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Putnam
1957 Putnam
1957 Putnam
Part of
Putnam
Subcontests
(14)
B7
1
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Putnam 1957 B7
Let
C
C
C
consist of a regular polygon and its interior. Show that for each positive integer
n
n
n
, there exists a set of points
S
(
n
)
S(n)
S
(
n
)
in the plane such that every
n
n
n
points can be covered by
C
C
C
, but
S
(
n
)
S(n)
S
(
n
)
cannot be covered by
C
.
C.
C
.
B6
1
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Putnam 1957 B6
The curve
y
=
y
(
x
)
y=y(x)
y
=
y
(
x
)
satisfies
y
′
(
0
)
=
1.
y'(0)=1.
y
′
(
0
)
=
1.
It satisfies the differential equation
(
x
2
+
9
)
y
′
′
+
(
x
2
+
4
)
y
=
0.
(x^2 +9)y'' +(x^2 +4)y=0.
(
x
2
+
9
)
y
′′
+
(
x
2
+
4
)
y
=
0.
Show that it crosses the
x
x
x
-axis between
x
=
3
2
π
and
x
=
63
53
π
.
x= \frac{3}{2} \pi \;\;\; \text{and} \;\;\; x= \sqrt{\frac{63}{53}} \pi.
x
=
2
3
π
and
x
=
53
63
π
.
B4
1
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Putnam 1957 B4
Let
a
(
n
)
a(n)
a
(
n
)
be the number of representations of the positive integer
n
n
n
as an ordered sum of
1
1
1
's and
2
2
2
's. Let
b
(
n
)
b(n)
b
(
n
)
be the number of representations of the positive integer
n
n
n
as an ordered sum of integers greater than
1.
1.
1.
Show that
a
(
n
)
=
b
(
n
+
2
)
a(n)=b(n+2)
a
(
n
)
=
b
(
n
+
2
)
for each
n
n
n
.
B3
1
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Putnam 1957 B3
For
f
(
x
)
f(x)
f
(
x
)
a positive , monotone decreasing function defined in
[
0
,
1
]
,
[0,1],
[
0
,
1
]
,
prove that
∫
0
1
f
(
x
)
d
x
⋅
∫
0
1
x
f
(
x
)
2
d
x
≤
∫
0
1
f
(
x
)
2
d
x
⋅
∫
0
1
x
f
(
x
)
d
x
.
\int_{0}^{1} f(x) dx \cdot \int_{0}^{1} xf(x)^{2} dx \leq \int_{0}^{1} f(x)^{2} dx \cdot \int_{0}^{1} xf(x) dx.
∫
0
1
f
(
x
)
d
x
⋅
∫
0
1
x
f
(
x
)
2
d
x
≤
∫
0
1
f
(
x
)
2
d
x
⋅
∫
0
1
x
f
(
x
)
d
x
.
B2
1
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Putnam 1957 B2
In order to determine
1
A
\frac{1}{A}
A
1
for
A
>
0
A>0
A
>
0
, one can use the iteration
X
k
+
1
=
X
k
(
2
−
A
X
k
)
,
X_{k+1}=X_{k}(2-AX_{k}),
X
k
+
1
=
X
k
(
2
−
A
X
k
)
,
where
X
0
X_0
X
0
is a selected starting value. Find the limitation, if any, on the starting value
X
0
X_0
X
0
so that the above iteration converges to
1
A
.
\frac{1}{A}.
A
1
.
B1
1
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Putnam 1957 B1
Consider the determinant of the matrix
(
a
i
j
)
i
j
(a_{ij})_{ij}
(
a
ij
)
ij
with
1
≤
i
,
j
≤
100
1\leq i,j \leq 100
1
≤
i
,
j
≤
100
and
a
i
j
=
i
j
.
a_{ij}=ij.
a
ij
=
ij
.
Prove that if the absolute value of each of the
100
!
100!
100
!
terms in the expansion of this determinant is divided by
101
,
101,
101
,
then the remainder is always
1.
1.
1.
A7
1
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Putnam 1957 A7
Each member of a set of circles in the
x
y
xy
x
y
-plane is tangent to the
x
x
x
-axis and no two of the circles intersect. Show that (a) the points of tangency can include all rational points on the axis. (b) the points of tangency cannot include all the irrational points.
A6
1
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Putnam 1957 A6
Let
a
>
0
a>0
a
>
0
,
S
1
=
ln
a
S_1 =\ln a
S
1
=
ln
a
and
S
n
=
∑
i
=
1
n
−
1
ln
(
a
−
S
i
)
S_n = \sum_{i=1 }^{n-1} \ln( a- S_i )
S
n
=
∑
i
=
1
n
−
1
ln
(
a
−
S
i
)
for
n
>
1.
n >1.
n
>
1.
Show that
lim
n
→
∞
S
n
=
a
−
1.
\lim_{n \to \infty} S_n = a-1.
n
→
∞
lim
S
n
=
a
−
1.
A5
1
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Putnam 1957 A5
Given
n
n
n
points in the plane, show that the largest distance determined by these points cannot occur more than
n
n
n
times.
A3
1
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Putnam 1957 A3
Let
a
,
b
a,b
a
,
b
be real numbers and
k
k
k
a positive integer. Show that
∣
cos
k
b
cos
a
−
cos
k
a
cos
b
cos
b
−
cos
a
∣
<
k
2
−
1
\left| \frac{ \cos kb \cos a - \cos ka \cos b}{\cos b -\cos a} \right|<k^2 -1
cos
b
−
cos
a
cos
kb
cos
a
−
cos
ka
cos
b
<
k
2
−
1
whenever the left side is defined.
A2
1
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Putnam 1957 A2
Let
a
>
1.
a>1.
a
>
1.
A uniform wire is bent into a form coinciding with the portion of the curve
y
=
e
x
y=e^x
y
=
e
x
for
x
∈
[
0
,
a
]
x\in [0,a]
x
∈
[
0
,
a
]
, and the line segment
y
=
e
a
y=e^a
y
=
e
a
for
x
∈
[
a
−
1
,
a
]
.
x\in [a-1,a].
x
∈
[
a
−
1
,
a
]
.
The wire is then suspended from the point
(
a
−
1
,
e
a
)
(a-1, e^a)
(
a
−
1
,
e
a
)
and a horizontal force
F
F
F
is applied to the point
(
0
,
1
)
(0,1)
(
0
,
1
)
to hold the wire in coincidence with the curve and segment. Show that the force
F
F
F
is directed to the right.
A1
1
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Putnam 1957 A1
The normals to a surface all intersect a fixed straight line. Show that the surface is a portion of a surface of revolution.
A4
1
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Roots of polynomial covered by round disk
Let
P
(
z
)
P(z)
P
(
z
)
be a polynomial with real coefficients whose roots are covered by a disk of radius R. Prove that for any real number
k
k
k
, the roots of the polynomial
n
P
(
z
)
−
k
P
′
(
z
)
nP(z)-kP'(z)
n
P
(
z
)
−
k
P
′
(
z
)
can be covered by a disk of radius
R
+
∣
k
∣
R+|k|
R
+
∣
k
∣
, where
n
n
n
is the degree of
P
(
z
)
P(z)
P
(
z
)
, and
P
′
(
z
)
P'(z)
P
′
(
z
)
is the derivative of
P
(
z
)
P(z)
P
(
z
)
. can anyone help me? It would also be extremely helpful if anyone could tell me where they've seen this type of problems.............Has it appeared in any mathematics competitions? Or are there any similar questions for me to attempt? Thanks in advance!
B5
1
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increasing mapping on set power
Let
f
f
f
be an increasing mapping from the family of subsets of a given finite set
H
H
H
into itself, i.e. such that for every
X
⊆
Y
⊆
H
X \subseteq Y\subseteq H
X
⊆
Y
⊆
H
we have
f
(
X
)
⊆
f
(
Y
)
⊆
H
.
f (X )\subseteq f (Y )\subseteq H .
f
(
X
)
⊆
f
(
Y
)
⊆
H
.
Prove that there exists a subset
H
0
H_{0}
H
0
of
H
H
H
such that
f
(
H
0
)
=
H
0
.
f (H_{0}) = H_{0}.
f
(
H
0
)
=
H
0
.