MathDB
Problems
Contests
Undergraduate contests
Putnam
1972 Putnam
1972 Putnam
Part of
Putnam
Subcontests
(12)
A1
1
Hide problems
Putnam 1972 A1
Show that
(
n
m
)
,
(
n
m
+
1
)
,
(
n
m
+
2
)
\binom{n}{m},\binom{n}{m+1},\binom{n}{m+2}
(
m
n
)
,
(
m
+
1
n
)
,
(
m
+
2
n
)
and
(
n
m
+
3
)
\binom{n}{m+3}
(
m
+
3
n
)
cannot be in arithmetic progression, where
n
,
m
>
0
n,m>0
n
,
m
>
0
and
n
≥
m
+
3
n\geq m+3
n
≥
m
+
3
.
A6
1
Hide problems
Putnam 1972 A6
Let
f
f
f
be an integrable real-valued function on the closed interval
[
0
,
1
]
[0, 1]
[
0
,
1
]
such that
∫
0
1
x
m
f
(
x
)
d
x
=
{
0
for
m
=
0
,
1
,
…
,
n
−
1
;
1
for
m
=
n
.
\int_{0}^{1} x^{m}f(x) dx=\begin{cases} 0 \;\; \text{for}\; m=0,1,\ldots,n-1;\\ 1\;\; \text{for}\; m=n. \end{cases}
∫
0
1
x
m
f
(
x
)
d
x
=
{
0
for
m
=
0
,
1
,
…
,
n
−
1
;
1
for
m
=
n
.
Show that
∣
f
(
x
)
∣
≥
2
n
(
n
+
1
)
|f(x)|\geq2^{n}(n+1)
∣
f
(
x
)
∣
≥
2
n
(
n
+
1
)
on a set of positive measure.
A3
1
Hide problems
Putnam 1972 A3
A sequence
(
x
i
)
(x_{i})
(
x
i
)
is said to have a Cesaro limit exactly if
lim
n
→
∞
x
1
+
…
+
x
n
n
\lim_{n\to\infty} \frac{x_{1}+\ldots+x_{n}}{n}
lim
n
→
∞
n
x
1
+
…
+
x
n
exists. Find all real-valued functions
f
f
f
on the closed interval
[
0
,
1
]
[0, 1]
[
0
,
1
]
such that
(
f
(
x
i
)
)
(f(x_i))
(
f
(
x
i
))
has a Cesaro limit if and only if
(
x
i
)
(x_i)
(
x
i
)
has a Cesaro limit.
A2
1
Hide problems
Putnam 1972 A2
Let
S
S
S
be a set with a binary operation
∗
\ast
∗
such that 1)
a
∗
(
a
∗
b
)
=
b
a \ast(a\ast b)=b
a
∗
(
a
∗
b
)
=
b
for all
a
,
b
∈
S
a,b\in S
a
,
b
∈
S
. 2)
(
a
∗
b
)
∗
b
=
a
(a\ast b)\ast b=a
(
a
∗
b
)
∗
b
=
a
for all
a
,
b
∈
S
a,b\in S
a
,
b
∈
S
. Show that
∗
\ast
∗
is commutative and give an example where
∗
\ast
∗
is not associative.
A4
1
Hide problems
Putnam 1972 A4
Show that a circle inscribed in a square has a larger perimeter than any other ellipse inscribed in the square.
B5
1
Hide problems
Putnam 1972 B5
Let
A
,
B
,
C
A,B,C
A
,
B
,
C
and
D
D
D
be non-coplanar points such that
∠
A
B
C
=
∠
A
D
C
\angle ABC=\angle ADC
∠
A
BC
=
∠
A
D
C
and
∠
B
A
D
=
∠
B
C
D
\angle BAD=\angle BCD
∠
B
A
D
=
∠
BC
D
. Show that
A
B
=
C
D
AB=CD
A
B
=
C
D
and
A
D
=
B
C
AD=BC
A
D
=
BC
.
B4
1
Hide problems
Putnam 1972 B4
Show that for
n
>
1
n > 1
n
>
1
we can find a polynomial
P
(
a
,
b
,
c
)
P(a, b, c)
P
(
a
,
b
,
c
)
with integer coefficients such that
P
(
x
n
,
x
n
+
1
,
x
+
x
n
+
2
)
=
x
.
P(x^{n},x^{n+1},x+x^{n+2})=x.
P
(
x
n
,
x
n
+
1
,
x
+
x
n
+
2
)
=
x
.
B3
1
Hide problems
Putnam 1972 B3
A group
G
G
G
has elements
g
,
h
g,h
g
,
h
satisfying
g
h
g
=
h
g
2
h
,
g
3
=
1
ghg=hg^{2}h, g^{3}=1
g
h
g
=
h
g
2
h
,
g
3
=
1
and
h
n
=
1
h^n=1
h
n
=
1
for some odd integer
n
n
n
. Prove that
h
=
e
h=e
h
=
e
, where
e
e
e
is the identity element.
B2
1
Hide problems
Putnam 1972 B2
A particle moves in a straight line with monotonically decreasing acceleration. It starts from rest and has velocity
v
v
v
a distance
d
d
d
from the start. What is the maximum time it could have taken to travel the distance
d
d
d
?
B1
1
Hide problems
Putnam 1972 B1
Let
∑
n
=
0
∞
x
n
(
x
−
1
)
2
n
n
!
=
∑
n
=
0
∞
a
n
x
n
\sum_{n=0}^{\infty} \frac{x^n (x-1)^{2n}}{n!}=\sum_{n=0}^{\infty} a_{n}x^{n}
∑
n
=
0
∞
n
!
x
n
(
x
−
1
)
2
n
=
∑
n
=
0
∞
a
n
x
n
. Show that no three consecutive
a
n
a_n
a
n
can be equal to
0
0
0
.
A5
1
Hide problems
n| 2^{n} -1 is not possible for n>1
Prove that there is no positive integer
n
>
1
n>1
n
>
1
such that
n
∣
2
n
−
1.
n\mid2^{n} -1.
n
∣
2
n
−
1.
B6
1
Hide problems
Putnam 1972 B6
Let
n
1
<
n
2
<
n
3
<
⋯
<
n
k
n_1<n_2<n_3<\cdots <n_k
n
1
<
n
2
<
n
3
<
⋯
<
n
k
be a set of positive integers. Prove that the polynomial 1\plus{}z^{n_1}\plus{}z^{n_2}\plus{}\cdots \plus{}z^{n_k} has no roots inside the circle |z|<\frac{\sqrt{5}\minus{}1}{2}.