MathDB
Problems
Contests
Undergraduate contests
Putnam
1964 Putnam
1964 Putnam
Part of
Putnam
Subcontests
(12)
B5
1
Hide problems
Putnam 1964 B5
Let
u
n
u_n
u
n
denote the least common multiple of the first
n
n
n
terms of a strictly increasing sequence of positive integers. Prove that the series
∑
n
=
1
∞
1
u
n
\sum_{n=1}^{\infty} \frac{1}{ u_n }
n
=
1
∑
∞
u
n
1
is convergent
B2
1
Hide problems
Putnam 1964 B2
Let
S
S
S
be a set of
n
>
0
n>0
n
>
0
elements, and let
A
1
,
A
2
,
…
A
k
A_1 , A_2 , \ldots A_k
A
1
,
A
2
,
…
A
k
be a family of distinct subsets such that any two have a non-empty intersection. Assume that no other subset of
S
S
S
intersects all of the
A
i
.
A_i.
A
i
.
Prove that
k
=
2
n
−
1
.
k=2^{n-1}.
k
=
2
n
−
1
.
B1
1
Hide problems
Putnam 1964 B1
Let
u
k
u_k
u
k
be a sequence of integers, and let
V
n
V_n
V
n
be the number of those which are less than or equal to
n
n
n
. Show that if
∑
k
=
1
∞
1
u
k
<
∞
,
\sum_{k=1}^{\infty} \frac{1}{u_k } < \infty,
k
=
1
∑
∞
u
k
1
<
∞
,
then
lim
n
→
∞
V
n
n
=
0.
\lim_{n \to \infty} \frac{ V_{n}}{n}=0.
n
→
∞
lim
n
V
n
=
0.
A6
1
Hide problems
Putnam 1964 A6
Let
S
S
S
be a finite subset of a straight line. Say that
S
S
S
has the repeated distance property if every value of the distance between two points of
S
S
S
(except the longest) occurs at least twice. Show that if
S
S
S
has the repeated distance property then the ratio of any two distances between two points of
S
S
S
is rational.
A5
1
Hide problems
Putnam 1964 A5
Prove that there exists a constant
K
K
K
such that the following inequality holds for any sequence of positive numbers
a
1
,
a
2
,
a
3
,
…
:
a_1 , a_2 , a_3 , \ldots:
a
1
,
a
2
,
a
3
,
…
:
∑
n
=
1
∞
n
a
1
+
a
2
+
…
+
a
n
≤
K
∑
n
=
1
∞
1
a
n
.
\sum_{n=1}^{\infty} \frac{n}{a_1 + a_2 +\ldots + a_n } \leq K \sum_{n=1}^{\infty} \frac{1}{a_{n}}.
n
=
1
∑
∞
a
1
+
a
2
+
…
+
a
n
n
≤
K
n
=
1
∑
∞
a
n
1
.
A4
1
Hide problems
Putnam 1964 A4
Let
p
n
p_n
p
n
be a bounded sequence of integers which satisfies the recursion
p
n
=
p
n
−
1
+
p
n
−
2
+
p
n
−
3
p
n
−
4
p
n
−
1
p
n
−
2
+
p
n
−
3
+
p
n
−
4
.
p_n =\frac{p_{n-1} +p_{n-2} + p_{n-3}p _{n-4}}{p_{n-1} p_{n-2}+ p_{n-3} +p_{n-4}}.
p
n
=
p
n
−
1
p
n
−
2
+
p
n
−
3
+
p
n
−
4
p
n
−
1
+
p
n
−
2
+
p
n
−
3
p
n
−
4
.
Show that the sequence eventually becomes periodic.
A3
1
Hide problems
Putnam 1964 A3
Let
P
1
,
P
2
,
…
P_1 , P_2 , \ldots
P
1
,
P
2
,
…
be a sequence of distinct points which is dense in the interval
(
0
,
1
)
(0,1)
(
0
,
1
)
. The points
P
1
,
…
,
P
n
−
1
P_1 , \ldots , P_{n-1}
P
1
,
…
,
P
n
−
1
decompose the interval into
n
n
n
parts, and
P
n
P_n
P
n
decomposes one of these into two parts. Let
a
n
a_n
a
n
and
b
n
b_n
b
n
be the length of these two intervals. Prove that \sum_{n=1}^{\infty} a_n b_n (a_n +b_n) =1 \slash 3.
A2
1
Hide problems
Putnam 1964 A2
Find all continuous positive functions
f
(
x
)
f(x)
f
(
x
)
, for
0
≤
x
≤
1
0\leq x \leq 1
0
≤
x
≤
1
, such that
∫
0
1
f
(
x
)
d
x
=
1
,
\int_{0}^{1} f(x)\; dx =1,
∫
0
1
f
(
x
)
d
x
=
1
,
∫
0
1
x
f
(
x
)
d
x
=
α
,
\int_{0}^{1} xf(x)\; dx =\alpha,
∫
0
1
x
f
(
x
)
d
x
=
α
,
∫
0
1
x
2
f
(
x
)
d
x
=
α
2
,
\int_{0}^{1} x^2 f(x)\; dx =\alpha^2,
∫
0
1
x
2
f
(
x
)
d
x
=
α
2
,
where
α
\alpha
α
is a given real number.
A1
1
Hide problems
six points in a plane, an inequality
Given
6
6
6
points in a plane, assume that each two of them are connected by a segment. Let
D
D
D
be the length of the longest, and
d
d
d
the length of the shortest of these segments. Prove that
D
d
≥
3
\frac Dd\ge\sqrt3
d
D
≥
3
.
B4
1
Hide problems
Great Circles on a Sphere
Into how many regions do
n
n
n
great circles, no three of which meet at a point, divide a sphere?
B3
1
Hide problems
f(na) goes to 0 for all a - Paenza 2010
Let
f
:
R
→
R
f: \mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
be a continuous function with the following property: for all
α
∈
R
>
0
\alpha \in \mathbb{R}_{>0}
α
∈
R
>
0
, the sequence
(
a
n
)
n
∈
N
(a_n)_{n \in \mathbb{N}}
(
a
n
)
n
∈
N
defined as
a
n
=
f
(
n
α
)
a_n = f(n\alpha)
a
n
=
f
(
n
α
)
satisfies
lim
n
→
∞
a
n
=
0
\lim_{n \to \infty} a_n = 0
lim
n
→
∞
a
n
=
0
. Is it necessarily true that
lim
x
→
+
∞
f
(
x
)
=
0
\lim_{x \to +\infty} f(x) = 0
lim
x
→
+
∞
f
(
x
)
=
0
?
B6
1
Hide problems
partition of a disk
This is rather simple, but I liked it :). Show that a disk cannot be partitioned into two congruent subsets.