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Putnam
1988 Putnam
1988 Putnam
Part of
Putnam
Subcontests
(12)
B6
1
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Putnam 1988 B6
Prove that there exist an infinite number of ordered pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
of integers such that for every positive integer
t
t
t
, the number
a
t
+
b
at+b
a
t
+
b
is a triangular number if and only if
t
t
t
is a triangular number. (The triangular numbers are the
t
n
=
n
(
n
+
1
)
/
2
t_n = n(n+1)/2
t
n
=
n
(
n
+
1
)
/2
with
n
n
n
in
{
0
,
1
,
2
,
…
}
\{0,1,2,\dots\}
{
0
,
1
,
2
,
…
}
.)
B5
1
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Putnam 1988 B5
For positive integers
n
n
n
, let
M
n
M_n
M
n
be the
2
n
+
1
2n+1
2
n
+
1
by
2
n
+
1
2n+1
2
n
+
1
skew-symmetric matrix for which each entry in the first
n
n
n
subdiagonals below the main diagonal is 1 and each of the remaining entries below the main diagonal is -1. Find, with proof, the rank of
M
n
M_n
M
n
. (According to one definition, the rank of a matrix is the largest
k
k
k
such that there is a
k
×
k
k \times k
k
×
k
submatrix with nonzero determinant.)One may note that \begin{align*} M_1 &= \left( \begin{array}{ccc} 0 & -1 & 1 \\ 1 & 0 & -1 \\ -1 & 1 & 0 \end{array}\right) \\ M_2 &= \left( \begin{array}{ccccc} 0 & -1 & -1 & 1 & 1 \\ 1 & 0 & -1 & -1 & 1 \\ 1 & 1 & 0 & -1 & -1 \\ -1 & 1 & 1 & 0 & -1 \\ -1 & -1 & 1 & 1 & 0 \end{array} \right). \end{align*}
B4
1
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Putnam 1988 B4
Prove that if
∑
n
=
1
∞
a
n
\sum_{n=1}^\infty a_n
∑
n
=
1
∞
a
n
is a convergent series of positive real numbers, then so is
∑
n
=
1
∞
(
a
n
)
n
/
(
n
+
1
)
\sum_{n=1}^\infty (a_n)^{n/(n+1)}
∑
n
=
1
∞
(
a
n
)
n
/
(
n
+
1
)
.
B3
1
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Putnam 1988 B3
For every
n
n
n
in the set
N
=
{
1
,
2
,
…
}
\mathrm{N} = \{1,2,\dots \}
N
=
{
1
,
2
,
…
}
of positive integers, let
r
n
r_n
r
n
be the minimum value of
∣
c
−
d
3
∣
|c-d\sqrt{3}|
∣
c
−
d
3
∣
for all nonnegative integers
c
c
c
and
d
d
d
with
c
+
d
=
n
c+d=n
c
+
d
=
n
. Find, with proof, the smallest positive real number
g
g
g
with
r
n
≤
g
r_n \leq g
r
n
≤
g
for all
n
∈
N
n \in \mathbb{N}
n
∈
N
.
B2
1
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Putnam 1988 B2
Prove or disprove: If
x
x
x
and
y
y
y
are real numbers with
y
≥
0
y\geq0
y
≥
0
and
y
(
y
+
1
)
≤
(
x
+
1
)
2
y(y+1) \leq (x+1)^2
y
(
y
+
1
)
≤
(
x
+
1
)
2
, then
y
(
y
−
1
)
≤
x
2
y(y-1)\leq x^2
y
(
y
−
1
)
≤
x
2
.
B1
1
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Putnam 1988 B1
A composite (positive integer) is a product
a
b
ab
ab
with
a
a
a
and
b
b
b
not necessarily distinct integers in
{
2
,
3
,
4
,
…
}
\{2,3,4,\dots\}
{
2
,
3
,
4
,
…
}
. Show that every composite is expressible as
x
y
+
x
z
+
y
z
+
1
xy+xz+yz+1
x
y
+
x
z
+
yz
+
1
, with
x
,
y
,
z
x,y,z
x
,
y
,
z
positive integers.
A6
1
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Putnam 1988 A6
If a linear transformation
A
A
A
on an
n
n
n
-dimensional vector space has
n
+
1
n+1
n
+
1
eigenvectors such that any
n
n
n
of them are linearly independent, does it follow that
A
A
A
is a scalar multiple of the identity? Prove your answer.
A5
1
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Putnam 1988 A5
Prove that there exists a unique function
f
f
f
from the set
R
+
\mathrm{R}^+
R
+
of positive real numbers to
R
+
\mathrm{R}^+
R
+
such that
f
(
f
(
x
)
)
=
6
x
−
f
(
x
)
f(f(x)) = 6x-f(x)
f
(
f
(
x
))
=
6
x
−
f
(
x
)
and
f
(
x
)
>
0
f(x)>0
f
(
x
)
>
0
for all
x
>
0
x>0
x
>
0
.
A4
1
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Putnam 1988 A4
(a) If every point of the plane is painted one of three colors, do there necessarily exist two points of the same color exactly one inch apart? (b) What if "three'' is replaced by "nine''?
A3
1
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Putnam 1988 A3
Determine, with proof, the set of real numbers
x
x
x
for which
∑
n
=
1
∞
(
1
n
csc
1
n
−
1
)
x
\sum_{n=1}^\infty \left( \frac{1}{n} \csc \frac{1}{n} - 1 \right)^x
n
=
1
∑
∞
(
n
1
csc
n
1
−
1
)
x
converges.
A2
1
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Putnam 1988 A2
A not uncommon calculus mistake is to believe that the product rule for derivatives says that
(
f
g
)
′
=
f
′
g
′
(fg)' = f'g'
(
f
g
)
′
=
f
′
g
′
. If
f
(
x
)
=
e
x
2
f(x)=e^{x^2}
f
(
x
)
=
e
x
2
, determine, with proof, whether there exists an open interval
(
a
,
b
)
(a,b)
(
a
,
b
)
and a nonzero function
g
g
g
defined on
(
a
,
b
)
(a,b)
(
a
,
b
)
such that this wrong product rule is true for
x
x
x
in
(
a
,
b
)
(a,b)
(
a
,
b
)
.
A1
1
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Putnam 1988 A1
Let
R
R
R
be the region consisting of the points
(
x
,
y
)
(x,y)
(
x
,
y
)
of the cartesian plane satisfying both
∣
x
∣
−
∣
y
∣
≤
1
|x|-|y| \leq 1
∣
x
∣
−
∣
y
∣
≤
1
and
∣
y
∣
≤
1
|y| \leq 1
∣
y
∣
≤
1
. Sketch the region
R
R
R
and find its area.