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Subcontests
(10)
10
3
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2010 Algebra #10: Functional Compositional Equation
Let
p
(
x
)
p(x)
p
(
x
)
and
q
(
x
)
q(x)
q
(
x
)
be two cubic polynomials such that
p
(
0
)
=
−
24
p(0)=-24
p
(
0
)
=
−
24
,
q
(
0
)
=
30
q(0)=30
q
(
0
)
=
30
, and
p
(
q
(
x
)
)
=
q
(
p
(
x
)
)
p(q(x))=q(p(x))
p
(
q
(
x
))
=
q
(
p
(
x
))
for all real numbers
x
x
x
. Find the ordered pair
(
p
(
3
)
,
q
(
6
)
)
(p(3),q(6))
(
p
(
3
)
,
q
(
6
))
.
2010 Calculus #10: Summation with Constants
Let
f
(
n
)
=
∑
k
=
1
n
1
k
f(n)=\displaystyle\sum_{k=1}^n \dfrac{1}{k}
f
(
n
)
=
k
=
1
∑
n
k
1
. Then there exists constants
γ
\gamma
γ
,
c
c
c
, and
d
d
d
such that
f
(
n
)
=
ln
(
x
)
+
γ
+
c
n
+
d
n
2
+
O
(
1
n
3
)
,
f(n)=\ln(x)+\gamma+\dfrac{c}{n}+\dfrac{d}{n^2}+O\left(\dfrac{1}{n^3}\right),
f
(
n
)
=
ln
(
x
)
+
γ
+
n
c
+
n
2
d
+
O
(
n
3
1
)
,
where the
O
(
1
n
3
)
O\left(\dfrac{1}{n^3}\right)
O
(
n
3
1
)
means terms of order
1
n
3
\dfrac{1}{n^3}
n
3
1
or lower. Compute the ordered pair
(
c
,
d
)
(c,d)
(
c
,
d
)
.
2010 Geometry #10
Circles
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
intersect at points
A
A
A
and
B
B
B
. Segment
P
Q
PQ
PQ
is tangent to
ω
1
\omega_1
ω
1
at
P
P
P
and to
ω
2
\omega_2
ω
2
at
Q
Q
Q
, and
A
A
A
is closer to
P
Q
PQ
PQ
than
B
B
B
. Point
X
X
X
is on
ω
1
\omega_1
ω
1
such that
P
X
∥
Q
B
PX\parallel QB
PX
∥
QB
, and point
Y
Y
Y
is on
ω
2
\omega_2
ω
2
such that
Q
Y
∥
P
B
QY\parallel PB
Q
Y
∥
PB
. Given that
∠
A
P
Q
=
3
0
∘
\angle APQ=30^\circ
∠
A
PQ
=
3
0
∘
and
∠
P
Q
A
=
1
5
∘
\angle PQA=15^\circ
∠
PQ
A
=
1
5
∘
, find the ratio
A
X
/
A
Y
AX/AY
A
X
/
A
Y
.
9
3
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2010 Algebra #9: Real Recursive Roots
Let
f
(
x
)
=
c
x
(
x
−
1
)
f(x)=cx(x-1)
f
(
x
)
=
c
x
(
x
−
1
)
, where
c
c
c
is a positive real number. We use
f
n
(
x
)
f^n(x)
f
n
(
x
)
to denote the polynomial obtained by composing
f
f
f
with itself
n
n
n
times. For every positive integer
n
n
n
, all the roots of
f
n
(
x
)
f^n(x)
f
n
(
x
)
are real. What is the smallest possible value of
c
c
c
?
2010 Calculus #9: Trigonometric Differential Equation
Let
x
(
t
)
x(t)
x
(
t
)
be a solution to the differential equation
(
x
+
x
′
)
2
+
x
⋅
x
′
′
=
cos
t
\left(x+x^\prime\right)^2+x\cdot x^{\prime\prime}=\cos t
(
x
+
x
′
)
2
+
x
⋅
x
′′
=
cos
t
with
x
(
0
)
=
x
′
(
0
)
=
2
5
x(0)=x^\prime(0)=\sqrt{\frac{2}{5}}
x
(
0
)
=
x
′
(
0
)
=
5
2
. Compute
x
(
π
4
)
x\left(\dfrac{\pi}{4}\right)
x
(
4
π
)
.
2010 Geometry #9
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral with an inscribed circle centered at
I
I
I
. Let
C
I
CI
C
I
intersect
A
B
AB
A
B
at
E
E
E
. If
∠
I
D
E
=
3
5
∘
\angle IDE=35^\circ
∠
I
D
E
=
3
5
∘
,
∠
A
B
C
=
7
0
∘
\angle ABC=70^\circ
∠
A
BC
=
7
0
∘
, and
∠
B
C
D
=
6
0
∘
\angle BCD=60^\circ
∠
BC
D
=
6
0
∘
, then what are all possible measures of
∠
C
D
A
\angle CDA
∠
C
D
A
?
8
3
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2010 Algebra #8: Polynomial Permutations
How many polynomials of degree exactly
5
5
5
with real coefficients send the set
{
1
,
2
,
3
,
4
,
5
,
6
}
\{1, 2, 3, 4, 5, 6\}
{
1
,
2
,
3
,
4
,
5
,
6
}
to a permutation of itself?
2010 Calculus #8: Factorial and Power Summation
Let
f
(
n
)
=
∑
k
=
2
∞
1
k
n
⋅
k
!
.
f(n)=\displaystyle\sum_{k=2}^\infty \dfrac{1}{k^n\cdot k!}.
f
(
n
)
=
k
=
2
∑
∞
k
n
⋅
k
!
1
.
Calculate
∑
n
=
2
∞
f
(
n
)
\displaystyle\sum_{n=2}^\infty f(n)
n
=
2
∑
∞
f
(
n
)
.
2010 Geometry #8: Maximum Area of Triangle
Let
O
O
O
be the point
(
0
,
0
)
(0,0)
(
0
,
0
)
. Let
A
A
A
,
B
B
B
,
C
C
C
be three points in the plane such that
A
O
=
15
AO=15
A
O
=
15
,
B
O
=
15
BO = 15
BO
=
15
, and
C
O
=
7
CO = 7
CO
=
7
, and such that the area of triangle
A
B
C
ABC
A
BC
is maximal. What is the length of the shortest side of
A
B
C
ABC
A
BC
?
7
3
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2010 Algebra #7: System of Complex Equations
Let
a
,
b
,
c
,
x
,
y
,
a,b,c,x,y,
a
,
b
,
c
,
x
,
y
,
and
z
z
z
be complex numbers such that
a
=
b
+
c
x
−
2
,
b
=
c
+
a
y
−
2
,
c
=
a
+
b
z
−
2
.
a=\dfrac{b+c}{x-2},\qquad b=\dfrac{c+a}{y-2},\qquad c=\dfrac{a+b}{z-2}.
a
=
x
−
2
b
+
c
,
b
=
y
−
2
c
+
a
,
c
=
z
−
2
a
+
b
.
If
x
y
+
y
z
+
x
z
=
67
xy+yz+xz=67
x
y
+
yz
+
x
z
=
67
and
x
+
y
+
z
=
2010
x+y+z=2010
x
+
y
+
z
=
2010
, find the value of
x
y
z
xyz
x
yz
.
2010 Calculus #7: Minimization of Multivariable Expression
Let
a
1
a_1
a
1
,
a
2
a_2
a
2
, and
a
3
a_3
a
3
be nonzero complex numbers with non-negative real and imaginary parts. Find the minimum possible value of
∣
a
1
+
a
2
+
a
3
∣
∣
a
1
a
2
a
3
∣
3
.
\dfrac{|a_1+a_2+a_3|}{\sqrt[3]{|a_1a_2a_3|}}.
3
∣
a
1
a
2
a
3
∣
∣
a
1
+
a
2
+
a
3
∣
.
2010 Geometry #7
You are standing in an infinitely long hallway with sides given by the lines
x
=
0
x=0
x
=
0
and
x
=
6
x=6
x
=
6
. You start at
(
3
,
0
)
(3,0)
(
3
,
0
)
and want to get to
(
3
,
6
)
(3,6)
(
3
,
6
)
. Furthermore, at each instant you want your distance to
(
3
,
6
)
(3,6)
(
3
,
6
)
to either decrease or stay the same. What is the area of the set of points that you could pass through on your journey from
(
3
,
0
)
(3,0)
(
3
,
0
)
to
(
3
,
6
)
(3,6)
(
3
,
6
)
?
6
3
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2010 Algebra #6: Fill in the Signs
Suppose that a polynomial of the form
p
(
x
)
=
x
2010
±
x
2009
±
⋯
±
x
±
1
p(x)=x^{2010}\pm x^{2009}\pm \cdots \pm x \pm 1
p
(
x
)
=
x
2010
±
x
2009
±
⋯
±
x
±
1
has no real roots. What is the maximum possible number of coefficients of
−
1
-1
−
1
in
p
p
p
?
2010 Calculus #6: Splitting the Plane
Let
f
(
x
)
=
x
3
−
x
2
f(x)=x^3-x^2
f
(
x
)
=
x
3
−
x
2
. For a given value of
x
x
x
, the graph of
f
(
x
)
f(x)
f
(
x
)
, together with the graph of the line
c
+
x
c+x
c
+
x
, split the plane up into regions. Suppose that
c
c
c
is such that exactly two of these regions have finite area. Find the value of
c
c
c
that minimizes the sum of the areas of these two regions.
2010 Geometry #6
Three unit circles
ω
1
\omega_1
ω
1
,
ω
2
\omega_2
ω
2
, and
ω
3
\omega_3
ω
3
in the plane have the property that each circle passes through the centers of the other two. A square
S
S
S
surrounds three circles in such a way that each of its four sides is tangent to at least one of
ω
1
\omega_1
ω
1
,
ω
2
\omega_2
ω
2
, and
ω
3
\omega_3
ω
3
. Find the side length of the square
S
S
S
.
5
3
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2010 Algebra #5: Symmetrical Powers
Suppose that
x
x
x
and
y
y
y
are complex numbers such that
x
+
y
=
1
x+y=1
x
+
y
=
1
and
x
20
+
y
20
=
20
x^{20}+y^{20}=20
x
20
+
y
20
=
20
. Find the sum of all possible values of
x
2
+
y
2
x^2+y^2
x
2
+
y
2
.
2010 Calculus #5: Two Defined Functions
Let the functions
f
(
α
,
x
)
f(\alpha,x)
f
(
α
,
x
)
and
g
(
α
)
g(\alpha)
g
(
α
)
be defined as
f
(
α
,
x
)
=
(
x
2
)
α
x
−
1
g
(
α
)
=
d
4
f
d
x
4
∣
x
=
2
f(\alpha,x)=\dfrac{(\frac{x}{2})^\alpha}{x-1}\qquad\qquad\qquad g(\alpha)=\,\dfrac{d^4f}{dx^4}|_{x=2}
f
(
α
,
x
)
=
x
−
1
(
2
x
)
α
g
(
α
)
=
d
x
4
d
4
f
∣
x
=
2
Then
g
(
α
)
g(\alpha)
g
(
α
)
is a polynomial is
α
\alpha
α
. Find the leading coefficient of
g
(
α
)
g(\alpha)
g
(
α
)
.
2010 Geometry #5
A sphere is the set of points at a fixed positive distance
r
r
r
from its center. Let
S
\mathcal{S}
S
be a set of
2010
2010
2010
-dimensional spheres. Suppose that the number of points lying on every element of
S
\mathcal{S}
S
is a finite number
n
n
n
. Find the maximal possible value of
n
n
n
.
4
3
Hide problems
2010 Algebra #4: Common Roots of Common Coefficients
Suppose that there exist nonzero complex numbers
a
a
a
,
b
b
b
,
c
c
c
, and
d
d
d
such that
k
k
k
is a root of both the equations
a
x
3
+
b
x
2
+
c
x
+
d
=
0
ax^3+bx^2+cx+d=0
a
x
3
+
b
x
2
+
c
x
+
d
=
0
and
b
x
3
+
c
x
2
+
d
x
+
a
=
0
bx^3+cx^2+dx+a=0
b
x
3
+
c
x
2
+
d
x
+
a
=
0
. Find all possible values of
k
k
k
(including complex values).
2010 Calculus #4: Limit of Summation
Compute
lim
n
→
∞
∑
k
=
1
n
∣
cos
(
k
)
∣
n
\displaystyle\lim_{n\to\infty}\dfrac{\sum_{k=1}^n|\cos(k)|}{n}
n
→
∞
lim
n
∑
k
=
1
n
∣
cos
(
k
)
∣
.
2010 Geometry #4
Let
A
B
C
D
ABCD
A
BC
D
be an isosceles trapezoid such that
A
B
=
10
AB=10
A
B
=
10
,
B
C
=
15
BC=15
BC
=
15
,
C
D
=
28
CD=28
C
D
=
28
, and
D
A
=
15
DA=15
D
A
=
15
. There is a point
E
E
E
such that
△
A
E
D
\triangle AED
△
A
E
D
and
△
A
E
B
\triangle AEB
△
A
EB
have the same area and such that
E
C
EC
EC
is minimal. Find
E
C
EC
EC
.
3
3
Hide problems
2010 Algebra #3: Recursive Sequence
Let
S
0
=
0
S_0=0
S
0
=
0
and let
S
k
S_k
S
k
equal
a
1
+
2
a
2
+
…
+
k
a
k
a_1+2a_2+\ldots+ka_k
a
1
+
2
a
2
+
…
+
k
a
k
for
k
≥
1
k\geq 1
k
≥
1
. Define
a
i
a_i
a
i
to be
1
1
1
if
S
i
−
1
<
i
S_{i-1}<i
S
i
−
1
<
i
and
−
1
-1
−
1
if
S
i
−
1
≥
i
S_{i-1}\geq i
S
i
−
1
≥
i
. What is the largest
k
≤
2010
k\leq 2010
k
≤
2010
such that
S
k
=
0
S_k=0
S
k
=
0
?
2010 Calculus #3: Common Zeroes
Let
p
p
p
be a monic cubic polynomial such that
p
(
0
)
=
1
p(0)=1
p
(
0
)
=
1
and such that all the zeroes of
p
′
(
x
)
p^\prime (x)
p
′
(
x
)
are also zeroes of
p
(
x
)
p(x)
p
(
x
)
. Find
p
p
p
. Note: monic means that the leading coefficient is
1
1
1
.
2010 Geometry #3
For
0
≤
y
≤
2
0\leq y\leq 2
0
≤
y
≤
2
, let
D
y
D_y
D
y
be the half-disk of diameter 2 with one vertex at
(
0
,
y
)
(0,y)
(
0
,
y
)
, the other vertex on the positive
x
x
x
-axis, and the curved boundary further from the origin than the straight boundary. Find the area of the union of
D
y
D_y
D
y
for all
0
≤
y
≤
2
0\leq y\leq 2
0
≤
y
≤
2
.
2
3
Hide problems
2010 Algebra #2: Number Ranks
The rank of a rational number
q
q
q
is the unique
k
k
k
for which
q
=
1
a
1
+
⋯
+
1
a
k
q=\frac{1}{a_1}+\cdots+\frac{1}{a_k}
q
=
a
1
1
+
⋯
+
a
k
1
, where each
a
i
a_i
a
i
is the smallest positive integer
q
q
q
such that
q
≥
1
a
1
+
⋯
+
1
a
i
q\geq \frac{1}{a_1}+\cdots+\frac{1}{a_i}
q
≥
a
1
1
+
⋯
+
a
i
1
. Let
q
q
q
be the largest rational number less than
1
4
\frac{1}{4}
4
1
with rank
3
3
3
, and suppose the expression for
q
q
q
is
1
a
1
+
1
a
2
+
1
a
3
\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}
a
1
1
+
a
2
1
+
a
3
1
. Find the ordered triple
(
a
1
,
a
2
,
a
3
)
(a_1,a_2,a_3)
(
a
1
,
a
2
,
a
3
)
.
2010 Calculus #2: Differential Equation
Let
f
f
f
be a function such that
f
(
0
)
=
1
f(0)=1
f
(
0
)
=
1
,
f
′
(
0
)
=
2
f^\prime (0)=2
f
′
(
0
)
=
2
, and
f
′
′
(
t
)
=
4
f
′
(
t
)
−
3
f
(
t
)
+
1
f^{\prime\prime}(t)=4f^\prime(t)-3f(t)+1
f
′′
(
t
)
=
4
f
′
(
t
)
−
3
f
(
t
)
+
1
for all
t
t
t
. Compute the
4
4
4
th derivative of
f
f
f
, evaluated at
0
0
0
.
2010 Geometry #2
A rectangular piece of paper is folded along its diagonal (as depicted below) to form a non-convex pentagon that has an area of
7
10
\tfrac{7}{10}
10
7
of the area of the original rectangle. Find the ratio of the longer side of the rectangle to the shorter side of the rectangle. [asy] size(150); pair A = (-5,0); pair B = (5,0); pair C = (-3,4); pair D = (3,4); pair E = intersectionpoint(B--C,A--D); draw(A--B--D--cycle); draw(A--C); draw(C--E); draw(E--B,dashed); markscalefactor=0.06; draw(rightanglemark(A,C,B)); [/asy]
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3
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2010 Algebra #1: System of Positive Reals
Suppose that
x
x
x
and
y
y
y
are positive reals such that
x
−
y
2
=
3
,
x
2
+
y
4
=
13.
x-y^2=3, \qquad x^2+y^4=13.
x
−
y
2
=
3
,
x
2
+
y
4
=
13.
Find
x
x
x
.
2010 Calculus #1: Subtract the Derivative
Suppose that
p
(
x
)
p(x)
p
(
x
)
is a polynomial and that
p
(
x
)
−
p
′
(
x
)
=
x
2
+
2
x
+
1
p(x)-p^\prime (x)=x^2+2x+1
p
(
x
)
−
p
′
(
x
)
=
x
2
+
2
x
+
1
. Compute
p
(
5
)
p(5)
p
(
5
)
.
2010 Geometry #1
Below is pictured a regular seven-pointed star. Find the measure of angle
a
a
a
in radians. [asy] size(150); draw(unitcircle, white); pair A = dir(180/7); pair B = dir(540/7); pair C = dir(900/7); pair D = dir(180); pair E = dir(-900/7); pair F = dir(-540/7); pair G = dir (-180/7); draw(A--D); draw(B--E); draw(C--F); draw(D--G); draw(E--A); draw(F--B); draw(G--C); label((-0.1,0.5), "
a
a
a
"); [/asy]