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Putnam
1975 Putnam
1975 Putnam
Part of
Putnam
Subcontests
(12)
A1
1
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Putnam 1975 A1
Show that a positive integer
m
m
m
is a sum of two triangular numbers if and only if
4
m
+
1
4m+1
4
m
+
1
is a sum of two squares.
B6
1
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Putnam 1975 B6
Let
H
n
=
∑
r
=
1
n
1
r
H_n=\sum_{r=1}^{n} \frac{1}{r}
H
n
=
∑
r
=
1
n
r
1
. Show that n-(n-1)n^{-1\slash (n-1)}>H_n>n(n+1)^{1\slash n}-n for
n
>
2
n>2
n
>
2
.
B5
1
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Putnam 1975 B5
Define
f
0
(
x
)
=
e
x
f_{0}(x)=e^x
f
0
(
x
)
=
e
x
and
f
n
+
1
(
x
)
=
x
f
n
′
(
x
)
f_{n+1}(x)=x f_{n}'(x)
f
n
+
1
(
x
)
=
x
f
n
′
(
x
)
. Show that
∑
n
=
0
∞
f
n
(
1
)
n
!
=
e
e
\sum_{n=0}^{\infty} \frac{f_{n}(1)}{n!}=e^e
∑
n
=
0
∞
n
!
f
n
(
1
)
=
e
e
.
B4
1
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Putnam 1975 B4
Does a circle have a subset which is topologically closed and which contains exactly one point of each pair of diametrically opposite points?
B3
1
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Putnam 1975 B3
Let
n
n
n
be a positive integer. Let
S
=
{
a
1
,
…
,
a
k
}
S=\{a_1,\ldots, a_{k}\}
S
=
{
a
1
,
…
,
a
k
}
be a finite collection of at least
n
n
n
not necessarily distinct positive real numbers. Let
f
(
S
)
=
(
∑
i
=
1
k
a
i
)
n
f(S)=\left(\sum_{i=1}^{k} a_{i}\right)^{n}
f
(
S
)
=
(
i
=
1
∑
k
a
i
)
n
and
g
(
S
)
=
∑
1
≤
i
1
<
…
<
i
n
≤
k
a
i
1
⋅
…
⋅
a
i
n
.
g(S)=\sum_{1\leq i_{1}<\ldots<i_{n} \leq k} a_{i_{1}}\cdot\ldots\cdot a_{i_{n}}.
g
(
S
)
=
1
≤
i
1
<
…
<
i
n
≤
k
∑
a
i
1
⋅
…
⋅
a
i
n
.
Determine
sup
S
g
(
S
)
f
(
S
)
\sup_{S} \frac{g(S)}{f(S)}
sup
S
f
(
S
)
g
(
S
)
.
B2
1
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Putnam 1975 B2
A slab is the set of points strictly between two parallel planes. Prove that a countable sequence of slabs, the sum of whose thicknesses converges, cannot fill space.
B1
1
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Putnam 1975 B1
Consider the additive group
Z
2
\mathbb{Z}^{2}
Z
2
. Let
H
H
H
be the smallest subgroup containing
(
3
,
8
)
,
(
4
,
−
1
)
(3,8), (4,-1)
(
3
,
8
)
,
(
4
,
−
1
)
and
(
5
,
4
)
(5,4)
(
5
,
4
)
. Let
H
x
y
H_{xy}
H
x
y
be the smallest subgroup containing
(
0
,
x
)
(0,x)
(
0
,
x
)
and
(
1
,
y
)
(1,y)
(
1
,
y
)
. Find some pair
(
x
,
y
)
(x,y)
(
x
,
y
)
with
x
>
0
x>0
x
>
0
such that
H
=
H
x
y
H=H_{xy}
H
=
H
x
y
.
A6
1
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Putnam 1975 A6
Given three points in space forming an acute-angled triangle, show that we can find two further points such that no three of the five points are collinear and the line through any two is normal to the plane through the other three.
A5
1
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Putnam 1975 A5
Let
I
⊂
R
I\subset \mathbb{R}
I
⊂
R
be an interval and
f
(
x
)
f(x)
f
(
x
)
a continuous real-valued function on
I
I
I
. Let
y
1
y_1
y
1
and
y
2
y_2
y
2
be linearly independent solutions of
y
′
′
=
f
(
x
)
y
y''=f(x)y
y
′′
=
f
(
x
)
y
taking positive values on
I
I
I
. Show that for some positive number
k
k
k
the function
k
⋅
y
1
y
2
k\cdot\sqrt{y_1 y_2}
k
⋅
y
1
y
2
is a solution of
y
′
′
+
1
y
3
=
f
(
x
)
y
y''+\frac{1}{y^{3}}=f(x)y
y
′′
+
y
3
1
=
f
(
x
)
y
.
A4
1
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Putnam 1975 A4
Let
m
>
1
m>1
m
>
1
be an odd integer. Let
n
=
2
m
n=2m
n
=
2
m
and \theta=e^{2\pi i\slash n}. Find integers
a
1
,
…
,
a
k
a_{1},\ldots,a_{k}
a
1
,
…
,
a
k
such that
∑
i
=
1
k
a
i
θ
i
=
1
1
−
θ
\sum_{i=1}^{k}a_{i}\theta^{i}=\frac{1}{1-\theta}
∑
i
=
1
k
a
i
θ
i
=
1
−
θ
1
.
A3
1
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Putnam 1975 A3
Let
0
<
α
<
β
<
γ
∈
R
0<\alpha<\beta <\gamma\in \mathbb{R}
0
<
α
<
β
<
γ
∈
R
. Let
K
=
{
(
x
,
y
,
z
)
∈
R
3
∣
x
,
y
,
z
≥
0
and
x
β
+
y
β
+
z
β
=
1
}
K=\{(x,y,z)\in \mathbb{R}^{3}\;|\; x,y,z\geq 0\; \text{and}\; x^{\beta}+y^{\beta}+z^{\beta}=1\}
K
=
{(
x
,
y
,
z
)
∈
R
3
∣
x
,
y
,
z
≥
0
and
x
β
+
y
β
+
z
β
=
1
}
. Define
f
:
K
→
R
,
(
x
,
y
,
z
)
↦
x
α
+
y
β
+
z
γ
f:K\rightarrow \mathbb{R},\; (x,y,z)\mapsto x^{\alpha}+y^{\beta}+z^{\gamma}
f
:
K
→
R
,
(
x
,
y
,
z
)
↦
x
α
+
y
β
+
z
γ
. At what points of
K
K
K
does
f
f
f
assume its minimal and maximal values?
A2
1
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Putnam 1975 A2
Describe the region
R
R
R
consisting of the points
(
a
,
b
)
(a,b)
(
a
,
b
)
of the cartesian plane for which both (possibly complex) roots of the polynomial
z
2
+
a
z
+
b
z^2+az+b
z
2
+
a
z
+
b
have absolute value smaller than
1
1
1
.