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Putnam
1966 Putnam
1966 Putnam
Part of
Putnam
Subcontests
(12)
A5
1
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Putnam 1966 A5
Let
C
C
C
denote the family of continuous functions on the real axis. Let
T
T
T
be a mapping of
C
C
C
into
C
C
C
which has the following properties:1.
T
T
T
is linear, i.e.
T
(
c
1
ψ
1
+
c
2
ψ
2
)
=
c
1
T
ψ
1
+
c
2
T
ψ
2
T(c_1\psi _1+c_2\psi _2)= c_1T\psi _1+c_2T\psi _2
T
(
c
1
ψ
1
+
c
2
ψ
2
)
=
c
1
T
ψ
1
+
c
2
T
ψ
2
for
c
1
c_1
c
1
and
c
2
c_2
c
2
real and
ψ
1
\psi_1
ψ
1
and
ψ
2
\psi_2
ψ
2
in
C
C
C
.2.
T
T
T
is local, i.e. if
ψ
1
≡
ψ
2
\psi_1 \equiv \psi_2
ψ
1
≡
ψ
2
in some interval
I
I
I
then also
T
ψ
1
≡
T
ψ
2
T\psi_1 \equiv T\psi_2
T
ψ
1
≡
T
ψ
2
holds in
I
I
I
.Show that
T
T
T
must necessarily be of the form
T
ψ
(
x
)
=
f
(
x
)
ψ
(
x
)
T\psi(x)=f(x)\psi(x)
T
ψ
(
x
)
=
f
(
x
)
ψ
(
x
)
where
f
(
x
)
f(x)
f
(
x
)
is a suitable continuous function.
B3
1
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Putnam 1966 B3
Show that if the series
∑
n
=
1
∞
1
p
n
\sum_{n=1}^{\infty} \frac{1}{p_n}
n
=
1
∑
∞
p
n
1
is convergent, where
p
1
,
p
2
,
p
3
,
…
,
p
n
,
…
p_1,p_2,p_3,\dots, p_n, \dots
p
1
,
p
2
,
p
3
,
…
,
p
n
,
…
are positive real numbers, then the series
∑
n
=
1
∞
n
2
(
p
1
+
p
2
+
⋯
+
p
n
)
2
p
n
\sum_{n=1}^{\infty} \frac{n^2}{(p_1+p_2+\dots +p_n)^2}p_n
n
=
1
∑
∞
(
p
1
+
p
2
+
⋯
+
p
n
)
2
n
2
p
n
is also convergent.
B6
1
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Putnam 1966 B6
Show that all the solutions of the differential equation
y
′
′
+
e
x
y
=
0
y''+e^xy=0
y
′′
+
e
x
y
=
0
remain bounded as
x
→
∞
.
x\to \infty.
x
→
∞.
B5
1
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Putnam 1966 B5
Given
n
(
≥
3
)
n(\geq 3)
n
(
≥
3
)
distinct points in the plane, no three of which are on the same straight line, prove that there exists a simple closed polygon with these points as vertices.
B4
1
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Putnam 1966 B4
Let
0
<
a
1
<
a
2
<
⋯
<
a
m
n
+
1
0<a_1<a_2< \dots < a_{mn+1}
0
<
a
1
<
a
2
<
⋯
<
a
mn
+
1
be
m
n
+
1
mn+1
mn
+
1
integers. Prove that you can select either
m
+
1
m+1
m
+
1
of them no one of which divides any other, or
n
+
1
n+1
n
+
1
of them each dividing the following one.
B2
1
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Putnam 1966 B2
Prove that among any ten consecutive integers at least one is relatively prime to each of the others.
B1
1
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Putnam 1966 B1
Let a convex polygon
P
P
P
be contained in a square of side one. Show that the sum of the sides of
P
P
P
is less than or equal to
4
4
4
.
A6
1
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Putnam 1966 A6
Justify the statement that
3
=
1
+
2
1
+
3
1
+
4
1
+
5
1
+
…
.
3=\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{1+\dots}}}}}.
3
=
1
+
2
1
+
3
1
+
4
1
+
5
1
+
…
.
A4
1
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Putnam 1966 A4
Prove that after deleting the perfect squares from the list of positive integers the number we find in the
n
t
h
n^{th}
n
t
h
position is equal to
n
+
{
n
}
,
n+\{\sqrt{n}\},
n
+
{
n
}
,
where
{
n
}
\{\sqrt{n}\}
{
n
}
denotes the integer closest to
n
.
\sqrt{n}.
n
.
A3
1
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Putnam 1966 A3
Let
0
<
x
1
<
1
0<x_1<1
0
<
x
1
<
1
and
x
n
+
1
=
x
n
(
1
−
x
n
)
,
n
=
1
,
2
,
3
,
…
x_{n+1}=x_n(1-x_n), n=1,2,3, \dots
x
n
+
1
=
x
n
(
1
−
x
n
)
,
n
=
1
,
2
,
3
,
…
. Show that
lim
n
→
∞
n
x
n
=
1.
\lim_{n \to \infty} nx_n=1.
n
→
∞
lim
n
x
n
=
1.
A2
1
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Putnam 1966 A2
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be the lengths of the sides of a triangle, let
p
=
(
a
+
b
+
c
)
/
2
p=(a+b+c)/2
p
=
(
a
+
b
+
c
)
/2
, and
r
r
r
be the radius of the inscribed circle. Show that
1
(
p
−
a
)
2
+
1
(
p
−
b
)
2
+
1
(
p
−
c
)
2
≥
1
r
2
.
\frac{1}{(p-a)^2}+ \frac{1}{(p-b)^2}+\frac{1}{(p-c)^2} \geq \frac{1}{r^2}.
(
p
−
a
)
2
1
+
(
p
−
b
)
2
1
+
(
p
−
c
)
2
1
≥
r
2
1
.
A1
1
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Putnam 1966 A1
Let
f
(
n
)
f(n)
f
(
n
)
be the sum of the first
n
n
n
terms of the sequence
0
,
1
,
1
,
2
,
2
,
3
,
3
,
4
,
…
,
0,1,1,2,2,3,3,4, \dots,
0
,
1
,
1
,
2
,
2
,
3
,
3
,
4
,
…
,
where the
n
n
n
th term is given by
a
n
=
{
n
/
2
if
n
is even,
(
n
−
1
)
/
2
if
n
is odd.
a_n= \begin{cases} n/2 & \text{if } n \text{ is even,} \\ (n-1)/2 & \text{if } n \text{ is odd.} \end{cases}
a
n
=
{
n
/2
(
n
−
1
)
/2
if
n
is even,
if
n
is odd.
Show that if
x
x
x
and
y
y
y
are positive integers and
x
>
y
x>y
x
>
y
then
x
y
=
f
(
x
+
y
)
−
f
(
x
−
y
)
xy=f(x+y)-f(x-y)
x
y
=
f
(
x
+
y
)
−
f
(
x
−
y
)
.