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Undergraduate contests
Putnam
2009 Putnam
2009 Putnam
Part of
Putnam
Subcontests
(12)
B6
1
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Putnam 2009 B6
Prove that for every positive integer
n
,
n,
n
,
there is a sequence of integers
a
0
,
a
1
,
…
,
a
2009
a_0,a_1,\dots,a_{2009}
a
0
,
a
1
,
…
,
a
2009
with a_0\equal{}0 and a_{2009}\equal{}n such that each term after
a
0
a_0
a
0
is either an earlier term plus
2
k
2^k
2
k
for some nonnnegative integer
k
,
k,
k
,
or of the form
b
m
o
d
c
b\mod{c}
b
mod
c
for some earlier positive terms
b
b
b
and
c
.
c.
c
.
[Here
b
m
o
d
c
b\mod{c}
b
mod
c
denotes the remainder when
b
b
b
is divided by
c
,
c,
c
,
so
0
≤
(
b
m
o
d
c
)
<
c
.
0\le(b\mod{c})<c.
0
≤
(
b
mod
c
)
<
c
.
]
B5
1
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Putnam 2009 B5
Let
f
:
(
1
,
∞
)
→
R
f: (1,\infty)\to\mathbb{R}
f
:
(
1
,
∞
)
→
R
be a differentiable function such that f'(x)\equal{}\frac{x^2\minus{}\left(f(x)\right)^2}{x^2\left(\left(f(x)\right)^2\plus{}1\right)} \text{for all }x>1. Prove that \displaystyle\lim_{x\to\infty}f(x)\equal{}\infty.
B4
1
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Putnam 2009 B4
Say that a polynomial with real coefficients in two variable,
x
,
y
,
x,y,
x
,
y
,
is balanced if the average value of the polynomial on each circle centered at the origin is
0.
0.
0.
The balanced polynomials of degree at most
2009
2009
2009
form a vector space
V
V
V
over
R
.
\mathbb{R}.
R
.
Find the dimension of
V
.
V.
V
.
B3
1
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Putnam 2009 B3
Call a subset
S
S
S
of
{
1
,
2
,
…
,
n
}
\{1,2,\dots,n\}
{
1
,
2
,
…
,
n
}
mediocre if it has the following property: Whenever
a
a
a
and
b
b
b
are elements of
S
S
S
whose average is an integer, that average is also an element of
S
.
S.
S
.
Let
A
(
n
)
A(n)
A
(
n
)
be the number of mediocre subsets of
{
1
,
2
,
…
,
n
}
.
\{1,2,\dots,n\}.
{
1
,
2
,
…
,
n
}
.
[For instance, every subset of
{
1
,
2
,
3
}
\{1,2,3\}
{
1
,
2
,
3
}
except
{
1
,
3
}
\{1,3\}
{
1
,
3
}
is mediocre, so A(3)\equal{}7.] Find all positive integers
n
n
n
such that A(n\plus{}2)\minus{}2A(n\plus{}1)\plus{}A(n)\equal{}1.
B2
1
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Putnam 2009 B2
A game involves jumping to the right on the real number line. If
a
a
a
and
b
b
b
are real numbers and
b
>
a
,
b>a,
b
>
a
,
the cost of jumping from
a
a
a
to
b
b
b
is b^3\minus{}ab^2. For what real numbers
c
c
c
can one travel from
0
0
0
to
1
1
1
in a finite number of jumps with total cost exactly
c
?
c?
c
?
B1
1
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Putnam 2009 B1
Show that every positive rational number can be written as a quotient of products of factorials of (not necessarily distinct) primes. For example, \frac{10}9\equal{}\frac{2!\cdot 5!}{3!\cdot 3!\cdot 3!}.
A6
1
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Putnam 2009 A6
Let
f
:
[
0
,
1
]
2
→
R
f: [0,1]^2\to\mathbb{R}
f
:
[
0
,
1
]
2
→
R
be a continuous function on the closed unit square such that
∂
f
∂
x
\frac{\partial f}{\partial x}
∂
x
∂
f
and
∂
f
∂
y
\frac{\partial f}{\partial y}
∂
y
∂
f
exist and are continuous on the interior of
(
0
,
1
)
2
.
(0,1)^2.
(
0
,
1
)
2
.
Let a\equal{}\int_0^1f(0,y)\,dy,\ b\equal{}\int_0^1f(1,y)\,dy,\ c\equal{}\int_0^1f(x,0)\,dx and d\equal{}\int_0^1f(x,1)\,dx. Prove or disprove: There must be a point
(
x
0
,
y
0
)
(x_0,y_0)
(
x
0
,
y
0
)
in
(
0
,
1
)
2
(0,1)^2
(
0
,
1
)
2
such that \frac{\partial f}{\partial x}(x_0,y_0)\equal{}b\minus{}a and \frac{\partial f}{\partial y}(x_0,y_0)\equal{}d\minus{}c.
A5
1
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Putnam 2009 A5
Is there a finite abelian group
G
G
G
such that the product of the orders of all its elements is
2
2009
?
2^{2009}?
2
2009
?
A4
1
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Putnam 2009 A4
Let
S
S
S
be a set of rational numbers such that (a)
0
∈
S
;
0\in S;
0
∈
S
;
(b) If
x
∈
S
x\in S
x
∈
S
then x\plus{}1\in S and x\minus{}1\in S; and (c) If
x
∈
S
x\in S
x
∈
S
and
x
∉
{
0
,
1
}
,
x\notin\{0,1\},
x
∈
/
{
0
,
1
}
,
then \frac{1}{x(x\minus{}1)}\in S. Must
S
S
S
contain all rational numbers?
A3
1
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Putnam 2009 A3
Let
d
n
d_n
d
n
be the determinant of the
n
×
n
n\times n
n
×
n
matrix whose entries, from left to right and then from top to bottom, are
cos
1
,
cos
2
,
…
,
cos
n
2
.
\cos 1,\cos 2,\dots,\cos n^2.
cos
1
,
cos
2
,
…
,
cos
n
2
.
(For example, d_3 \equal{} \begin{vmatrix}\cos 1 & \cos2 & \cos3 \\ \cos4 & \cos5 & \cos 6 \\ \cos7 & \cos8 & \cos 9\end{vmatrix}. The argument of
cos
\cos
cos
is always in radians, not degrees.) Evaluate
lim
n
→
∞
d
n
.
\lim_{n\to\infty}d_n.
lim
n
→
∞
d
n
.
A2
1
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Putnam 2009 A2
Functions
f
,
g
,
h
f,g,h
f
,
g
,
h
are differentiable on some open interval around
0
0
0
and satisfy the equations and initial conditions \begin{align*}f'&=2f^2gh+\frac1{gh},\ f(0)=1,\\ g'&=fg^2h+\frac4{fh},\ g(0)=1,\\ h'&=3fgh^2+\frac1{fg},\ h(0)=1.\end{align*} Find an explicit formula for
f
(
x
)
,
f(x),
f
(
x
)
,
valid in some open interval around
0.
0.
0.
A1
1
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Putnam 2009 A1
Let
f
f
f
be a real-valued function on the plane such that for every square
A
B
C
D
ABCD
A
BC
D
in the plane, f(A)\plus{}f(B)\plus{}f(C)\plus{}f(D)\equal{}0. Does it follow that f(P)\equal{}0 for all points
P
P
P
in the plane?