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2005 Harvard-MIT Mathematics Tournament
2005 Harvard-MIT Mathematics Tournament
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Harvard-MIT Mathematics Tournament
Subcontests
(10)
10
3
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2005 Algebra #10: Absolute Values of Roots
Find the sum of the absolute values of the roots of
x
4
−
4
x
3
−
4
x
2
+
16
x
−
8
=
0
x^4 - 4x^3 - 4x^2 + 16x - 8 = 0
x
4
−
4
x
3
−
4
x
2
+
16
x
−
8
=
0
.
2005 Calculus #10: Smooth Function at 1
Let
f
:
R
→
R
f : \mathbf{R} \to \mathbf{R}
f
:
R
→
R
be a smooth function such that
f
′
(
x
)
=
f
(
1
−
x
)
f'(x)=f(1-x)
f
′
(
x
)
=
f
(
1
−
x
)
for all
x
x
x
and
f
(
0
)
=
1
f(0)=1
f
(
0
)
=
1
. Find
f
(
1
)
f(1)
f
(
1
)
.
2005 Geometry #10: Circle Traffic
Let
A
B
AB
A
B
be a diameter of a semicircle
Γ
\Gamma
Γ
. Two circles,
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
, externally tangent to each other and internally tangent to
Γ
\Gamma
Γ
, are tangent to the line
A
B
AB
A
B
at
P
P
P
and
Q
Q
Q
, respectively, and to semicircular arc
A
B
AB
A
B
at
C
C
C
and
D
D
D
, respectively, with
A
P
<
A
Q
AP<AQ
A
P
<
A
Q
. Suppose
F
F
F
lies on
Γ
\Gamma
Γ
such that
∠
F
Q
B
=
∠
C
Q
A
\angle FQB = \angle CQA
∠
FQB
=
∠
CQ
A
and that
∠
A
B
F
=
8
0
∘
\angle ABF = 80^\circ
∠
A
BF
=
8
0
∘
. Find
∠
P
D
Q
\angle PDQ
∠
P
D
Q
in degrees.
9
3
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2005 Algebra #9: 27,000,001
The number
27
,
000
,
001
27,\,000,\,001
27
,
000
,
001
has exactly four prime factors. Find their sum.
2005 Calculus #9: A sum!
Compute
∑
k
=
0
∞
4
(
4
k
)
!
.
\displaystyle\sum_{k=0}^{\infty} \dfrac {4}{(4k)!}.
k
=
0
∑
∞
(
4
k
)!
4
.
2005 Geometry #9: Circle and Point Mumbo Jumbo
Let
A
C
AC
A
C
be a diameter of a circle
ω
\omega
ω
of radius
1
1
1
, and let
D
D
D
be a point on
A
C
AC
A
C
such that
C
D
=
1
5
CD=\frac{1}{5}
C
D
=
5
1
. Let
B
B
B
be the point on
ω
\omega
ω
such that
D
B
DB
D
B
is perpendicular to
A
C
AC
A
C
, and
E
E
E
is the midpoint of
D
B
DB
D
B
. The line tangent to
ω
\omega
ω
at
B
B
B
intersects line
C
E
CE
CE
at the point
X
X
X
. Compute
A
X
AX
A
X
.
8
3
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2005 Calculus #8: Functional Function
If
f
f
f
is a continuous real function such that
f
(
x
−
1
)
+
f
(
x
+
1
)
≥
x
+
f
(
x
)
f(x-1) + f(x+1) \ge x + f(x)
f
(
x
−
1
)
+
f
(
x
+
1
)
≥
x
+
f
(
x
)
for all
x
x
x
, what is the minimum possible value of
∫
1
2005
f
(
x
)
d
x
\displaystyle\int_{1}^{2005} f(x) \, \mathrm{d}x
∫
1
2005
f
(
x
)
d
x
?
2005 Algebra #8: Infinite Sum of Rational Function
Compute
∑
n
=
0
∞
n
n
4
+
n
2
+
1
.
\displaystyle\sum_{n=0}^{\infty} \dfrac {n}{n^4 + n^2 + 1}.
n
=
0
∑
∞
n
4
+
n
2
+
1
n
.
2005 Geometry #8: 26-51-73 Triangle
Let
T
T
T
be a triangle with side lengths
26
26
26
,
51
51
51
, and
73
73
73
. Let
S
S
S
be the set of points inside
T
T
T
which do not lie within a distance of
5
5
5
of any side of
T
T
T
. Find the area of
S
S
S
.
7
3
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2005 Algebra #7: max(funny expression)
Let
x
x
x
be a positive real number. Find the maximum possible value of
x
2
+
2
−
x
4
+
4
x
.
\frac{x^2+2-\sqrt{x^4+4}}{x}.
x
x
2
+
2
−
x
4
+
4
.
2005 Calculus #7: Ants
Two ants, one starting at
(
−
1
,
1
)
(-1, 1)
(
−
1
,
1
)
, the other at
(
1
,
1
)
(1, 1)
(
1
,
1
)
, walk to the right along the parabola
y
=
x
2
y = x^2
y
=
x
2
such that their midpoint moves along the line
y
=
1
y = 1
y
=
1
with constant speed
1
1
1
. When the left ant first hits the line
y
=
1
2
y = \frac {1}{2}
y
=
2
1
, what is its speed?
2005 Geometry #7: Tetrahedron with _|_ sides
Let
A
B
C
D
ABCD
A
BC
D
be a tetrahedron such that edges
A
B
AB
A
B
,
A
C
AC
A
C
, and
A
D
AD
A
D
are mutually perpendicular. Let the areas of triangles
A
B
C
ABC
A
BC
,
A
C
D
ACD
A
C
D
, and
A
D
B
ADB
A
D
B
be denoted by
x
x
x
,
y
y
y
, and
z
z
z
, respectively. In terms of
x
x
x
,
y
y
y
, and
z
z
z
, find the area of triangle
B
C
D
BCD
BC
D
.
6
3
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2005 Algebra #6: Points of Intersection
Find the sum of the x-coordinates of the distinct points of intersection of the plane curves given by
x
2
=
x
+
y
+
4
x^2 = x + y + 4
x
2
=
x
+
y
+
4
and
y
2
=
y
−
15
x
+
36
y^2 = y - 15x + 36
y
2
=
y
−
15
x
+
36
.
2005 Calculus #6: Polar Coordinates and an Area
The graph of
r
=
2
+
cos
2
θ
r=2+\cos2\theta
r
=
2
+
cos
2
θ
and its reflection over the line
y
=
x
y=x
y
=
x
bound five regions in the plane. Find the area of the region containing the origin.
2005 Geometry #6: Paper Triangle
A triangular piece of paper of area
1
1
1
is folded along a line parallel to one of the sides and pressed flat. What is the minimum possible area of the resulting figure?
5
3
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2005 Algebra #5: The Round Table of Positive Integers
Ten positive integers are arranged around a circle. Each number is one more than the greatest common divisor of its two neighbors. What is the sum of the ten numbers?
2005 Calculus #5: Towers of x - limit
Calculate
lim
x
→
0
+
(
x
x
x
−
x
x
)
.
\lim_{x \to 0^+} \left( x^{x^x} - x^x \right).
x
→
0
+
lim
(
x
x
x
−
x
x
)
.
2005 Geometry #5: Two Cubes and a Sphere
A cube with side length
2
2
2
is inscribed in a sphere. A second cube, with faces parallel to the first, is inscribed between the sphere and one face of the first cube. What is the length of a side of the smaller cube?
4
3
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2005 Algebra #4: Floor Sum
If
a
,
b
,
c
>
0
a,b,c>0
a
,
b
,
c
>
0
, what is the smallest possible value of
⌊
a
+
b
c
⌋
+
⌊
b
+
c
a
⌋
+
⌊
c
+
a
b
⌋
\left\lfloor \dfrac {a+b}{c} \right\rfloor + \left\lfloor \dfrac {b+c}{a} \right\rfloor + \left\lfloor \dfrac {c+a}{b} \right\rfloor
⌊
c
a
+
b
⌋
+
⌊
a
b
+
c
⌋
+
⌊
b
c
+
a
⌋
? (Note that
⌊
x
⌋
\lfloor x \rfloor
⌊
x
⌋
denotes the greatest integer less than or equal to
x
x
x
.)
2005 Calculus #4: Smooth function - derivative at 0
Let
f
:
R
→
R
f : \mathbf {R} \to \mathbf {R}
f
:
R
→
R
be a smooth function such that
f
′
(
x
)
2
=
f
(
x
)
f
′
′
(
x
)
f'(x)^2 = f(x) f''(x)
f
′
(
x
)
2
=
f
(
x
)
f
′′
(
x
)
for all
x
x
x
. Suppose
f
(
0
)
=
1
f(0)=1
f
(
0
)
=
1
and
f
(
4
)
(
0
)
=
9
f^{(4)} (0) = 9
f
(
4
)
(
0
)
=
9
. Find all possible values of
f
′
(
0
)
f'(0)
f
′
(
0
)
.
2005 Geometry #4: Crazy Circle Watermarked on Triangle
Let
X
Y
Z
XYZ
X
Y
Z
be a triangle with
∠
X
=
6
0
∘
\angle X = 60^\circ
∠
X
=
6
0
∘
and
∠
Y
=
4
5
∘
\angle Y = 45^\circ
∠
Y
=
4
5
∘
. A circle with center
P
P
P
passes through points
A
A
A
and
B
B
B
on side
X
Y
XY
X
Y
,
C
C
C
and
D
D
D
on side
Y
Z
YZ
Y
Z
, and
E
E
E
and
F
F
F
on side
Z
X
ZX
ZX
. Suppose
A
B
=
C
D
=
E
F
AB=CD=EF
A
B
=
C
D
=
EF
. Find
∠
X
P
Y
\angle XPY
∠
XP
Y
in degrees.
3
3
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2005 Algebra #3: Maximum Value of Symmetric Equation
Let
x
x
x
,
y
y
y
, and
z
z
z
be distinct real numbers that sum to
0
0
0
. Find the maximum possible value of
x
y
+
y
z
+
z
x
x
2
+
y
2
+
z
2
.
\dfrac {xy+yz+zx}{x^2+y^2+z^2}.
x
2
+
y
2
+
z
2
x
y
+
yz
+
z
x
.
2005 Calculus #3: Scary-looking Integrals
Let
f
:
R
→
R
f : \mathbf{R} \to \mathbf{R}
f
:
R
→
R
be a continuous function with
∫
0
1
f
(
x
)
f
′
(
x
)
d
x
=
0
\displaystyle\int_{0}^{1} f(x) f'(x) \, \mathrm{d}x = 0
∫
0
1
f
(
x
)
f
′
(
x
)
d
x
=
0
and
∫
0
1
f
(
x
)
2
f
′
(
x
)
d
x
=
18
\displaystyle\int_{0}^{1} f(x)^2 f'(x) \, \mathrm{d}x = 18
∫
0
1
f
(
x
)
2
f
′
(
x
)
d
x
=
18
. What is
∫
0
1
f
(
x
)
4
f
′
(
x
)
d
x
\displaystyle\int_{0}^{1} f(x)^4 f'(x) \, \mathrm{d} x
∫
0
1
f
(
x
)
4
f
′
(
x
)
d
x
?
2005 Geometry #3: Centroided Triangle Nest
Let
A
B
C
D
ABCD
A
BC
D
be a rectangle with area
1
1
1
, and let
E
E
E
lie on side
C
D
CD
C
D
. What is the area of the triangle formed by the centroids of triangles
A
B
E
ABE
A
BE
,
B
C
E
BCE
BCE
, and
A
D
E
ADE
A
D
E
?
2
3
Hide problems
2005 Algebra #2: Power Inage
How many real numbers
x
x
x
are solutions to the following equation?
200
3
x
+
200
4
x
=
200
5
x
2003^x + 2004^x = 2005^x
200
3
x
+
200
4
x
=
200
5
x
2005 Calculus #2: Parameterized Stuff - find the length
A plane curve is parameterized by
x
(
t
)
=
∫
t
∞
cos
u
u
d
u
x(t)=\displaystyle\int_{t}^{\infty} \dfrac {\cos u}{u} \, \mathrm{d}u
x
(
t
)
=
∫
t
∞
u
cos
u
d
u
and
y
(
t
)
=
∫
t
∞
sin
u
u
d
u
y(t) = \displaystyle\int_{t}^{\infty} \dfrac {\sin u}{u} \, \mathrm{d}u
y
(
t
)
=
∫
t
∞
u
sin
u
d
u
for
1
≤
t
≤
2
1 \le t \le 2
1
≤
t
≤
2
. What is the length of the curve?
2005 Geometry #2: Tetrahedron Surface Area
Let
A
B
C
D
ABCD
A
BC
D
be a regular tetrahedron with side length
2
2
2
. The plane parallel to edges
A
B
AB
A
B
and
C
D
CD
C
D
and lying halfway between them cuts
A
B
C
D
ABCD
A
BC
D
into two pieces. Find the surface area of one of these pieces.
1
3
Hide problems
2005 Algebra #1: Absolute Value Equation
How many real numbers
x
x
x
are solutions to the following equation?
∣
x
−
1
∣
=
∣
x
−
2
∣
+
∣
x
−
3
∣
|x-1| = |x-2| + |x-3|
∣
x
−
1∣
=
∣
x
−
2∣
+
∣
x
−
3∣
2005 Calculus #1: Parallel Tangent Lines
Let
f
(
x
)
=
x
3
+
a
x
+
b
f(x) = x^3 + ax + b
f
(
x
)
=
x
3
+
a
x
+
b
, with
a
≠
b
a \ne b
a
=
b
, and suppose the tangent lines to the graph of
f
f
f
at
x
=
a
x=a
x
=
a
and
x
=
b
x=b
x
=
b
are parallel. Find
f
(
1
)
f(1)
f
(
1
)
.
2005 Geometry #1: Cube Stuff
The volume of a cube (in cubic inches) plus three times the total length of its edges (in inches) is equal to twice its surface area (in square inches). How many inches long is its long diagonal?