2007 Guts #21: Bob Diffusing Bombs
Bob the bomb-defuser has stumbled upon an active bomb. He opens it up, and finds the red and green wires conveniently located for him to cut. Being a seasoned member of the bomb-squad, Bob quickly determines that it is the green wire that he should cut, and puts his wirecutters on the green wire. But just before he starts to cut, the bomb starts to count down, ticking every second. Each time the bomb ticks, starting at time
t
=
15
t=15
t
=
15
seconds, Bob panics and has a certain chance to move his wirecutters to the other wire. However, he is a rational man even when panicking, and has a
1
2
t
2
\frac{1}{2t^2}
2
t
2
1
chance of switching wires at time t, regardless of which wire he is about to cut. When the bomb ticks at
t
=
1
t=1
t
=
1
, Bob cuts whatever wire his wirecutters are on, without switching wires. What is the probability that Bob cuts the green wire?
11
1
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2007 Guts #11: Two Circles in a Plane
Let
A
10
A_{10}
A
10
denote the answer to problem
10
10
10
. Two circles lie in the plane; denote the lengths of the internal and external tangents between these two circles by
x
x
x
and
y
y
y
, respectively. Given that the product of the radii of these two circles is
15
/
2
15/2
15/2
, and that the distance between their centers is
A
10
A_{10}
A
10
, determine
y
2
−
x
2
y^2-x^2
y
2
−
x
2
.
24
1
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2007 Guts #24: How Many Ordered Triples
Let
x
,
y
,
n
x,y,n
x
,
y
,
n
be positive integers with
n
>
1
n>1
n
>
1
. How many ordered triples
(
x
,
y
,
n
)
(x, y, n)
(
x
,
y
,
n
)
of solutions are there to the equation
x
n
−
y
n
=
2
100
x^n-y^n=2^{100}
x
n
−
y
n
=
2
100
?
25
1
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2007 Guts #25: Finding All Possible Values
Two real numbers
x
x
x
and
y
y
y
are such that
8
y
4
+
4
x
2
y
2
+
4
x
y
2
+
2
x
3
+
2
y
2
+
2
x
=
x
2
+
1
8y^4+4x^2y^2+4xy^2+2x^3+2y^2+2x=x^2+1
8
y
4
+
4
x
2
y
2
+
4
x
y
2
+
2
x
3
+
2
y
2
+
2
x
=
x
2
+
1
. Find all possible values of
x
+
2
y
2
x+2y^2
x
+
2
y
2
10
4
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9
4
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8
4
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7
4
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6
4
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5
4
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4
4
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3
4
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2
4
Show problems
1
4
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