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Putnam
1968 Putnam
1968 Putnam
Part of
Putnam
Subcontests
(12)
B6
1
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Putnam 1968 B6
Show that one cannot find compact sets
A
1
,
A
2
,
A
3
,
…
A_1, A_2, A_3, \ldots
A
1
,
A
2
,
A
3
,
…
in
R
\mathbb{R}
R
such that (1) All elements of
A
n
A_n
A
n
are rational. (2) Any compact set
K
⊂
R
K\subset \mathbb{R}
K
⊂
R
which only contains rational numbers is contained in some
A
m
A_{m}
A
m
.
B5
1
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Putnam 1968 B5
Let
S
S
S
be the set of
2
×
2
2\times2
2
×
2
-matrices over
F
p
\mathbb{F}_{p}
F
p
with trace
1
1
1
and determinant
0
0
0
. Determine
∣
S
∣
|S|
∣
S
∣
.
B4
1
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Putnam 1968 B4
Suppose that
f
:
R
→
R
f:\mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
is continuous and
L
=
∫
−
∞
∞
f
(
x
)
d
x
L=\int_{-\infty}^{\infty} f(x) dx
L
=
∫
−
∞
∞
f
(
x
)
d
x
exists. Show that
∫
−
∞
∞
f
(
x
−
1
x
)
d
x
=
L
.
\int_{-\infty}^{\infty}f\left(x-\frac{1}{x}\right)dx=L.
∫
−
∞
∞
f
(
x
−
x
1
)
d
x
=
L
.
B3
1
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Putnam 1968 B3
Given that a
6
0
∘
60^{\circ}
6
0
∘
angle cannot be trisected with ruler and compass, prove that a
12
0
∘
n
\frac{120^{\circ}}{n}
n
12
0
∘
angle cannot be trisected with ruler and compass for
n
=
1
,
2
,
…
n=1,2,\ldots
n
=
1
,
2
,
…
B2
1
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Putnam 1968 B2
Let
G
G
G
be a finite group with
n
n
n
elements and
K
K
K
a subset of
G
G
G
with more than
n
2
\frac{n}{2}
2
n
elements. Show that for any
g
∈
G
g\in G
g
∈
G
one can find
h
,
k
∈
K
h,k\in K
h
,
k
∈
K
such that
g
=
h
⋅
k
g=h\cdot k
g
=
h
⋅
k
.
B1
1
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Putnam 1968 B1
The random variables
X
,
Y
X, Y
X
,
Y
can each take a finite number of integer values. They are not necessarily independent. Express
P
(
min
(
X
,
Y
)
=
k
)
P(\min(X,Y)=k)
P
(
min
(
X
,
Y
)
=
k
)
in terms of
p
1
=
P
(
X
=
k
)
p_1=P(X=k)
p
1
=
P
(
X
=
k
)
,
p
2
=
P
(
Y
=
k
)
p_2=P(Y=k)
p
2
=
P
(
Y
=
k
)
and
p
3
=
P
(
max
(
X
,
Y
)
=
k
)
p_3=P(\max(X,Y)=k)
p
3
=
P
(
max
(
X
,
Y
)
=
k
)
.
A6
1
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Putnam 1968 A6
Find all polynomials whose coefficients are all
±
1
\pm1
±
1
and whose roots are all real.
A5
1
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Putnam 1968 A5
Find the smallest possible
α
∈
R
\alpha\in \mathbb{R}
α
∈
R
such that if
P
(
x
)
=
a
x
2
+
b
x
+
c
P(x)=ax^2+bx+c
P
(
x
)
=
a
x
2
+
b
x
+
c
satisfies
∣
P
(
x
)
∣
≤
1
|P(x)|\leq1
∣
P
(
x
)
∣
≤
1
for
x
∈
[
0
,
1
]
x\in [0,1]
x
∈
[
0
,
1
]
, then we also have
∣
P
′
(
0
)
∣
≤
α
|P'(0)|\leq \alpha
∣
P
′
(
0
)
∣
≤
α
.
A4
1
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Putnam 1968 A4
Let
S
2
⊂
R
3
S^{2}\subset \mathbb{R}^{3}
S
2
⊂
R
3
be the unit sphere. Show that for any
n
n
n
points on
S
2
S^{2}
S
2
, the sum of the squares of the
n
(
n
−
1
)
2
\frac{n(n-1)}{2}
2
n
(
n
−
1
)
distances between them is at most
n
2
n^{2}
n
2
.
A3
1
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Putnam 1968 A3
Let
S
S
S
be a finite set and
P
P
P
the set of all subsets of
S
S
S
. Show that one can label the elements of
P
P
P
as
A
i
A_i
A
i
such that (1)
A
1
=
∅
A_1 =\emptyset
A
1
=
∅
. (2) For each
n
≥
1
n\geq1
n
≥
1
we either have
A
n
−
1
⊂
A
n
A_{n-1}\subset A_{n}
A
n
−
1
⊂
A
n
and
∣
A
n
∖
A
n
−
1
∣
=
1
|A_{n} \setminus A_{n-1}|=1
∣
A
n
∖
A
n
−
1
∣
=
1
or
A
n
⊂
A
n
−
1
A_{n}\subset A_{n-1}
A
n
⊂
A
n
−
1
and
∣
A
n
−
1
∖
A
n
∣
=
1.
|A_{n-1} \setminus A_{n}|=1.
∣
A
n
−
1
∖
A
n
∣
=
1.
A2
1
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Putnam 1968 A2
Given integers
a
,
b
,
c
,
d
,
m
,
n
a,b,c,d,m,n
a
,
b
,
c
,
d
,
m
,
n
such that
a
d
−
b
c
≠
0
ad-bc\ne 0
a
d
−
b
c
=
0
and any real
ε
>
0
\varepsilon >0
ε
>
0
, show that one can find rational numbers
x
,
y
x,y
x
,
y
such that
0
<
∣
a
x
+
b
y
−
m
∣
<
ε
0<|ax+by-m|<\varepsilon
0
<
∣
a
x
+
b
y
−
m
∣
<
ε
and
0
<
∣
c
x
+
d
y
−
n
∣
<
ε
0<|cx+dy-n|<\varepsilon
0
<
∣
c
x
+
d
y
−
n
∣
<
ε
.
A1
1
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Approximate Value of Pi
Prove \ \ \ \frac{22}{7}\minus{}\pi \equal{}\int_0^1 \frac{x^4(1\minus{}x)^4}{1\plus{}x^2}\ dx.