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2009 Harvard-MIT Mathematics Tournament
2009 Harvard-MIT Mathematics Tournament
Part of
Harvard-MIT Mathematics Tournament
Subcontests
(10)
10
5
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9
4
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8
4
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7
4
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5
4
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6
5
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4
3
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2009 Algebra #4 - LCM and GCD of Polynomials
Suppose
a
a
a
,
b
b
b
and
c
c
c
are integers such that the greatest common divisor of
x
2
+
a
x
+
b
x^2+ax+b
x
2
+
a
x
+
b
and
x
2
+
b
x
+
c
x^2+bx+c
x
2
+
b
x
+
c
is
x
+
1
x+1
x
+
1
(in the set of polynomials in
x
x
x
with integer coefficients), and the least common multiple of
x
2
+
a
x
+
b
x^2+ax+b
x
2
+
a
x
+
b
and
x
2
+
b
x
+
c
x^2+bx+c
x
2
+
b
x
+
c
x
3
−
4
x
2
+
x
+
6
x^3-4x^2+x+6
x
3
−
4
x
2
+
x
+
6
. Find
a
+
b
+
c
a+b+c
a
+
b
+
c
.
2009 Calculus #4: Fourth Degree Polynomial
Let
P
P
P
be a fourth degree polynomial, with derivative
P
′
P^\prime
P
′
, such that
P
(
1
)
=
P
(
3
)
=
P
(
5
)
=
P
′
(
7
)
=
0
P(1)=P(3)=P(5)=P^\prime (7)=0
P
(
1
)
=
P
(
3
)
=
P
(
5
)
=
P
′
(
7
)
=
0
. Find the real number
x
≠
1
,
3
,
5
x\neq 1,3,5
x
=
1
,
3
,
5
such that
P
(
x
)
=
0
P(x)=0
P
(
x
)
=
0
.
2009 Combinatorics #4 - Number of Functions
How many functions
f
:
f
{
1
,
2
,
3
,
4
,
5
}
⟶
{
1
,
2
,
3
,
4
,
5
}
f : f\{1, 2, 3, 4, 5\}\longrightarrow\{1, 2, 3, 4, 5\}
f
:
f
{
1
,
2
,
3
,
4
,
5
}
⟶
{
1
,
2
,
3
,
4
,
5
}
satisfy
f
(
f
(
x
)
)
=
f
(
x
)
f(f(x)) = f(x)
f
(
f
(
x
))
=
f
(
x
)
for all
x
∈
{
1
,
2
,
3
,
4
,
5
}
x\in\{ 1,2, 3, 4, 5\}
x
∈
{
1
,
2
,
3
,
4
,
5
}
?
3
5
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2
4
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1
3
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2009 Algebra #1 - Difference of Squares
If
a
a
a
and
b
b
b
are positive integers such that
a
2
−
b
4
=
2009
a^2-b^4= 2009
a
2
−
b
4
=
2009
, find
a
+
b
a+b
a
+
b
.
2009 Calculus #1: One Unit Lower
Let
f
f
f
be a diff erentiable real-valued function defi ned on the positive real numbers. The tangent lines to the graph of
f
f
f
always meet the
y
y
y
-axis 1 unit lower than where they meet the function. If
f
(
1
)
=
0
f(1)=0
f
(
1
)
=
0
, what is
f
(
2
)
f(2)
f
(
2
)
?
2009 Combinatorics #1 - Arranging Integers from -7 to 7
How many ways can the integers from
−
7
-7
−
7
to
7
7
7
inclusive be arranged in a sequence such that the absolute value of the numbers in the sequence does not decrease?